Recent zbMATH articles in MSC 34https://zbmath.org/atom/cc/342021-11-25T18:46:10.358925ZWerkzeugBook review of: F. Brauer et al., Mathematical models in epidemiologyhttps://zbmath.org/1472.000162021-11-25T18:46:10.358925Z"Hickson, Roslyn"https://zbmath.org/authors/?q=ai:hickson.roslyn-iReview of [Zbl 1433.92001].Modelling with ordinary differential equations. A comprehensive approachhttps://zbmath.org/1472.340012021-11-25T18:46:10.358925Z"Borzì, Alfio"https://zbmath.org/authors/?q=ai:borzi.alfioOrdinary differential equations (ODEs) are important for modelling and simulating many time-dependent phenomena in physics, chemistry, biology and other areas. This book is not only about the modelling aspect as suggested by the title, but covers the entire theory of ODEs, including the existence and uniqueness of solutions, stability aspects as well as boundary and eigenvalue problems. Also numerical solutions methods are introduced. After the first eight chpater on these topics there are six chapters about more advanced topics, such as optimal control for ODE models inverse problems, differential games, ODEs for stochastic processes, and the use of neural network for solving ODEs. The book contains many examples but is written in a dense and rigirous mathemathical way, which might be too advanced for undergraduates. However, due to the immense set of contents on less than 400 pages, starting with fundamental material abaut ODEs and including the advaned topics, the book is a very good source for advanced lectures or self-studies. The book has almost 150 references and a good index.The center and focus problem. Algebraic solutions and hypotheseshttps://zbmath.org/1472.340022021-11-25T18:46:10.358925Z"Popa, M. N."https://zbmath.org/authors/?q=ai:popa.mihail-n"Pricop, V. V."https://zbmath.org/authors/?q=ai:pricop.victor-vPublisher's description: This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades.
A rather essential result is obtained in solving Poincaré's problem. Namely, there are given the upper estimations of the number of Poincaré-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.Linear algebra to differential equationshttps://zbmath.org/1472.340032021-11-25T18:46:10.358925Z"Vasundhara Devi, J."https://zbmath.org/authors/?q=ai:vasundhara-devi.jonnalagadda"Deo, Sadashiv G."https://zbmath.org/authors/?q=ai:deo.sadashiv-g"Khandeparkar, Ramakrishna"https://zbmath.org/authors/?q=ai:khandeparkar.ramakrishnaPublisher's description: Linear Algebra to Differential Equations concentrates on the essential topics necessary for all engineering students in general and computer science branch students, in particular. Specifically, the topics dealt will help the reader in applying linear algebra as a tool.
The advent of high-speed computers has paved the way for studying large systems of linear equations as well as large systems of linear differential equations. Along with the standard numerical methods, methods that curb the progress of error are given for solving linear systems of equations.
The topics of linear algebra and differential equations are linked by Kronecker products and calculus of matrices. These topics are useful in dealing with linear systems of differential equations and matrix differential equations. Differential equations are treated in terms of vector and matrix differential systems, as they naturally arise while formulating practical problems. The essential concepts dealing with the solutions and their stability are briefly presented to motivate the reader towards further investigation.
This book caters to the needs of Engineering students in general and in particular, to students of Computer Science \& Engineering, Artificial Intelligence, Machine Learning and Robotics. Further, the book provides a quick and complete overview of linear algebra and introduces linear differential systems, serving the basic requirements of scientists and researchers in applied fields.
Features
\begin{itemize}
\item Provides complete basic knowledge of the subject
\item Exposes the necessary topics lucidly
\item Introduces the abstraction and at the same time is down to earth
\item Highlights numerical methods and approaches that are more useful
\item Essential techniques like SVD and PCA are given
\item Applications (both classical and novel) bring out similarities in various disciplines:
\item Illustrative examples for every concept: A brief overview of techniques that hopefully serves the present and future needs of students and scientists.
\end{itemize}Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implementedhttps://zbmath.org/1472.340042021-11-25T18:46:10.358925Z"Bérard, Pierre"https://zbmath.org/authors/?q=ai:berard.pierre-h"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernardThis paper gives an account of results emanating from Sturm's seminal work on eigenvalues and eigenfunctions of the Dirichlet problem
\[
-y''(x)+q(x)y(x) = \lambda y(x)\text{ in }]0,1[,\ y(0)=y(1)=0.
\]
The authors are particularly interested in the statement that the number of the zeros of nontrivial real linear combinations of the first \(n\) real eigenfunctions is bounded by \(n-1\), where \(n\) is any positive integer. They present different proofs, interwoven with detailed accounts of the history of the presented approaches. This historical overview also briefly discusses the generalization to the Laplace operator \(-\Delta \) with Dirichlet boundary conditions on a bounded domain in \(\mathbb{R}^n\) and counterexamples thereof.
First, the authors present Liouville's proof. Then they give an alternative proof using the strategy proposed by Gelfand, considering the first eigenfunction of the \(n\)-particle Hamiltonian restricted to Fermions. This apprears to be the first complete proof following Gelfand's suggestion. Finally, Gelfand's strategy is compared with Kellog's approach and the resulting approach via oscillation kernels.
This paper is highly recommended to anyone interested in the history of Stum-Liouville theory or working in oscillation theory.A novel analytical technique to obtain kink solutions for higher order nonlinear fractional evolution equationshttps://zbmath.org/1472.340052021-11-25T18:46:10.358925Z"Ul Hassan, Qazi Mahmood"https://zbmath.org/authors/?q=ai:hassan.qazi-mahmood-ul"Ahmad, Jamshad"https://zbmath.org/authors/?q=ai:ahmad.jamshad"Shakeel, Muhammad"https://zbmath.org/authors/?q=ai:shakeel.muhammadSummary: We use the fractional derivatives in Caputo's sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.Travelling wave solutions of the general regularized long wave equationhttps://zbmath.org/1472.340062021-11-25T18:46:10.358925Z"Zheng, Hang"https://zbmath.org/authors/?q=ai:zheng.hang"Xia, Yonghui"https://zbmath.org/authors/?q=ai:xia.yonghui"Bai, Yuzhen"https://zbmath.org/authors/?q=ai:bai.yuzhen"Wu, Luoyi"https://zbmath.org/authors/?q=ai:wu.luoyiThis paper studies the model of the general regularized long wave (GRLW) equation. The main contribution of this paper is to find that GRLW equation has extra kink and anti-kink wave solutions when $p = 2n + 1$, while it's not for $p = 2n$. The authors give the phase diagram and obtained possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.Stability, asymptotic and exponential stability for various types of equations with discontinuous solutions via Lyapunov functionalshttps://zbmath.org/1472.340072021-11-25T18:46:10.358925Z"Gallegos, Claudio A."https://zbmath.org/authors/?q=ai:gallegos.claudio-a"Grau, Rogelio"https://zbmath.org/authors/?q=ai:grau.rogelio"Mesquita, Jaqueline G."https://zbmath.org/authors/?q=ai:mesquita.jaqueline-godoySummary: In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales.Implicit fractional differential equation involving \(\psi\)-Caputo with boundary conditionshttps://zbmath.org/1472.340082021-11-25T18:46:10.358925Z"Abdellatif, Boutiara"https://zbmath.org/authors/?q=ai:abdellatif.boutiara"Benbachir, Maamar"https://zbmath.org/authors/?q=ai:benbachir.maamarThis paper deals with the existence and uniqueness of solutions for boundary-value problems of the nonlinear \(\psi\)-Caputo fractional differential equations
\[
\begin{aligned} ^CD^{\alpha, \psi}_{a^+}u(t) &= f(t, u(t),^CD^{\alpha, \psi}_{a^+}u(t)), \quad t\in [a, T],\\
u(T) &= \lambda u(\eta). \end{aligned}
\]
where \(^CD^{\alpha, \psi}_{a^+}\) is the \(\psi\)-Caputo fractional derivative of order \(\alpha \in (0, 1]\), \(f : [a, T]\times \mathbb{R} \to \mathbb{R}\) is a given continuous function, \(\lambda\) is a real constant and \(\eta\in (a, T).\) The results are obtained by using standard fixed point theorems. Further some types of fractional Ulam-Hyers stability are established. An example is given to illustrate the existence results.Existence and uniqueness of tripled fixed points for mixed monotone operators with perturbations and applicationhttps://zbmath.org/1472.340092021-11-25T18:46:10.358925Z"Afshari, Hojjat"https://zbmath.org/authors/?q=ai:afshari.hojjat"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alireza(no abstract)Existence of solution to fractional differential equation with fractional integral type boundary conditionshttps://zbmath.org/1472.340102021-11-25T18:46:10.358925Z"Ali, Anwar"https://zbmath.org/authors/?q=ai:ali.anwar"Sarwar, Muhammad"https://zbmath.org/authors/?q=ai:sarwar.muhammad"Zada, Mian Bahadur"https://zbmath.org/authors/?q=ai:zada.mian-bahadur"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamalSummary: This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problem
\[
D^{\gamma} \mathfrak{u} (\ell ) = \psi (\ell , \mathfrak{u} (\lambda \ell )), \ell \in \mathfrak{Z} = [0 , 1],
\]
with fractional integral boundary conditions
\[
\mathfrak{p}_1 \mathfrak{u} (0) + \mathfrak{q}_1 \mathfrak{u} (1) = \frac{1}{\Gamma (\gamma )} \int_0^1 (1 - \rho )^{\gamma - 1} g_1 (\rho , \mathfrak{u} (\rho )) d \rho ,
\]
and
\[
\mathfrak{p}_2 \mathfrak{u}^{\prime} (0) + \mathfrak{q}_2 \mathfrak{u}^{\prime} (1) = \frac{1}{\Gamma (\gamma )} \int_0^1 (1 - \rho )^{\gamma - 1} g_2 (\rho , \mathfrak{u} (\rho )) d \rho ,
\]
where \(\gamma \in (1, 2], 0 < \lambda < 1, D\) denotes the Caputo fractional derivative (in short CFD), \(\psi, g_1, g_2 : \mathfrak{Z} \times \mathfrak{R} \to \mathfrak{R}\) are continuous functions and \(\mathfrak{p}_i, \mathfrak{q}_i (i = 1, 2)\) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end.A new generalized Gronwall inequality with a double singularity and its applications to fractional stochastic differential equationshttps://zbmath.org/1472.340112021-11-25T18:46:10.358925Z"Ding, Xiao-Li"https://zbmath.org/authors/?q=ai:ding.xiaoli"Daniel, Cao-Labora"https://zbmath.org/authors/?q=ai:daniel.cao-labora"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseThis paper focuses primarily on a new generalized Gronwall inequality with a double singularity and its applications to fractional stochastic differential equations
The first few pages introduce all the necessary notation and terminology to understand the paper: the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations is discussed.
The main results are presented in Section 3 (Generalized Gronwall inequalities with weakly singular kernels) with the help of Theorem 3.1 and 3.2. In Section 4, the properties and integral inequalities obtained in Section 3 to discuss the well-posedness of semilinear fractional stochastic differential equations are studied.An analytical solution for forcing nonlinear fractional delayed Duffing oscillatorhttps://zbmath.org/1472.340122021-11-25T18:46:10.358925Z"El-Dib, Yusry O."https://zbmath.org/authors/?q=ai:el-dib.yusry-oSummary: Stability analysis of motions in a nonlinear periodically forced, nonlinear fractional time-delayed, is investigated. An enhanced perturbation method is developed to study the stability behavior for the nonlinear oscillator. The basic idea of the method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to the very high accuracy of the obtained solution.Two efficient methods for solving fractional Lane-Emden equations with conformable fractional derivativehttps://zbmath.org/1472.340132021-11-25T18:46:10.358925Z"Malik, Adyan M."https://zbmath.org/authors/?q=ai:malik.adyan-m"Mohammed, Osama H."https://zbmath.org/authors/?q=ai:mohammed.osama-hApplication of conformable residual power series method, namely CRP, and conformable Homotopy-Adomian decomposition method, namely CH-A, have attracted considerable attention in recent years to attain numerical solution of fractional partial differential equations. Nevertheless, it is clear that interest of scientists in fractional calculus also has been steadily increasing. It may also be pointed out that this paper is concerned with an interesting application of CRP and CH-A, to obtain approximate solutions of fractional Lane-Emden equation which has applications in physics and astrophysics. Additionally, the problem is formulated in conformable fractional derivative (CL-M) which is local by its nature such as the standard derivative function.
In this paper, the authors present two different numerical method to obtain solution of equation which has general form as following; \[CD^{2\alpha}y + \frac{2\alpha}{x^\alpha} + CD^{\alpha}y + f(y) = 0 \quad x>0,\quad 0<\alpha\leq1. \] The main result of the paper is the improvement for the approximate analytic solution of nonlinear fractional differential equations, by making use of CRP and CH-A methods. Accordingly, it is clear from results that the proposed methods lead to a efficacious way of obtaining solution of nonlinear CL-M. In addition, graphical representation of solutions are given for the different values of \(\alpha\).A coupled system of nonlinear Caputo-Hadamard Langevin equations associated with nonperiodic boundary conditionshttps://zbmath.org/1472.340142021-11-25T18:46:10.358925Z"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-m"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanSummary: In this paper, we study the coupled system of nonlinear Langevin equations involving Caputo-Hadamard fractional derivative and subject to nonperiodic boundary conditions. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the features of the Hadamard fractional derivative, the implementation of fixed point theorems, and the employment of Urs's stability approach. An example is introduced to facilitate the understanding of the theoretical findings.Existence theory and Ulam's stabilities of fractional Langevin equationhttps://zbmath.org/1472.340152021-11-25T18:46:10.358925Z"Rizwan, Rizwan"https://zbmath.org/authors/?q=ai:rizwan.rizwan"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbarSummary: In this paper, we consider fractional Langevin equation and derive a formula of solutions for fractional Langevin equation involving two Caputo fractional derivatives. Secondly, we implement the concept of Ulam-Hyers as well as Ulam-Hyers-Rassias stability. Then, we choose Generalized Diaz-Margolis's fixed point approach to derive Ulam-Hyers as well as Ulam-Hyers-Rassias stability results for our proposed model, over generalized complete metric space. We give several examples which support our main results.A nonlinear integro-differential equation with fractional order and nonlocal conditionshttps://zbmath.org/1472.340162021-11-25T18:46:10.358925Z"Wahash, Hanan A."https://zbmath.org/authors/?q=ai:wahash.hanan-abdulrahman"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Panchal, Satish K."https://zbmath.org/authors/?q=ai:panchal.satish-kushabaSummary: This paper deals with a nonlinear integro-differential equation of fractional order \(\alpha\in(0,1)\) with nonlocal conditions involving fractional derivative in the Caputo sense. Under a new approach and minimal assumptions on the function \(f\), we prove the existence, uniqueness, estimates on solutions and continuous dependence of the solutions. The used techniques in analysis rely on fractional calculus, Banach contraction mapping principle, and Pachpatte's inequality. At the end, some numerical examples to justify our results are illustrated.A new algorithm for fractional differential equation based on fractional order reproducing kernel spacehttps://zbmath.org/1472.340172021-11-25T18:46:10.358925Z"Zhang, Ruimin"https://zbmath.org/authors/?q=ai:zhang.ruimin"Lin, Yingzhen"https://zbmath.org/authors/?q=ai:lin.yingzhenSummary: This paper develops an effective and new method to solve a class of fractional differential equations. The method is based on a fractional order reproducing kernel space. First, depending on some theories, a fractional order reproducing kernel space \(W_{\alpha} [0, 1]\) is constructed. The fractional order reproducing kernel space is a very suitable space to solve a class of fractional differential equations. Then, we calculate the reproducing kernel \(R_y (x)\) of the space \(W_{\alpha} [0, 1]\) skilfully in {\S}3. And convergence order and time complexity of this algorithm are discussed. We prove that the approximate solution converges to its exact solution \(u\) is not less than the second order. The time complexity of the algorithm is equal to the polynomial time of the third degree. Finally, three experiments support the algorithm strongly from the aspect of theory and technique.Geometric analysis of differential-algebraic equations via linear control theoryhttps://zbmath.org/1472.340182021-11-25T18:46:10.358925Z"Chen, Yahao"https://zbmath.org/authors/?q=ai:chen.yahao"Respondek, Witold"https://zbmath.org/authors/?q=ai:respondek.witoldIn the paper under review, the authors demonstrate the connection between the analysis of linear differential-algebraic equations (DAEs) of the form
\[
E \dot{x}=Hx,
\]
where \(x\in \mathcal{X}\cong {\mathbb R}^n\), \(E\in {\mathbb R}^{l\times n}\) and \(H\in {\mathbb R}^{l\times n}\), and linear control systems of the form
\[
\left\{\begin{array}{l} \dot{z}=Az+Bu, \\
y=Cz+Du, \end{array}\right.
\]
where \(z\in {\mathbb R}^q\), \(u\in {\mathbb R}^m\), \(y\in {\mathbb R}^p\) and \(A\), \(B\), \(C\), \(D\) are real matrices of appropriate sizes. Denote this DAE and this control system by \(\Delta_{l,n}=(E, H)\) and \(\Lambda_{q,m,p}=(A,B,C,D)\) respectively. The structure of the linear DAE is determined by the Kronecker canonical form (KCF) of the corresponding matrix pencil \(sE-H\) and the structure of the linear control system is determined by the its Morse canonical form (MCF).
Two DAEs \(\Delta_{l,n}=(E, H)\) and \(\tilde{\Delta}_{l,n}=(\tilde{E}, \tilde{H})\) are called externally equivalent, briefly, ex-equivalent, if there exist \(Q\in Gl(l,{\mathbb R})\) and \(P\in Gl(n,{\mathbb R})\) (\(Gl(n,{\mathbb R})\) is a group of nonsigular real \(n\times n\) matrices) such that \(\tilde{E}= QEP^{-1}\) and \(\tilde{H}= QHP^{-1}\) (see Definition 2.1). This means that the corresponding matrix pencils \(sE-H\) and \(s\tilde{E}-\tilde{H}\) are strictly equivalent (see [\textit{F. Gantmacher}, The Theory of Matrices, New York: Chelsea (1959; Zbl 0085.01001)]). Two linear control systems \(\Lambda_{q,m,p}=(A,B,C,D)\) and \(\tilde{\Lambda}_{q,m,p}=(\tilde{A},\tilde{B},\tilde{C},\tilde{D})\) are called Morse equivalent, briefly, M-equivalent, if there exist \(T_s\in Gl(q,{\mathbb R})\), \(T_i \in Gl(m,{\mathbb R})\), \(T_o\in Gl(p,{\mathbb R})\), \(F\in{\mathbb R}^{m\times q}\), \(K\in{\mathbb R}^{q\times p}\) such that \(\left[\begin{array}{cc} \tilde{A} & \tilde{B} \\
\tilde{C} & \tilde{D} \end{array}\right] = \left[\begin{array}{cc} T_s & T_s K \\
0 & T_o \end{array}\right] \left[\begin{array}{cc} A & B \\
C & D \end{array}\right] \left[\begin{array}{cc} T_s^{-1} & 0 \\
F T_s^{-1} & T_i^{-1}\end{array}\right]\) (see Definition 2.3).
The main results are the following.
For a linear control system \(\Lambda_{q,m,p}=(A,B,C,D)\), by setting the output \(y\) to be zero, the authors define the DAE with ``generalized'' states \((z, u)\in{\mathbb R}^{q+m}\) which is called the implicitation of \(\Lambda_{q,m,p}\) (see Definition 3.1). Such an output zeroing procedure was called the implicitation procedure. Moreover, the authors propose a converse procedure called explicitation, which ``attaches'' to a linear DAE a linear control system defined up to a coordinates change, a feedback transformation, and an output injection. Thus, the explicitation of the DAE is a class of control systems (see Definition 3.2 and Remark 3.3).
Theorem 3.4 describes the relationship between linear DAEs and control systems, using the introduced notions of implicitation, explicitation, ex-equivalence and M-equivalence. In particular, it is shown that Morse equivalent control systems give ex-equivalent DAEs and ex-equivalent DAEs produce Morse equivalent control systems. Theorem 5.3 establishes a correspondence between the KCF of the DAE and the MCF of its explicitation systems.
In section 4, geometric connections between linear DAEs and control systems are demonstrated by comparing the Wong sequences of DAEs (including the limits of the Wong sequences) and invariant subspaces of control systems. In section 6, the authors introduce the notion of internal equivalence for DAEs (see Definition 6.8) and show that the existence and uniqueness of the DAE solutions can be explained using the explicitation procedure and a concept of internal equivalence.Theory of index-one nonlinear complementarity systemshttps://zbmath.org/1472.340192021-11-25T18:46:10.358925Z"Stechlinski, Peter"https://zbmath.org/authors/?q=ai:stechlinski.peter-gThe paper is devoted to the initial value problem for a special type of differential-algebraic equations, so-called nonlinear complementarity systems:
\[
\begin{aligned} & \dot x(t,p) = f(t,p, x(t,p),u(t,p)), \\
& 0 \le u(t,p) \perp h(t,p, x(t,p),u(t,p)) \ge 0, \\
& x(t_0,p) = f_0(p), \end{aligned}
\]
where \(x \in \mathbb{R}^{n_x}\) and \(u \in \mathbb{R}^{n_u}\) are the vectors of state variables depending on the time \(t\) (dot over \(x\) means the differentiation by \(t\)) and the parameter \(p \in \mathbb{R}^{n_p}\). For vectors \(u,h \in \mathbb{R}^n\), the notation \(0 \le u \perp h \ge 0\) denotes that \(u \ge 0\), \(h \ge 0\) and the standard dot product \((u,h)=0\). The participating vector-functions are assumed to be \(C^1\)-smooth. Systems of this type appear in mechanics, electrical circuits, process systems engineering, etc.
The author studies well-posedness of such problems, the existence, uniqueness, and continuation of their solutions, the strong regularity of solutions, Lipschitzian dependence on parameters, and some other properties. One of the main tools used in the paper is the theory of piecewise continuous differentiable functions and generalized derivatives.Existence and properties of solutions for a class of fractional differential equationshttps://zbmath.org/1472.340202021-11-25T18:46:10.358925Z"Xu, Yong-qiang"https://zbmath.org/authors/?q=ai:xu.yongqiang"Chen, Shu-hong"https://zbmath.org/authors/?q=ai:chen.shuhong"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhongSummary: In this paper, we consider the initial value problem of a class of fractional differential equations. Firstly, we obtain the existence and uniqueness of the solutions by using Picard's method of successive approximation. Then we discuss the dependence of the solutions on the initial value.Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equationshttps://zbmath.org/1472.340212021-11-25T18:46:10.358925Z"Cao, Dat"https://zbmath.org/authors/?q=ai:cao.dat-t"Hoang, Luan"https://zbmath.org/authors/?q=ai:hoang.luan-thachSummary: This paper develops further and systematically the asymptotic expansion theory that was initiated by \textit{C. Foias} and \textit{J. C. Saut} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 1--47 (1987; Zbl 0635.35075)]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.On asymptotic series expansions of solutions to the Riccati equationhttps://zbmath.org/1472.340222021-11-25T18:46:10.358925Z"Samovol, V. S."https://zbmath.org/authors/?q=ai:samovol.v-sSummary: We consider scalar real Riccati equations with coefficients expanding in convergent power series in a neighborhood of infinity. Continued solutions of such equations are studied. Power geometry methods are used to obtain conditions for expanding these solutions in asymptotic series.Computation of periodic solution of linear constant coefficients ordinary differential equationhttps://zbmath.org/1472.340232021-11-25T18:46:10.358925Z"Kzaz, M."https://zbmath.org/authors/?q=ai:kzaz.m"Issaoui, M."https://zbmath.org/authors/?q=ai:issaoui.m"Maach, F."https://zbmath.org/authors/?q=ai:maach.fSummary: The aim of this paper is to determinate the periodic solution of linear constant coefficients ordinary differential equation, of any order. First, we give the exact solution in the form of a sum of integrals twice the number of roots of the characteristic polynomial of the differential equation. Then, we propose a numerical method to approximate the solution, when at least one of the roots of the characteristic polynomial is not available. Finally, we will present numerical examples to illustrate the effectiveness of the proposed method.A degenerate planar piecewise linear differential system with three zoneshttps://zbmath.org/1472.340242021-11-25T18:46:10.358925Z"Chen, Hebai"https://zbmath.org/authors/?q=ai:chen.hebai"Jia, Man"https://zbmath.org/authors/?q=ai:jia.man"Tang, Yilei"https://zbmath.org/authors/?q=ai:tang.yileiConsider the following continuous piecewise linear system defined in three zones: \[ \frac{d x}{d t}=F(x)-y, \quad \frac{d y}{d t}=g(x)-\alpha, \] where \begin{eqnarray*} F(x) & = & \left \{ \begin{array}{ll} t_r (x-1)+t_c, & \mbox{if } x>1, \\
t_c x, & \mbox{if } -1\leq x \leq 1, \\
t_l (x+1)-t_c, & \mbox{if } x<-1, \\
\end{array} \right.\\
g(x) & = & \left \{ \begin{array}{ll} d_r (x-1)+d_c, & \mbox{if } x>1, \\
d_c x, & \mbox{if } -1\leq x \leq 1, \\
d_l (x+1)-d_c, & \mbox{if } x<-1, \\
\end{array} \right.\\
\end{eqnarray*} where \(\alpha\), \(t_r\), \(t_c\), \(t_l\), \(d_r\), \(d_c\) and \(d_l\) are real parameters. In particular, the case of \(d_c=0\) is called the degenerate case of the system. This case has been extensively studied for the existence of limit cycles and its global phase portraits in the Poincaré disc except for the extreme case, i. e. the case when \((\alpha, d_l, d_c, d_r, t_l, t_c, t_r)\in \mathcal{G}^*\subset \mathbb{R}^7\), where \begin{eqnarray*} \mathcal{G}^* &=& \{(\alpha, d_l, d_c, d_r, t_l, t_c, t_r)\in \mathbb{R}^7: \ d_l>0, d_r>0, d_c=0, t_r>0, t_l<0, \\
& & t_l^2-4d_l<0, t_r^2-4d_r<0, \gamma_l+\gamma_r=0 \}, \end{eqnarray*} where \[ \gamma_l=\frac{t_l}{2\sqrt{d_l-\frac{t_l^2}{4}}}, \quad \gamma_r=\frac{t_r}{2\sqrt{d_r-\frac{t_r^2}{4}}}. \] In this paper, the authors investigate the dynamics of the extreme case of the system. They find that a generalized degenerate Hopf bifurcation occurs for points at infinity. Furthermore, the bifurcation diagram and all topologically different global phase portraits of the system for the extreme case are obtained.Heteroclinic chaotic threshold in a nonsmooth system with jump discontinuitieshttps://zbmath.org/1472.340252021-11-25T18:46:10.358925Z"Tian, R. L."https://zbmath.org/authors/?q=ai:tian.runlan|tian.ruilin|tian.ruiling|tian.ruilan|tian.runli"Wang, T."https://zbmath.org/authors/?q=ai:wang.thomas|wang.tianen|wang.tiqang|wang.tianxiu|wang.tienan|wang.tao.6|wang.tixang|wang.tianyin|wang.tiang|wang.tingdong|wang.ti|wang.tianran|wang.tianxi|wang.tichun|wang.tiejun|wang.tianrui|wang.taiyong|wang.tiangming|wang.tie|wang.tingchun|wang.tianxiong|wang.tengfi|wang.te|wang.tianyue|wang.tiancheng|wang.tingchang|wang.tianyou|wang.tingyan|wang.tengjiao|wang.tsejun|wang.tengjiang|wang.tianqi|wang.tangrui|wang.tianbo|wang.tianshu|wang.tiechao|wang.tinghua|wang.tianlei|wang.tingfu|wang.tangyu|wang.tianfei|wang.tingming|wang.tsaipei|wang.taotao|wang.tianming|wang.teng|wang.tixian|wang.tuo|wang.tiansong|wang.tieguang|wang.tangming|wang.tairan|wang.tianjun|wang.tzuchiang|wang.tiezhu|wang.tongxin|wang.tiecheng|wang.tiandong|wang.tan|wang.taiyuan|wang.taining|wang.tianhua|wang.tanshang|wang.tankin|wang.tifu|wang.tonghuan|wang.tianzhi|wang.tiexing|wang.taochun|wang.tiefang|wang.tianlu|wang.ting|wang.tongke|wang.taoli|wang.tiansheng|wang.tianshi|wang.tongya|wang.tengli|wang.tiantian|wang.tianying|wang.tonghe|wang.tianmiao|wang.tingqi|wang.tieping|wang.tiaoming|wang.tianzhen|wang.tianwei|wang.tonghui|wang.tuo.1|wang.tao.2|wang.tony|wang.tsunkuei|wang.tsunghai|wang.tianwen|wang.tianjiang|wang.tiancai|wang.tifo|wang.terence|wang.tianxiao|wang.taoyi|wang.tong|wang.tingxiu|wang.tingting|wang.tixiang|wang.taiwei|wang.tao.4|wang.tunglu|wang.taipeng|wang.tieying|wang.tao.9|wang.tongmin|wang.tianping|wang.tingzhen|wang.tiebang|wang.taychang|wang.tingsheng|wang.tongtong|wang.tao.3|wang.tianzhu|wang.tianhui|wang.tian|wang.tianze|wang.tiedan|wang.tsusheng|wang.tianyu|wang.tingfang|wang.tianhao|wang.tongzheng|wang.tieshen|wang.tinran|wang.tianshuang|wang.tianyi.1|wang.tianxing|wang.tingjing|wang.tingjian|wang.tianxiang|wang.tianbiao|wang.tun|wang.tianjiao|wang.taiqi|wang.tonglin|wang.tengfei|wang.tongyu|wang.taozheng|wang.tao.8|wang.tongge|wang.taiyue|wang.tianqin|wang.tao.5|wang.tingsong|wang.towe|wang.tiane|wang.tengyao|wang.tai|wang.tianlin|wang.tianhong|wang.tiangmin|wang.tower|wang.tianjing|wang.tingkai|wang.tianzhong|wang.taige|wang.tianyang|wang.tianyuan|wang.tianfu|wang.tianqing|wang.tao.7|wang.tsewei|wang.taijun|wang.tsili|wang.tongguang|wang.tianheng|wang.taofen|wang.tongqiang|wang.tonggen"Zhou, Y. F."https://zbmath.org/authors/?q=ai:zhou.yingfang|zhou.yifan|zhou.yifu|zhou.yanfei|zhou.yifeng|zhou.yinfei|zhou.yangfan|zhou.yin-feng|zhou.yong-feng|zhou.yufen|zhou.yuanfeng|zhou.yongfang|zhou.yangfeng|zhou.yanfang|zhou.yafu|zhou.yufeng|zhou.yunfei|zhou.yunfeng|zhou.yufei|zhou.yufang|zhou.yafei"Li, J."https://zbmath.org/authors/?q=ai:li.ji.2|li.jinlu|li.jianxiang|li.jianlang|li.jing.6|li.jinhon|li.jufang|li.jinxiang|li.jinghan|li.jingyan|li.jinguo|li.jingru|li.juntao|li.jing.12|li.jiani|li.ju|li.jingli|li.jiekun|li.jinna|li.junfei|li.jiaorui|li.jialing|li.jinyan|li.jianming|li.jingye|li.jinzheng|li.jianshun|li.jiawen|li.jinsong|li.jianshi|li.jiguang|li.jinglei|li.jiayu|li.jinjing|li.jiangxiang|li.junbao|li.jiahong|li.jianlong|li.jikai|li.jingfei|li.jun.1|li.jingfan|li.jiankui|li.junlei|li.junkui|li.jingjing|li.jiqian|li.jitao|li.jingna|li.junyu|li.jianning|li.jingfeng|li.jinkai|li.jiqiang|li.jerry|li.jiachao|li.junhua|li.jingzhu|li.jianmin|li.jiazhi|li.jiongseng|li.jingzhen|li.jiayi|li.jinshui|li.jiyun|li.jiakai|li.junqin|li.jinwei|li.jiangang|li.jinyu|li.junying|li.jingling|li.jingchao|li.jinggao|li.jieyun|li.juanzi|li.jianqing|li.jing.11|li.jianqi|li.jichun.1|li.jihong|li.jimei|li.jianguo|li.jianmei|li.jiazhong|li.jikang|li.jiaojun|li.jinlong|li.jingxi|li.jingjun|li.jingxiang|li.jinmian|li.jun|li.jiakun|li.jingyu|li.jiankang|li.jingpei|li.jianye|li.jingyin|li.jianlin.1|li.jiangyu|li.jikun|li.jiquan|li.jianhua|li.jian.2|li.jinhai|li.jianbing|li.junmei|li.jingjiao|li.jiechao|li.jinwen|li.jiaojie|li.jianpei|li.jingzhe|li.jupeng|li.jianyong|li.jiemei|li.jinlin|li.jinling|li.jinming|li.jiachen|li.jiangtao|li.jing.10|li.jizi|li.jiayun|li.jiangcheng|li.juchun|li.jina|li.jiantang|li.jianghua|li.jia.2|li.jingchun|li.jiantong|li.jingyun|li.jianliang|li.jia.1|li.junyong|li.jiankun|li.jianjin|li.jialu|li.jing.2|li.jincheng|li.junqiu|li.jiaojiao|li.jiuyong|li.jinfa|li.jinjun.1|li.jinju|li.jia.3|li.jianzhi|li.jielin|li.jianyu|li.junlou|li.jiajia|li.jiechen|li.junfeng|li.jian.1|li.jian.3|li.junfang.1|li.jiangfeng|li.jiyanglin|li.junqiang|li.jingming|li.jueyou|li.jinhou|li.jintang|li.jessie|li.jiemin|li.jiangyan|li.jiawei|li.jiheng|li.jingwei|li.jianxiong|li.jiehong|li.junzhuang|li.junfeng.1|li.jiamei|li.jun.10|li.jiaqian|li.jiuping|li.jianian|li.juliang|li.jinge|li.jinzhu|li.jianjing|li.jiying|li.jing.7|li.jianhui|li.jie.2|li.jinghao|li.jinliang|li.jingxia|li.jing.5|li.jianfei|li.junfen|li.jigong|li.jingshan|li.jian-lei|li.jue|li.jinyang|li.jinglai|li.jianli|li.jingdi|li.jinyou|li.jiaxian|li.junbing|li.junyi|li.jinsheng|li.juane|li.jibao|li.juelong|li.jicheng|li.jianzhang|li.jingzi|li.jiangbo|li.jiahui|li.jianbin|li.jiangyi|li.jizhou|li.jiangxiong|li.jiangao|li.jingrun|li.jinggang|li.jingjie|li.jiyong|li.junru|li.jiarong|li.jianzeng|li.jing.3|li.junpu|li.jiexing|li.juqun|li.jinglu|li.jiexiang|li.jiajie|li.jianzhou|li.jinqi|li.jingshu|li.jiaxun|li.jinkun|li.jiannan|li.junyan|li.jiao|li.jansheng|li.jieliang|li.jinghai|li.jimeng|li.jianfu|li.jiusheng|li.juan.1|li.jiafeng|li.jinrui|li.jinping|li.jinchang|li.jiacheng|li.jiankeng|li.jinze|li.junquan|li.jimming|li.jieping|li.jingtao|li.jingying|li.jun.13|li.junbo|li.jingxue|li.junliang|li.jichao|li.juanfei|li.jianjie|li.jiqing|li.jinxian|li.jiatong|li.juiping|li.jianwei|li.jiankou|li.jinzhou|li.jinqiu|li.jilong|li.jianchun|li.jingi|li.jinxi|li.junxiang|li.jianghao|li.junpeng|li.junming|li.jiansun|li.jiabin|li.jiuhong|li.jiangfan|li.jianxia|li.jiayang|li.junning|li.jiuli|li.jingyang|li.jinghua|li.ji.3|li.jingrong|li.jiongyue|li.jianyun|li.jibin|li.juanru|li.jiangxin|li.junqiao|li.jianze|li.jintao|li.jiaoyan|li.jiuren|li.jingzheng|li.jixiang|li.jiaolong|li.junjun|li.jianwen|li.jinghuan|li.jinxiu|li.jianfeng|li.junshan|li.jianghuai|li.jishun|li.jinglan|li.jun.3|li.juexian|li.jinjin|li.jiangdan|li.jituo|li.jie|li.jiayin|li.jiangyuan|li.juxuan|li.jinggong|li.jiangzhong|li.junsheng|li.jianling|li.juanfang|li.jiongsheng|li.jianlin|li.junxian|li.jianqiao|li.jinbing|li.jinhong|li.jingran|li.jinzhong|li.jianghong|li.juncheng|li.jan|li.jialong|li.jianjuan|li.jiong|li.jinjie|li.jun.14|li.jingcui|li.jiaxiong|li.jiaofen|li.jiaqing|li.jiahao|li.jingfa|li.jingyuan|li.jinglong|li.junwei|li.jinyuan|li.junpeng.1|li.jianwu|li.jia|li.jingwu|li.jisheng|li.jiacui|li.junguo|li.jiyao|li.jiuxian|li.junlun|li.junbin|li.jun.8|li.jun-gang|li.jiangtao.2|li.jinglin|li.junqing|li.jingshi|li.jipeng|li.jinchuan|li.junhai|li.jiangeng|li.jianbao|li.jiangqi|li.junye|li.jinzhi|li.juxiang|li.jijun|li.jun.6|li.jinpeng|li.jinghui|li.jiangli|li.jiaxin|li.jingquan|li.junmin|li.jiequan|li.jiangrong|li.jialian|li.jinlan|li.jianke|li.jinbo|li.jingmei|li.jiaqi|li.jichun|li.ji.4|li.junxing|li.jianjiang|li.junzhi|li.jiandong|li.juanjuan|li.jiaona|li.ji.5|li.jinzhao|li.junjiang|li.jine|li.jingdong|li.jiajian|li.jianchang|li.jinke|li.jianju|li.juyi|li.junhao|li.jiarui|li.junxiong|li.jiajing|li.jiehao|li.jinning|li.jiyang|li.jiren|li.jiachang|li.jianshu|li.jinggai|li.jianbiao|li.jianqiu.1|li.jinbao.1|li.jiajin|li.jiehua|li.jin.4|li.jin.2|li.jin.5|li.jin.3|li.jingcheng|li.jiaying|li.jinnan|li.jihan|li.junxuan|li.jianping.1|li.jikuo|li.jianrong|li.jingzhao|li.jide|li.jiaheng|li.jiahan|li.jinxin|li.jianfen|li.jiangyun|li.jianglong|li.jingwen|li.jiahua|li.jingyue|li.jianyuan|li.jianbo|li.jiaming|li.junxia|li.jiming|li.jiping|li.jiabao|li.jinhua|li.joingsheng|li.junlin|li.jingjng|li.jingya|li.jingqun|li.jingrui|li.jingpeng|li.jiaren|li.jensen|li.junjie|li.jianxun|li.jinjiang|li.jinxia|li.jianxin|li.junhui|li.jianjun|li.jianxi|li.junqi|li.jiukun|li.jun.11|li.jing.13|li.jungong|li.jiamin|li.jinchao|li.jinshu|li.jixin|li.jiawang|li.jianzhen|li.jingbo|li.jingjuan|li.jinying|li.jianguang|li.jieru|li.jiang|li.jinhui|li.jiangping|li.jialiang|li.jiajun|li.jundong|li.jianing|li.junyang|li.juxin|li.jinxing|li.jun.12|li.jiexian|li.jinxuan|li.jingliang|li.jun.2|li.jianyao|li.jiantao|li.jing.1|li.jinghong|li.jingke|li.jiwei|li.jingjian|li.jinran|li.jiabo|li.jie.1|li.jinqian|li.jinqing|li.jiaao|li.jiansheng|li.jun.7|li.jifu|li.jinfeng|li.juling|li.jiaomei|li.jiaze|li.jingchang|li.junfang|li.jianan|li.jiachun|li.jiangmeng|li.jiu|li.junze|li.junzhao|li.jianzhong|li.jianhong|li.jiahe|li.jingzhi|li.jinshan|li.jinku|li.jingping|li.jinguang|li.jishen|li.juxi|li.jindong|li.juan|li.jiayong|li.jianxing|li.jianpeng|li.jingyao|li.jiadong|li.jingbin|li.jingyi|li.jialang|li.jiye|li.jiongshen|li.jing.4|li.jibo|li.jinfang|li.jiaru|li.jizhen|li.jifeng|li.jinmei|li.jiangjun|li.janchen|li.junhuai|li.jiangtao.1|li.junlong|li.jiaxu|li.jiguo|li.jiali|li.jalong|li.jing|li.jiegu|li.junping|li.jianquan|li.jiukai|li.jianshuo|li.jinjia|li.junling|li.jiangtao.3|li.jingshe|li.jinhuan|li.jiangnan|li.jianqiang|li.junhong|li.junli|li.joaquim|li.jiwen|li.jicai|li.jinquan|li.jimin|li.junlan|li.jingde|li.jiya|li.jiange|li.jiaqiang|li.jifang|li.jinsha|li.jinzong|li.jaegon"Zhu, S. T."https://zbmath.org/authors/?q=ai:zhu.shitong|zhu.suting|zhu.songtao|zhu.shaotaoIn this paper, the authors consider the problem of the heteroclinic Melnikov chaotic threshold for nonsmooth systems with jump discontinuities.
Concretely, the authors construct a kind of impulsive differential system, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. By using a new form of the nonsmooth heteroclinic Melnikov function, the authors obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops.
Their method is mainly some improvement of existing methods.On the number of limit cycles for a class of piecewise smooth Hamiltonian systems with discontinuous perturbationshttps://zbmath.org/1472.340262021-11-25T18:46:10.358925Z"Yang, Jihua"https://zbmath.org/authors/?q=ai:yang.jihua"Zhang, Erli"https://zbmath.org/authors/?q=ai:zhang.erliIn this paper, by analyzing the corresponding Picard-Fuchs equations, the authors obtain an upper bound of the number of limit cycles for a class of piecewise smooth Hamiltonian systems when they are perturbed inside discontinuous polynomials of degree \(n\) and present an example to illustrate an application of the theoretical results.Approximate rational solutions to the Thomas-Fermi equation based on dynamic consistencyhttps://zbmath.org/1472.340272021-11-25T18:46:10.358925Z"Mickens, Ronald E."https://zbmath.org/authors/?q=ai:mickens.ronald-e"Herron, Isom H."https://zbmath.org/authors/?q=ai:herron.isom-h-junSummary: We construct two rational approximate solutions to the Thomas-Fermi (TF) nonlinear differential equation. These expressions follow from an application of the principle of dynamic consistency. In addition to examining differences in the predicted numerical values of the two approximate solutions, we compare these values with an accurate numerical solution obtained using a fourth-order Runge-Kutta method. We also present several new integral relations satisfied by the bounded solutions of the TF equation.A partial inverse Sturm-Liouville problem on an arbitrary graphhttps://zbmath.org/1472.340282021-11-25T18:46:10.358925Z"Bondarenko, Natalia P."https://zbmath.org/authors/?q=ai:bondarenko.natalia-pSummary: The Sturm-Liouville operator with singular potentials of class \(W_2^{-1}\) on a graph of arbitrary geometrical structure is considered. We study the partial inverse problem, which consists in the recovery of the potential on a boundary edge of the graph from a subspectrum under the assumption that the potentials on the other edges are known a priori. We obtain (i) the uniqueness theorem, (ii) a reconstruction algorithm, (iii) global solvability, and (iv) local solvability and stability for this inverse problem. Our method is based on reduction of the partial inverse problem on a graph to the Sturm-Liouville problem on a finite interval with entire analytic functions in the boundary condition.Correction to: ``Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition''https://zbmath.org/1472.340292021-11-25T18:46:10.358925Z"Bondarenko, Natalia P."https://zbmath.org/authors/?q=ai:bondarenko.natalia-pCorrection to the author's paper [ibid. 43, No. 11, 7009--7021 (2020; Zbl 1456.34014)].Inverse problems for Sturm-Liouville operators on a compact equilateral graph by partial nodal datahttps://zbmath.org/1472.340302021-11-25T18:46:10.358925Z"Wang, Yu Ping"https://zbmath.org/authors/?q=ai:wang.yuping.1"Shieh, Chung-Tsun"https://zbmath.org/authors/?q=ai:shieh.chung-tsunSummary: Partial inverse nodal problems for Sturm-Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin-dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm-Liouville operator on the compact equilateral star graph.On the partial inverse problems for the transmission eigenvalue problem of the Schrödinger operatorhttps://zbmath.org/1472.340312021-11-25T18:46:10.358925Z"Xu, Qiao-Qiao"https://zbmath.org/authors/?q=ai:xu.qiao-qiao"Xu, Xiao-Chuan"https://zbmath.org/authors/?q=ai:xu.xiaochuan|xu.xiaochuan.1Summary: In this work, we study the partial inverse problems for the Schrödinger transmission eigenvalue problem. It is shown that if the potential is partially known a prior, then only partial eigenvalues can uniquely determine the potential. The relationship between the density of the known eigenvalues and the length of the subinterval on the given potential is revealed.Ambarzumyan theorems for Dirac operatorshttps://zbmath.org/1472.340322021-11-25T18:46:10.358925Z"Yang, Chuan-fu"https://zbmath.org/authors/?q=ai:yang.chuanfu"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng.3|wang.feng.2|wang.feng.4|wang.feng.1"Huang, Zhen-you"https://zbmath.org/authors/?q=ai:huang.zhenyouSummary: We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable self-adjoint matrix potential. The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators, which are subject to separation boundary conditions or periodic (semi-periodic) boundary conditions, and some analogs of Ambarzumyan's theorem are obtained. The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators, which are the second power of Dirac operators.Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulationhttps://zbmath.org/1472.340332021-11-25T18:46:10.358925Z"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Li, Yiqun"https://zbmath.org/authors/?q=ai:li.yiqun"Cheng, Jin"https://zbmath.org/authors/?q=ai:cheng.jin"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1In this paper, the authors prove the uniqueness of the determination of the variable fractional order in the homogeneous Dirichlet boundary-value problem of one-sided linear variable-order space-fractional diffusion equations in one space dimension with measurements of the unknown solutions near the boundary of the spatial domain. Based on the analysis, they develop a spectral Galerkin Levenberg-Marquardt method and a finite difference Levenberg-Marquardt method to numerically invert the variable order. Numerical experiments are carried out to investigate the numerical performance of these methods.Evolution differential inclusions associated to primal lower regular functionshttps://zbmath.org/1472.340342021-11-25T18:46:10.358925Z"Kecis, Ilyas"https://zbmath.org/authors/?q=ai:kecis.ilyas"Thibault, Lionel"https://zbmath.org/authors/?q=ai:thibault.lionelSummary: In this paper, we discuss the evolution inclusion governed by the subdifferential of a function \(f\) with a perturbation \(g\) depending on the time and the state. We prove the existence and uniqueness of local and global solution, assuming that \(f\) is primal lower regular and \(g\) satisfies some standard conditions. We give also many properties of the solution and study some particular cases of the inclusion, for example, the case when \(g\) depends only on the state and \(u_0\) belongs to the domain of the subdifferential of \(f\).The finite spectrum of fourth-order boundary value problems with transmission conditionshttps://zbmath.org/1472.340352021-11-25T18:46:10.358925Z"Bo, Fang-zhen"https://zbmath.org/authors/?q=ai:bo.fangzhen"Ao, Ji-jun"https://zbmath.org/authors/?q=ai:ao.jijunSummary: A class of fourth-order boundary value problems with transmission conditions are investigated. By constructing we prove that these class of fourth order problems consist of finite number of eigenvalues. Further, we show that the number of eigenvalues depend on the order of the equation, partition of the domain interval, and the boundary conditions (including the transmission conditions) given.Extremum estimates of the \(L^1\)-norm of weights for eigenvalue problems of vibrating string equations based on critical equationshttps://zbmath.org/1472.340362021-11-25T18:46:10.358925Z"Qi, Jiangang"https://zbmath.org/authors/?q=ai:qi.jiangang"Xie, Bing"https://zbmath.org/authors/?q=ai:xie.bingThe authors consider the boundary value problem
\begin{align*}
-y''(x)=\lambda w(x)y(x),\tag{1}\\
y(x)=0=y(1)-hy'(1), \quad 0<h<1\tag{2},
\end{align*}
where \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(mes \{x:w(x)>0, \, x\in [0,1]\}>0\) as well as \(mes \{x:w(x)<0, \, x\in [0,1]\}>0\). The expression of the infimum function \(E(\lambda, h)\) of \(||w||_1\) is given if (1) and (2) has nontrivial solution i.e \(\int_0^1|w|\ge E(\lambda,h)\) and
\[
E(\lambda, h)=\inf\{||w||_1: w\in \Omega(\lambda)\},\tag{3}
\]
where
\[
\Omega(\lambda)=\left\{w\in L^1[a,b]: \lambda\in \sigma(w)\cap\mathbb R, \int_0^1|w|>0\right\}.
\]
The authors apply the critical equation weights to solve the extremal problem (3). They obtain the following main result: ``If the problem (1) and (2) has nontrivial solution for \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(h\in(0,1)\), then
\[
E(\lambda, h)=\dfrac1{|\lambda|}\in\left\{\dfrac{1-h}h, \dfrac4{1-h}\right\}.
\]Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point boundary value problemshttps://zbmath.org/1472.340372021-11-25T18:46:10.358925Z"Cui, Yujun"https://zbmath.org/authors/?q=ai:cui.yujun"Zou, Yumei"https://zbmath.org/authors/?q=ai:zou.yumeiSummary: By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.Existence of solutions of \(\alpha \in(2,3]\) order fractional three-point boundary value problems with integral conditionshttps://zbmath.org/1472.340382021-11-25T18:46:10.358925Z"Mahmudov, N. I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu"Unul, S."https://zbmath.org/authors/?q=ai:unul.sinemSummary: Existence and uniqueness of solutions for \(\alpha \in(2,3]\) order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.Nonlocal integral boundary value problems with causal operators and fractional derivativeshttps://zbmath.org/1472.340392021-11-25T18:46:10.358925Z"Wang, G."https://zbmath.org/authors/?q=ai:wang.gangqiang|wang.gaoli|wang.guoye|wang.gaozhen|wang.guanglan|wang.guangyuan|wang.guangfu|wang.guidong|wang.guishen|wang.guofu|wang.guangcai|wang.gensen|wang.gaping|wang.guangbing|wang.gang.2|wang.guojiao|wang.guoqin|wang.guanmei|wang.guiping|wang.guanjun|wang.guocan|wang.guocong|wang.genqiang|wang.guochang|wang.guihe|wang.gui|wang.guanhui|wang.guanshu|wang.goufang|wang.guoqiang|wang.guoshuang|wang.guanjie|wang.guangchen|wang.guang-sheng|wang.gaoming|wang.geng|wang.guangding|wang.guijuan|wang.guanxiang|wang.guoxun|wang.guoqiang.2|wang.guoping|wang.guijin|wang.guangbin|wang.guangyu|wang.gongming|wang.guoli|wang.gang.4|wang.guofeng|wang.guangyang|wang.gehao|wang.guorongg|wang.ga|wang.guanhua|wang.guilan|wang.guangping|wang.guifang|wang.guobiao|wang.guanglie|wang.guochao|wang.gang.5|wang.guangju|wang.guanghui|wang.guofang|wang.guangxian|wang.gaosong|wang.guohua|wang.grace|wang.guihua|wang.guo|wang.gangfeng|wang.guojing|wang.guixiang|wang.guanmin|wang.guoying|wang.gu|wang.guangtao|wang.guoyan|wang.guifeng|wang.guangyin|wang.guizheng|wang.guxi|wang.guangbao|wang.guanghua|wang.guibao|wang.guangguang|wang.gaige|wang.gang.3|wang.guojuan|wang.guanchun|wang.guowu|wang.guotai|wang.guoren|wang.guilin|wang.guozhen|wang.guandong|wang.guoyuo|wang.guotao|wang.guang|wang.guojun.2|wang.guangqin|wang.ge|wang.guoiun|wang.guoqiu|wang.guangwu|wang.guohong|wang.guangbo|wang.guodong|wang.guozhu|wang.ganghua|wang.guoming|wang.gengjie|wang.gongbao|wang.gaixia|wang.gongshu|wang.guangxing|wang.guanzheng|wang.guide|wang.guozhong|wang.guangzhi|wang.guichang|wang.guanfa|wang.gengkun|wang.gengsheng|wang.guoshen|wang.geping|wang.guanying|wang.ganfu|wang.gaitang|wang.guanyi|wang.guolei|wang.gaohong|wang.guanyang|wang.guangqiu|wang.guizhou|wang.gicheol|wang.guoan|wang.gaizhen|wang.guijun|wang.guojin|wang.guanglu|wang.guangjing|wang.guanming|wang.gehua|wang.gewei|wang.guanqian|wang.guanbo|wang.guangyong|wang.guiyong|wang.guiquan|wang.guoru|wang.guancheng|wang.guoliang|wang.guangwa|wang.gaoxia|wang.guangjun|wang.guiye|wang.guoqiao|wang.guishuang|wang.guiling|wang.gangcheng|wang.gaihua|wang.guowei|wang.guangqian|wang.gaonan|wang.gaiming|wang.guanyu|wang.guanping|wang.guoqi|wang.guanglin|wang.guojun.1|wang.guizhu|wang.guofen|wang.guizhen|wang.guosheng|wang.guoxin|wang.guangqi|wang.geli|wang.guisheng|wang.guopeng|wang.gang.1|wang.guanglei|wang.gaoang|wang.guangde|wang.genglu|wang.gan|wang.gaofeng|wang.gen|wang.guangjie|wang.guomao|wang.guanqing|wang.gensheng|wang.gexia|wang.guobao|wang.gendi|wang.gourong|wang.guangchao|wang.guohui|wang.guiqiu|wang.guiyan|wang.gaocai|wang.guizhi|wang.guisong|wang.guixia|wang.guihong|wang.guannan|wang.guangmin|wang.guangxue|wang.guorong|wang.guolian|wang.gensgsheng|wang.guojun|wang.guangchun|wang.gaoxiong|wang.guangming|wang.guoquan|wang.gengshan|wang.gao|wang.guangrui|wang.gongbo|wang.gaowang|wang.grant|wang.guanghuai|wang.guizeng|wang.guilian|wang.guozun|wang.gai|wang.gangsheng|wang.gaimei|wang.guiying|wang.guichao|wang.guomin|wang.guobo|wang.guangyao|wang.genyuan|wang.guangfa|wang.gailing|wang.guiliang|wang.guoxing|wang.guoqiang.1|wang.guanpeng|wang.guogang|wang.gaolin|wang.guanzhong|wang.guoyin|wang.gengqiang|wang.guixi|wang.guoyou|wang.guoyu|wang.guanlin|wang.guangxuan|wang.guoqing.1|wang.gongpu|wang.guangyi|wang.guanghong|wang.guoqing.2|wang.gangwei|wang.guanxue|wang.guocheng|wang.guangqing|wang.guangxiong|wang.guanglun|wang.genxia|wang.gaowen|wang.guangshuai|wang.guoshun|wang.guozheng|wang.gang|wang.guobin|wang.guocai|wang.guozhao|wang.guiyun|wang.guanwei|wang.guangting|wang.gefei|wang.guojun.4|wang.gent|wang.guiyuan|wang.guoguang|wang.gongben|wang.guangwei|wang.guan"Zhang, L."https://zbmath.org/authors/?q=ai:zhang.lizhao|zhang.lingmi|zhang.le|zhang.lemei|zhang.lige|zhang.laiping|zhang.libao|zhang.linjing|zhang.lefei|zhang.lixiang|zhang.liabiao|zhang.lin|zhang.lianxing|zhang.lilian|zhang.lifan|zhang.lei.10|zhang.liuyang|zhang.lixun|zhang.licen|zhang.laobing|zhang.lisheng|zhang.lidan|zhang.longjie|zhang.liuyue|zhang.lingli|zhang.liguo|zhang.lei.25|zhang.lei|zhang.liwei|zhang.leiming|zhang.lidong|zhang.leiwu|zhang.liehui|zhang.liangyin|zhang.lvyuan|zhang.lichen|zhang.lumei|zhang.lele|zhang.leigang|zhang.lihao|zhang.ligang|zhang.longfei|zhang.linli|zhang.lingwei|zhang.longge|zhang.luping|zhang.lilin|zhang.lei.21|zhang.linxia|zhang.lingyan|zhang.lanfang|zhang.lingyuan|zhang.liruo|zhang.liyong|zhang.liangpei|zhang.lequn|zhang.lei.22|zhang.liyi|zhang.lin.3|zhang.linbo|zhang.luchao|zhang.lianhua|zhang.lingxi|zhang.lanxia|zhang.lei.14|zhang.leying|zhang.libing|zhang.lianhe|zhang.lijia|zhang.liangquan|zhang.lukai|zhang.linchuang|zhang.lequan|zhang.lailiang|zhang.lijiang|zhang.lianming|zhang.lina|zhang.liangdi|zhang.longting|zhang.lichuan|zhang.louxin|zhang.lixing|zhang.lianzhen|zhang.liren|zhang.letian|zhang.lai|zhang.li.6|zhang.lingmei|zhang.liqian|zhang.liansheng|zhang.li.4|zhang.liangyong|zhang.liuwei|zhang.laihui|zhang.lianmei|zhang.liya|zhang.lingfan|zhang.lingyun|zhang.lizhen|zhang.liangrui|zhang.lipeng|zhang.lingfu|zhang.longsheng|zhang.liangchi|zhang.lingping|zhang.liandi|zhang.likun|zhang.lingxiang|zhang.lei.24|zhang.lv|zhang.lyuou|zhang.liqiao|zhang.linmeng|zhang.lingye|zhang.luming|zhang.lulu|zhang.li.12|zhang.longyao|zhang.lianping|zhang.linda|zhang.lanhong|zhang.liangwei|zhang.leping|zhang.lei.18|zhang.lianfang|zhang.lizao|zhang.lixian|zhang.lunchuan|zhang.liping|zhang.li|zhang.li.5|zhang.liangqi|zhang.longxin|zhang.lei.17|zhang.lei.15|zhang.lijian|zhang.lieping|zhang.lianzheng|zhang.lu|zhang.lianying|zhang.lianshui|zhang.lianhai|zhang.lianwen|zhang.li.8|zhang.liaojun|zhang.lingyu|zhang.liangzhong|zhang.longbo|zhang.lixuan|zhang.luzou|zhang.lufang|zhang.linfeng|zhang.longteng|zhang.linghai|zhang.liwen|zhang.lingqi|zhang.liantang|zhang.linqing|zhang.lixin|zhang.lei.1|zhang.liancheng|zhang.liangjin|zhang.linru|zhang.longjun|zhang.lin.1|zhang.lipan|zhang.linsen|zhang.lianhong|zhang.lijuan|zhang.luojia|zhang.liangliang|zhang.lingsong|zhang.luyao|zhang.liangbin|zhang.liping.1|zhang.liyun|zhang.lei-hong|zhang.liangxiu|zhang.lingfeng|zhang.lanlan|zhang.licheng|zhang.leyan|zhang.lanhui|zhang.lei.9|zhang.lingyue|zhang.lingjuan|zhang.lingxin|zhang.lei.11|zhang.luona|zhang.lin.2|zhang.lihui|zhang.linyan|zhang.lingxian|zhang.lefeng|zhang.lizou|zhang.liqun|zhang.liangjun|zhang.lichun|zhang.laping|zhang.lingying|zhang.linan|zhang.lanzhu|zhang.lisa|zhang.linqiao|zhang.lifu|zhang.liuqing|zhang.lihong|zhang.lipai|zhang.lan|zhang.lirong|zhang.li.3|zhang.luyi|zhang.lixi|zhang.lianmeng|zhang.linrang|zhang.liumei|zhang.libang|zhang.lingrui|zhang.ling|zhang.linyang|zhang.lili|zhang.linnan|zhang.lei.2|zhang.lisha|zhang.lixu|zhang.linwen|zhang.lifang|zhang.lihai|zhang.lansheng|zhang.linjie|zhang.longhui|zhang.long|zhang.lipu|zhang.luying|zhang.liang|zhang.lian|zhang.li.9|zhang.liao|zhang.linmiao|zhang.lupeng|zhang.linyun|zhang.lei.16|zhang.liangzhe|zhang.linzi|zhang.lei.7|zhang.lijun|zhang.lingyi|zhang.liyu|zhang.langwen|zhang.limei|zhang.lingchen|zhang.luo|zhang.lianjun|zhang.liming|zhang.longxiang|zhang.liquing|zhang.lyuyuan|zhang.linyuan|zhang.liangchao|zhang.li-xin|zhang.lianyong|zhang.lunkai|zhang.longbing|zhang.lilong|zhang.lianfu|zhang.liyou|zhang.lixiu|zhang.limin|zhang.lei.5|zhang.lei.23|zhang.laiwu|zhang.liang.1|zhang.lingxia|zhang.laicheng|zhang.luyu|zhang.liangyun|zhang.liyang|zhang.lei.13|zhang.linke|zhang.lijie|zhang.lingming|zhang.lilun|zhang.liangxin|zhang.lanyong|zhang.liqiang|zhang.lida|zhang.liangying|zhang.liuhua|zhang.lingqin|zhang.letao|zhang.linna|zhang.luyang|zhang.lanling|zhang.liwei.1|zhang.leishi|zhang.luwan|zhang.liyan|zhang.liling|zhang.liu|zhang.lianzeng|zhang.linhua|zhang.libiao|zhang.li.11|zhang.lide|zhang.lei.20|zhang.lining|zhang.lizhi|zhang.lingchuan|zhang.litao|zhang.laixi|zhang.lufei|zhang.linlan|zhang.linfen|zhang.lixia|zhang.longchuan|zhang.lianmin|zhang.lingbo|zhang.liyuan|zhang.leilei|zhang.lejun|zhang.lanju|zhang.lanhua|zhang.lianyang|zhang.liufeng|zhang.lie|zhang.lingjun|zhang.lichao|zhang.lingzhong|zhang.leyou|zhang.liuping|zhang.liying|zhang.lijing|zhang.ledi|zhang.li.2|zhang.lihe|zhang.liqiong|zhang.luning|zhang.li.10|zhang.lianyi|zhang.lizhu|zhang.longbin|zhang.lishi|zhang.lingling|zhang.lianzhong|zhang.louzin|zhang.linjun|zhang.landing|zhang.lifa|zhang.luoping|zhang.lintao|zhang.lejie|zhang.liqin|zhang.lifei|zhang.liangyue|zhang.libin|zhang.liquan|zhang.li.1|zhang.lukun|zhang.lifeng|zhang.lingqian|zhang.liangcheng|zhang.lianfeng|zhang.liqing|zhang.linlin|zhang.leike|zhang.luyin|zhang.libo|zhang.lanyu|zhang.likai|zhang.lezhong|zhang.liting|zhang.lianzhu|zhang.linxi|zhang.linwan|zhang.liang.3|zhang.lun|zhang.liang.2|zhang.lei.4|zhang.li.7|zhang.luchan|zhang.lang|zhang.lingjie|zhang.liye|zhang.linghua|zhang.lihua|zhang.liqi|zhang.liangcai|zhang.liuliu|zhang.lingmin|zhang.linghong"Agarwal, R."https://zbmath.org/authors/?q=ai:agrawal.rajani|agarwal.rajshree|agarwal.rama-r|agarwal.ritu.1|agarwal.ravinder|agarwal.ramesh-c|agarwal.raja|agarwal.rishi|agarwal.ruchi|agarwal.reshu|agarwal.richa|agarwal.rachit|agarwal.ritu|agarwal.ramesh-k|agarwal.ritu.2|agarwal.r-k|agarwal.rajeev|agarwal.r-n|agarwal.r-d.1|agarwal.ravindra|agarwal.rishi-kumar|agarwal.rahul|agarwal.r-s|agarwal.ravi-p|agarwal.r-b|agarwal.riteshSummary: This paper investigates the existence of solutions for a class of nonlocal integral boundary value problems with causal operators and fractional derivatives. By applying the monotone iterative technique and the method of lower and upper solutions, we provide sufficient conditions under which such problems have the maximal and minimal solutions or quasisolutions in a corresponding sector. Finally, an example illustrating how the theory can be applied in practice is also included.Multiple solutions for mixed boundary value problems with \(\varphi\)-Laplacian operatorshttps://zbmath.org/1472.340402021-11-25T18:46:10.358925Z"Dallos Santos, Dionicio Pastor"https://zbmath.org/authors/?q=ai:santos.dionicio-pastor-dallosSummary: Using Leray-Schauder degree theory and the method of upper and lower solutions we establish existence and multiplicity of solutions for problems of the form \[\begin{aligned} (\varphi(u'))' = f(t,u,u') \\ u(0)= u(T)=u'(0), \end{aligned}\] where \(\varphi\) is an increasing homeomorphism such that \(\varphi(0)=0\), and \(f\) is a continuous function.Lyapunov-type inequalities for differential equation involving one-dimensional Minkowski-curvature operatorhttps://zbmath.org/1472.340412021-11-25T18:46:10.358925Z"Wang, Youyu"https://zbmath.org/authors/?q=ai:wang.youyu"Wang, Yameng"https://zbmath.org/authors/?q=ai:wang.yameng"Liu, Jing"https://zbmath.org/authors/?q=ai:liu.jing.1|liu.jing\textit{R. Yang} et al. [Appl. Math. Lett. 91, 188--193 (2019; Zbl 1472.34043)] first obtained Lyapunov inequalities for the one-dimensional Minkowski-curvature problems of the form
\[
\begin{aligned} &(\phi(u'(t)))'+r(t)u(t)=0,\qquad a<t<b \\
&u(a)=u(b)=0, \end{aligned}
\]
and for the systems of the one-dimensional Minkowski-curvature problems. Motivated by this paper, the authors obtain three types of Lyapunov inequalities for one-dimensional Minkowski-curvature equation involving anti-periodic and Sturm-Liouville boundary conditions
\[
\begin{aligned} &(\phi(u'(t)))'+r(t)u(t)=0,\qquad a<t<b \\
&u(a)+u(b)=0,\quad u'(a)+u'(b)=0, \end{aligned}
\]
and
\[
\begin{aligned} &(\phi(u'(t)))'+r(t)u(t)=0,\qquad a<t<b \\
&\alpha u(a)-\beta u'(b)=0,\quad \gamma u(b)+\delta u'(b)=0 \end{aligned}
\]
respectively, in the current paper, where
\[
\phi(z):=\frac{z}{\sqrt{1-|z|^2}},\qquad z\in(-1,1),
\]
the weight function \(r(t)\) is continuous on \([a,b]\), and \(\alpha,\beta,\gamma,\delta\geq0\) with \[ \beta\gamma+\alpha\delta+\alpha\gamma(b-a)>0. \] Two examples are given to illustrate the application of the main results.Existence and iterative method for some fourth order nonlinear boundary value problemshttps://zbmath.org/1472.340422021-11-25T18:46:10.358925Z"Wei, Yongfang"https://zbmath.org/authors/?q=ai:wei.yongfang"Song, Qilin"https://zbmath.org/authors/?q=ai:song.qilin"Bai, Zhanbing"https://zbmath.org/authors/?q=ai:bai.zhanbingThere is a known method for solving two-point boundary value problems, which allows to reduce the problem to two second-order problems and to define an operator \(A\) for the right-hand side function with solutions of the second-order problems. The authors offer a novel efficient method includes more clearly the definition of the operator \(A\). It describes an iterative process that generates a sequence of functions \(u_ {1, k}\) and \( u_ {2, k}\) converging to a unique solution to the problem. The convergence of the iterative method is proved by the use of the Banach contracting mapping principle. Also the authors give some numerical examples to illustrate the main result.Lyapunov-type inequalities for one-dimensional Minkowski-curvature problemshttps://zbmath.org/1472.340432021-11-25T18:46:10.358925Z"Yang, Rui"https://zbmath.org/authors/?q=ai:yang.rui"Sim, Inbo"https://zbmath.org/authors/?q=ai:sim.inbo"Lee, Yong-Hoon"https://zbmath.org/authors/?q=ai:lee.yong-hoon-leeIn this paper, some Lyapunov-type inequalities for the one-dimensional Minkowski-curvature problem of the form
\[
\begin{aligned}
-&(\phi(u'(t)))'=r(t)u(t),\qquad a<t<b\\
&u(a)=u(b)=0,
\end{aligned}
\]
where
\[
\phi(z):=\frac{z}{\sqrt{1-|z|^2}},\qquad z\in(-1,1),
\]
the weight function \(r(t)\) (\(\geq0\) for all \(t\in(a,b)\)) satisfies that \(r\not\equiv0\) in any compact subinterval of \([a,b]\) and
\[
r\in{\mathcal A}:=\{r\in L^1_{\mathrm{loc}}((a,b),[0,\infty)):\int^b_a(s-a)(b-s)r(s)\mathrm{d}s<\infty\}.
\]
The class \({\mathcal A}\) admits rather stronger singular functions at the boundary.
As far as it is known, Lyapunov-type inequalities of Minkowski-curvature problems have not been studied. In the current paper, Lyapunov-type inequalities are established to give necessary conditions for the existence of positive solutions for scalar equations as well as systems of one-dimensional Minkowski-curvature problems with singular weights which belong to \({\mathcal A}\).
In the paper, Lyapunov inequalities for one-dimensional Minkowski-curvature problem with a singular weight function, for one-dimensional Minkowski-curvature problem which possesses a singular coefficient and a singular weight which is transformed from problems to study radial solutions in exterior domain of a ball, and for a cycled and a strongly coupled system of one-dimensional Minkowski-curvature problem with singular weights are also derived.On the solvability of singular boundary value problems on the real line in the critical growth casehttps://zbmath.org/1472.340442021-11-25T18:46:10.358925Z"Biagi, Stefano"https://zbmath.org/authors/?q=ai:biagi.stefano"Isernia, Teresa"https://zbmath.org/authors/?q=ai:isernia.teresaThe paper is devoted to the study of the existence of at least one weak solution of the boundary value problem \[(\Phi(a(t,x(t))\,x'(t)))' = f(t,x(t), x'(t)), \; t \in {\mathbb R},\; x(-\infty) = \nu_1; \;x(+\infty) =\nu_2\] where \(\nu_1\), \(\nu_2 \in {\mathbb R}\), \(\Phi: {\mathbb R} \to {\mathbb R}\) is a strictly increasing homeomorphism extending the classical \(p\)-Laplacian, \(a\) is a nonnegative continuous function on \({\mathbb R}\times {\mathbb R}\) that can be zero on a set of Lebesgue measure equals to zero, and \(f\) is a Carathéodory function on \({\mathbb R}\times {\mathbb R}^2\).
The results follow by applying the technique of lower and upper solutions to related non homogeneous Dirichlet problems, defined on the intervals \([-n,n]\), with \(n \in {\mathbb N}\), and passing to the limit in \(n\). Some examples are given to point out the applicability of the obtained results.Existence of positive solutions for multi-point semi-positive boundary value problemshttps://zbmath.org/1472.340452021-11-25T18:46:10.358925Z"Su, Hua"https://zbmath.org/authors/?q=ai:su.huaThe author considers the existence of positive solutions for the nonlinear multi-point boundary value problem \[ - \big(u''(t) + a(t)u'(t) \big ) = \lambda f(t, u) + \mu g(t, u), \, t \in (0, 1), \] \[ u'(0) = 0, \, \, u(1) = \sum_{i=1}^k \alpha_i u(\eta_i) - \sum_{i = k+1}^{m-2} \alpha_i u(\eta_i). \] Using cone theoretic techniques, the author shows the existence of at least one positive solution when \(\lambda\) and \(\mu\) are small under various suitable conditions on \(f\) and \(g\). They also establish the existence of at least three positive solutions under other conditions on \(f\) and \(g\) when \(\lambda\) and \(\mu\) are small.Existence of positive solutions for a nonlinear higher-order multipoint boundary value problemhttps://zbmath.org/1472.340462021-11-25T18:46:10.358925Z"Zhao, Min"https://zbmath.org/authors/?q=ai:zhao.min"Sun, Yongping"https://zbmath.org/authors/?q=ai:sun.yongpingSummary: We study the existence of positive solutions for a nonlinear higher-order multipoint boundary value problem. By applying a monotone iterative method, some existence results of positive solutions are obtained. The main result is illustrated with an example.On a nonlocal Sturm-Liouville problem with composite fractional derivativeshttps://zbmath.org/1472.340472021-11-25T18:46:10.358925Z"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.13"Qi, Jiangang"https://zbmath.org/authors/?q=ai:qi.jiangangSummary: In this paper, we obtain the existence of solutions for a nonlocal Sturm-Liouville problem with composite fractional derivatives under some initial value conditions. Furthermore, applying above results and operator theory, we specifically research the geometric multiplicity of eigenvalues for the nonlocal Sturm-Liouville eigenvalue problem.Two classes of conformable fractional Sturm-Liouville problems: theory and applicationshttps://zbmath.org/1472.340482021-11-25T18:46:10.358925Z"Mortazaasl, Hamid"https://zbmath.org/authors/?q=ai:mortazaasl.hamid"Akbarfam, Ali Asghar Jodayree"https://zbmath.org/authors/?q=ai:akbarfam.aliasghar-jodayreeSummary: In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of \(r\)-CFSLPs, we discuss two types of CFSLPs which include left- and right-sided CFDs, each of order \(\alpha \in (n,n+1]\), and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of \(s\)-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order \(\alpha \in (0,1]\) and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and -II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved.Construction of Green's functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficienthttps://zbmath.org/1472.340492021-11-25T18:46:10.358925Z"Özen, Kemal"https://zbmath.org/authors/?q=ai:ozen.kemalIn this paper, the author extends to third order equations the approach developed in [\textit{S. S. Akhiev}, Math. Comput. Appl. 9, No. 3, 349--358 (2004; Zbl 1093.34012)] to construct the Green's function for second order equations with nonlocal conditions.On the convergence of WKB approximations of the damped Mathieu equationhttps://zbmath.org/1472.340502021-11-25T18:46:10.358925Z"Nwaigwe, Dwight"https://zbmath.org/authors/?q=ai:nwaigwe.dwightSummary: The form of the fundamental set of solutions of the damped Mathieu equation is determined by Floquet theory. In the limit as \(m \rightarrow 0\), we can apply WKB theory to get first order approximations of the fundamental set. WKB theory states that this approximation gets better as \(m \rightarrow 0\) and \(T\) is fixed. However, convergence of the periodic part and characteristic exponent is not addressed. We show that they converge to those predicted by WKB theory. We also provide a rate of convergence that is not dependent on \(T\).
{\copyright 2021 American Institute of Physics}Fractional boundary value problem on the half-linehttps://zbmath.org/1472.340512021-11-25T18:46:10.358925Z"Khamessi, Bilel"https://zbmath.org/authors/?q=ai:khamessi.bilelThe author investigates a nonlinear boundary value problem of a fractional differential equation. Using results on the functions in Karamata's classes and the estimate on Green's function, the existence and the uniqueness of a positive solution of this equation are studied. Finally, a description of the global behavior of this solution is given.On heteroclinic solutions for BVPs involving \(\phi\)-Laplacian operators without asymptotic or growth assumptionshttps://zbmath.org/1472.340522021-11-25T18:46:10.358925Z"Minhós, Feliz"https://zbmath.org/authors/?q=ai:minhos.feliz-manuelSummary: In this paper we consider the second order discontinuous equation in the real line, \[\begin{aligned}(\phi(a(t)u'(t)))' &=f(t,u(t),u'(t)), \ \mathrm{a}.\mathrm{e} \ t \in \mathbb R \\ u(-\infty) &= A, u(+\infty)=B \end{aligned}\] with \(\phi\) an increasing homeomorphism such that \(\phi (0)=0\) and \(\phi (\mathbb R)=\mathbb R, a \in C(\mathbb R)\) with \(a(t)>0\), for \(t \in \mathbb R, f: \mathbb R^3 \to \mathbb R\), a \(L^1\)-Carathéodory function and \(A, B \in \mathbb R\) verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities \(\phi\) and \(f\). Moreover, as far as we know, our main result is even new when \(\phi(y)=y\), that is, for the equation \[(a(t)u'(t))'=f(t,u(t),u'(t)),\ \mathrm{a}.\mathrm{e} \ t \in \mathbb R. \]Spectra of regular quantum trees: rogue eigenvalues and dependence on vertex conditionhttps://zbmath.org/1472.340532021-11-25T18:46:10.358925Z"Hess, Zhaoxia W."https://zbmath.org/authors/?q=ai:hess.zhaoxia-w"Shipman, Stephen P."https://zbmath.org/authors/?q=ai:shipman.stephen-pSummary: We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to \(-\infty \), a narrow cluster of finitely many eigenvalues tends to \(-\infty \), while the eigenvalues above this cluster remain bounded from below. Certain ``rogue'' eigenvalues break away from this cluster and tend even faster toward \(-\infty \). The spectrum can be visualized as the intersection points of two objects in the plane -- a spiral curve depending on the Schrödinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter.A multipoint boundary value problem on a graphhttps://zbmath.org/1472.340542021-11-25T18:46:10.358925Z"Zavgorodnii, M. G."https://zbmath.org/authors/?q=ai:zavgorodnii.m-gSummary: We give a statement of a multipoint boundary value problem on a geometric graph. The characteristics of the graph admitting such a statement are described. The form of the boundary value problem adjoint to the problem posed is found. An analog of the Keldysh residue theorem is obtained for the Green's function of the spectral multipoint boundary value problem on the graph.Analytically integrable centers of perturbations of cubic homogeneous systemshttps://zbmath.org/1472.340552021-11-25T18:46:10.358925Z"Algaba, Antonio"https://zbmath.org/authors/?q=ai:algaba.antonio"García, Cristóbal"https://zbmath.org/authors/?q=ai:garcia.cristobal"Reyes, Manuel"https://zbmath.org/authors/?q=ai:reyes.manuelThe authors investigate analytic integrability and homogenization problem of planar polynomial differential systems, whose homogeneous component is a cubic homogeneous vector field. As a first step, they give complete norm forms of the cubic homogeneous vector fields having a polynomial first integral, using the results and methods in [\textit{A. Algaba} et al., Rocky Mt. J. Math. 41, No. 1, 1--22 (2011; Zbl 1213.37093)], which is a necessary condition for the analytic integrability of their considered perturbations of cubic homogeneous systems. Secondly, they use the notion of Lie symmetry to prove the vector field is analytically integrable if and only if it is orbitally equivalent to its cubic homogeneous component. They also point out that the analytic integrability of the considered vector field can be solved via the formal inverse integrating factor. Some examples are given to illustrate the results.Quadratic differential systems with a finite saddle-node and an infinite saddle-node (1, 1)\textit{SN} - (B)https://zbmath.org/1472.340562021-11-25T18:46:10.358925Z"Artés, Joan C."https://zbmath.org/authors/?q=ai:artes.joan-carles"Mota, Marcos C."https://zbmath.org/authors/?q=ai:mota.marcos-c"Rezende, Alex C."https://zbmath.org/authors/?q=ai:rezende.alex-cPhase portraits of (2;0) reversible vector fields with symmetrical singularitieshttps://zbmath.org/1472.340572021-11-25T18:46:10.358925Z"Buzzi, Claudio"https://zbmath.org/authors/?q=ai:buzzi.claudio-aguinaldo"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Santana, Paulo"https://zbmath.org/authors/?q=ai:santana.pauloSummary: In this paper we study the phase portraits in the Poincaré disk of the reversible vector fields of type (2;0) having generic bifurcations around a symmetric singular point \(p\). We also prove the nonexistence of any periodic orbit surrounding \(p\). We point out that some numerical computations were necessary in order to control the number of limit cycles.Periodic solutions and invariant torus in the Rössler systemhttps://zbmath.org/1472.340582021-11-25T18:46:10.358925Z"Cândido, Murilo R."https://zbmath.org/authors/?q=ai:candido.murilo-r"Novaes, Douglas D."https://zbmath.org/authors/?q=ai:novaes.douglas-duarte"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaThis interesting paper investigates the existence of periodic solutions and invariant tori for the Rössler system
\[
\dot x = - y - z, \quad \dot y = x + ay, \quad \dot x =bx - cz + xz.
\]
The parameters \(a,b,c\) are near to one of the following one-parameter families:
\begin{itemize}
\item[(A)] \((a,b,c) = (\bar{a},1,\bar{a})\) with \(\bar{a} \in (-\sqrt{2},\sqrt{2}) \setminus \{0\}\)
\item[(B)] \((a,b,c) = (0,\bar{b},0)\) with \(\bar{b} > -1\)
\end{itemize}
both presenting a zero-Hopf equilibrium at the origin.
In case (B) the existence of a periodic solution bifurcating from the zero-Hopf equilibrium is proved, using a recent result on averaging theory, based on Lyapunov-Schmidt reduction.
In case (A), instead, the existence of a periodic solution bifurcating from the equilibrium was already known and the authors improve this result by showing, again by the averaging theory, the existence of an invariant torus around this periodic solution.
The stability properties of the periodic solutions and invariant torus are analysed, as well. Finally, some numerical simulations are presented.Symmetric centers on planar cubic differential systemshttps://zbmath.org/1472.340592021-11-25T18:46:10.358925Z"Dukarić, Maša"https://zbmath.org/authors/?q=ai:dukaric.masa"Fernandes, Wilker"https://zbmath.org/authors/?q=ai:fernandes.wilker"Oliveira, Regilene"https://zbmath.org/authors/?q=ai:oliveira.regilene-d-sThe authors of this paper investigate the simultaneous existence of centers in symmetric planar polynomial differential systems. In particular, they present necessary and sufficient conditions for the existence of a bi-center of four families for real cubic symmetric systems and obtain the conditions for the isochronicity of such bi-centers.Periodic orbits in the Rössler systemhttps://zbmath.org/1472.340602021-11-25T18:46:10.358925Z"Gierzkiewicz, Anna"https://zbmath.org/authors/?q=ai:gierzkiewicz.anna"Zgliczyński, Piotr"https://zbmath.org/authors/?q=ai:zgliczynski.piotrSummary: We prove the existence of \(n\)-periodic orbits for almost all \(n\in\mathbb{N}\) in the Rössler system with an attracting periodic orbit, for two sets of parameters. The methods, using covering relations between two-dimensional sets, are similar to the ones used often in the proofs of the Sharkovskii Theorem. The proofs are computer-assisted with the use of C++ library containing modules for interval arithmetic, differentiation and integration of ODEs.An improved criterion for the unique existence of the limit cycle of a Liénard-type system with one parameterhttps://zbmath.org/1472.340612021-11-25T18:46:10.358925Z"Hayashi, Makoto"https://zbmath.org/authors/?q=ai:hayashi.makotoConsider the Liénard-type system
\[
\begin{array}{l}
\frac{{dx}}{{dt}}= \frac{{1}}{{a(x)}} [ y-\lambda F(x) ], \\
\frac{{dy}}{{dt}} = a(x) g(x)
\end{array}\tag{1}
\]
depending on the positive parameter \(\lambda\) and where \(a(x)\) is positive for all \(x\). The author derives conditions on the functions \(F, g\) and \( a\) such that system (1) for \(\lambda > 0\) has a unique limit cycle. The obtained result improves known results in the literature.Existence and stability of limit cycles in the model of a planar passive biped walking down a slopehttps://zbmath.org/1472.340622021-11-25T18:46:10.358925Z"Makarenkov, Oleg"https://zbmath.org/authors/?q=ai:makarenkov.olegSummary: We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula
[\textit{T. McGeer}, ``Passive dynamic walking'', Int. J. Robot. Res. 9, 62--82 (1990; \url{doi:10.1177/027836499000900206})].
Following the fundamental work by
\textit{M. Garcia} et al. [``The simplest walking model: stability, complexity, and scaling'', J. Biomech. Eng. 120, No. 2, 281--288 (1998; \url{doi:10.1115/1.2798313})],
we view the slope of the ground as a small parameter \(\gamma \geq 0\). When \(\gamma = 0\), the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in
[Garcia et al., loc. cit.],
the family of cycles disappears when \(\gamma\) increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no mathematically complete proofs of the existence and stability of walking cycles have been reported in the literature to date. The present paper proves the existence and stability of a walking cycle (long-period gait cycle, as termed by McGeer) by using the methods of perturbation theory for maps. In particular, we derive a perturbation theorem for the occurrence of stable fixed points from 1-parameter families in two-dimensional maps that can be of independent interest in applied sciences.Focus quantities with applications to some finite-dimensional systemshttps://zbmath.org/1472.340632021-11-25T18:46:10.358925Z"Sang, Bo"https://zbmath.org/authors/?q=ai:sang.boSummary: In studying small limit cycles of finite-dimensional systems, one of the central problem is the computation of focus quantities. In practice, the computation is a challenging problem even for some simple low-dimensional systems. This paper is devoted to the computation of focus quantities of all orders and to the study of Hopf bifurcations in some chaotic systems. A recursive formula for computing focus quantities is presented for a \(K + 2\)-dimensional system. The formula is a generalization of previous results on low-dimensional systems with \(K = 0\) and \(K = 1\). For a four-dimensional hyper-chaotic system, according to the sign of the first focus quantity, we prove that the simple Hopf bifurcation of the system is supercritical. For a five-dimensional chaotic system with four equilibria of Hopf type, according to the signs of the first focus quantities, we prove that the simple Hopf bifurcations of the system are subcritical.Reachability of maximal number of critical periods without independencehttps://zbmath.org/1472.340642021-11-25T18:46:10.358925Z"Chen, Xingwu"https://zbmath.org/authors/?q=ai:chen.xingwu"Wang, Zhaoxia"https://zbmath.org/authors/?q=ai:wang.zhaoxia"Zhang, Weinian"https://zbmath.org/authors/?q=ai:zhang.weinianIn this paper, the authors consider the following system
\[
\dot x=-y+M(x, y, \lambda), \quad \dot y=x+N(x, y, \lambda),
\]
where \(\lambda=(\lambda_1, \ldots, \lambda_n)\) and
\[
M(x, y, \lambda), N(x, y, \lambda)=O(|x,y|^2).
\]
Suppose the above system has a center at the origin, then we can define the period function
\[
P(r)=2\pi+\sum_{i=2}p_i(\lambda)r^i.
\]
A classic problem is to determine the number of critical periods, which are the zeros of \(P'(r)=0\) for all \(\lambda\).
It is well known that if for \(p_2(\lambda')=p_4(\lambda')=\dots=p_{2k}(\lambda')=0\), but \(p_{2k+2}(\lambda')\not=0\) for some \(\lambda'\), then \(P'(r)\) has at most \(k\) zeros near \(r=0\) when \(|\lambda-\lambda'|\) is small. A natural problem is to ask whether we can find \(\lambda\) so that \(P'(r)\) has exact \(k\) zeros near \(r=0\).
This is a difficult problem. Usually one need the condition that \(p_2, p_4, \dots, p_{2k+2}\) are independent of the parament \(\lambda\), which will imply that one can only find at most \(n-1\) critical periods for \(\lambda=(\lambda_1, \ldots, \lambda_n)\). The authors give a new method, which does not need the independence condition, so that they can find more critical periods than before.On the number of limit cycles in generalized Abel equationshttps://zbmath.org/1472.340652021-11-25T18:46:10.358925Z"Huang, Jianfeng"https://zbmath.org/authors/?q=ai:huang.jianfeng"Torregrosa, Joan"https://zbmath.org/authors/?q=ai:torregrosa.joan"Villadelprat, Jordi"https://zbmath.org/authors/?q=ai:villadelprat.jordiConsider the generalized Abel differential equation
\[
\frac{{dx}}{{d\theta}} = A(\theta) x^p + B(\theta) x^q,\tag{1}
\]
where \( p,q \) are natural numbers satisfying \( p \neq q, p,q \ge 2\), \(A\) and \(B \) are trigonometric polynomials of degree \(n \ge 1\) and \(m \ge 1\), respectively. Let the number \(H_{p,q}(n,m) \) denote the maximum number of isolated periodic solutions (limit cycles) of (1). By means of the second order Melnikov function the authors prove a lower bound for \(H_{p,q}(n,m) \), which is better than known ones. Especially, they obtain for the classical Abel equation (i.e. \( p=3, q=2\)) the estimate \( H_{3,2}(n,m) \geq 2(n+m)-1 \).Boundedness of solutions of a quasi-periodic sublinear Duffing equationhttps://zbmath.org/1472.340662021-11-25T18:46:10.358925Z"Peng, Yaqun"https://zbmath.org/authors/?q=ai:peng.yaqun"Zhang, Xinli"https://zbmath.org/authors/?q=ai:zhang.xinli"Piao, Daxiong"https://zbmath.org/authors/?q=ai:piao.daxiongIn this paper, the authors study the Lagrangian stability for the sublinear Duffing equations \(x''+e(t)|x|^{\alpha-1}x=p(t)\) with \(0<\alpha<1\), where \(e\) and \(p\) are real analytic quasi-periodic functions with frequency \(\omega\). By using the invariant curve theorem for quasi-periodic mappings established by \textit{P. Huang} et al. [Nonlinearity 29, No. 10, 3006--3030 (2016; Zbl 1378.37078)], they prove that every solution of the equation is bounded provided that the mean value of \(e\) is positive and the frequency \(\omega\) satisfies a Diophantine condition.On the use of Jacobi elliptic functions for modelling the response of antisymmetric oscillators with a constant restoring forcehttps://zbmath.org/1472.340672021-11-25T18:46:10.358925Z"Kovacic, Ivana"https://zbmath.org/authors/?q=ai:kovacic.ivanaThe paper is devoted to modelling the response of conservative antisymmetric constant force oscillator \[ \frac{d^2x}{dt^2}+\text{sgn}(x)=0 \] in terms of Jacobi elliptic functions.
Two different approaches are developed. The first one starts from the known period of vibrations and the solution for motion expressed in terms of Jacobi elliptic functions. The second approach derives the expression for the velocity and motion starting from the expression for the acceleration in terms of the Jacobi elliptic functions.Complex behavior of a hyperchaotic TNC oscillator: coexisting bursting, space magnetization, control of multistability and application in image encryption based on DNA codinghttps://zbmath.org/1472.340682021-11-25T18:46:10.358925Z"Tagne Mogue, R. L."https://zbmath.org/authors/?q=ai:tagne-mogue.r-l"Folifack Signing, V. R."https://zbmath.org/authors/?q=ai:signing.v-r-folifack"Kengne, J."https://zbmath.org/authors/?q=ai:kengne.jacques"Kountchou, M."https://zbmath.org/authors/?q=ai:kountchou.michaux"Njitacke, Z. T."https://zbmath.org/authors/?q=ai:njitacke.zeric-tabekouengTowards a classification of networks with asymmetric inputshttps://zbmath.org/1472.340692021-11-25T18:46:10.358925Z"Aguiar, Manuela"https://zbmath.org/authors/?q=ai:aguiar.manuela-a-d"Dias, Ana"https://zbmath.org/authors/?q=ai:dias.ana-paula-s"Soares, Pedro"https://zbmath.org/authors/?q=ai:soares.pedro-a-junZero-Hopf bifurcation in a 3D jerk systemhttps://zbmath.org/1472.340702021-11-25T18:46:10.358925Z"Braun, Francisco"https://zbmath.org/authors/?q=ai:braun.francisco"Mereu, Ana C."https://zbmath.org/authors/?q=ai:mereu.ana-cristinaThe authors consider the 3D jerk system
\begin{align*}
& \dot{x}=y, \\
& \dot{y}=z, \\
& \dot{z}=-az-bx+cy+x{{y}^{2}}-{{x}^{3}};\quad (x,y,z)\in {{\mathbb{R}}^{3}}, \\
\end{align*}
where \(a,b,c\in \mathbb{R}\) are system parameters. By applying averaging theory, the authors analyze the zero-Hopf bifurcation at the origin and prove the existence of bifurcating limit cycles depending on the system parameters. It was shown that up to three periodic orbits can be born at the equilibrium point.Bifurcation of nongeneric homoclinic orbit accompanied by pitchfork bifurcationhttps://zbmath.org/1472.340712021-11-25T18:46:10.358925Z"Geng, Fengjie"https://zbmath.org/authors/?q=ai:geng.fengjie"Li, Song"https://zbmath.org/authors/?q=ai:li.song.1Summary: The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.Global bifurcations in generic one-parameter families with a parabolic cycle on \(S^2\)https://zbmath.org/1472.340722021-11-25T18:46:10.358925Z"Goncharuk, N."https://zbmath.org/authors/?q=ai:goncharuk.n-b|goncharuk.nataliya-yu"Ilyashenko, Yu."https://zbmath.org/authors/?q=ai:ilyashenko.yulij-s"Solodovnikov, N."https://zbmath.org/authors/?q=ai:solodovnikov.nikita-aSummary: We classify global bifurcations in generic one-parameter local families of vector fields on \(S^2\) with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.Dynamic aspects of Sprott BC chaotic systemhttps://zbmath.org/1472.340732021-11-25T18:46:10.358925Z"Mota, Marcos C."https://zbmath.org/authors/?q=ai:mota.marcos-c"Oliveira, Regilene D. S."https://zbmath.org/authors/?q=ai:oliveira.regilene-d-sThe authors consider the 3D system
\begin{align*}
& \dot{x}=yz, \\
& \dot{y}=x-y, \\
& \dot{z}=1-x\left( \alpha y+\beta x \right);\quad (x,y,z)\in {{\mathbb{R}}^{3}}, \\
\end{align*}
where \(\alpha ,\beta \in [0,1]\) are system parameters. For \(\alpha =1\), \(\beta =0\), the considered system coincides with the Sprott B chaotic system and for \(\alpha =0\), \(\beta =1\), it coincides with the Sprott C chaotic system. It is shown that the considered system has two unstable equilibrium points. Further, the authors prove that this system passes through a subcritical Hopf bifurcation at \(\alpha =0\). The Poincaré compactification for a polynomial vector field is used for studying the behavior of the system at infinity. Moreover, it is proved that the system does not possess a polynomial first integral, nor algebraic invariant surfaces, neither a Darboux first integral.Periodic forcing on degenerate Hopf bifurcationhttps://zbmath.org/1472.340742021-11-25T18:46:10.358925Z"Yuan, Qigang"https://zbmath.org/authors/?q=ai:yuan.qigang"Ren, Jingli"https://zbmath.org/authors/?q=ai:ren.jingliIn this work, the authors deal with the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by the Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods. The method is novel and enriches the bifurcation theory of delayed differential equation to some degree.Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equationhttps://zbmath.org/1472.340752021-11-25T18:46:10.358925Z"Zhang, Lijun"https://zbmath.org/authors/?q=ai:zhang.lijun"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.6"Khalique, Chaudry Masood"https://zbmath.org/authors/?q=ai:khalique.chaudry-masood"Bai, Yuzhen"https://zbmath.org/authors/?q=ai:bai.yuzhenThis paper studies the model of the generalized Camassa-Holm-Novikov (gCHN) equation. The main contribution of this paper is to discuss the singular wave solutions including peakons and cuspons. They give the phase diagram and obtain the exact explicit parametric representation of the traveling wave solutions corresponding to these singular wave solutions.Multiple periodic solutions for one-sided sublinear systems: a refinement of the Poincaré-Birkhoff approachhttps://zbmath.org/1472.340762021-11-25T18:46:10.358925Z"Dondè, Tobia"https://zbmath.org/authors/?q=ai:donde.tobia"Zanolin, Fabio"https://zbmath.org/authors/?q=ai:zanolin.fabioThe paper investigates the existence of periodic solutions, both harmonic and subharmonic, for planar Hamiltonian systems of the type \[ x' = h(y), \qquad y' = - a(t)g(x), \] where \(a(t)\) is a sign-changing periodic function and at least one of \(g\) and \(h\) is bounded on \(\mathbb{R}^-\) or \(\mathbb{R}^+\).
At first, by further assuming the global continuability for the solutions, a multiplicity result is proved via the Poincaré-Birkhoff theorem; as usual, solutions are distinguished via their nodal properties. Then, a refinement of this result, obtained with the theory of topological horseshoses, is presented; here, the assumption of global continuability is replaced by a largeness condition on the weight function \(a(t)\) in its negativity intervals. In this latter case, the existence of chaotic dynamics is also ensured.
Applications of the results are finally described for a Minkowksi-curvature equation like \[ \left( \frac{u'}{\sqrt{1-(u')^2}} \right)' + a(t) g(u) = 0 \] as well as for the equation, with exponential nonlinearity, \[ u'' + k(t)e^u = p(t). \]Exact multiplicity and stability of periodic solutions for Duffing equation with bifurcation methodhttps://zbmath.org/1472.340772021-11-25T18:46:10.358925Z"Liang, Shuqing"https://zbmath.org/authors/?q=ai:liang.shuqingSummary: Under some \(L^p\)-norms \((p\in [1,\infty ])\) assumptions for the derivative of the restoring force, the exact multiplicity and the stability of \(2\pi\)-periodic solutions for Duffing equation are considered. The nontrivial \(2\pi\)-periodic solutions of it are positive or negative, and the bifurcation curve of it is a unique reversed \(S\)-shaped curve. The class of the restoring force is extended, comparing with the class of \(L^{\infty }\)-norm condition. The proof is based on the global bifurcation theorem, topological degree and the estimates for periodic eigenvalues of Hill's equation by \(L^p\)-norms\((p\in [1,\infty ])\).A new fixed point theorem and periodic solutions of nonconservative weakly coupled systemshttps://zbmath.org/1472.340782021-11-25T18:46:10.358925Z"Liu, Chunlian"https://zbmath.org/authors/?q=ai:liu.chunlian"Qian, Dingbian"https://zbmath.org/authors/?q=ai:qian.dingbianSummary: In this paper, we prove a new fixed point theorem for the coupling of twist conditions and Poincaré-Bohl type conditions, as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems.Non-resonance and double resonance for a planar system via rotation numbershttps://zbmath.org/1472.340792021-11-25T18:46:10.358925Z"Liu, Chunlian"https://zbmath.org/authors/?q=ai:liu.chunlian"Qian, Dingbian"https://zbmath.org/authors/?q=ai:qian.dingbian"Torres, Pedro J."https://zbmath.org/authors/?q=ai:torres.pedro-joseThe authors consider a general planar periodic system and propose two existence results.
In the first one they compare the nonlinearity with two positively homogeneous functions with ``rotation numbers'' larger than some \(n\) and smaller than \(n+1\). They thus prove that the system has a periodic solution, by the use of the Poincaré-Bohl fixed point theorem. This is a generalization of some classical ``nonresonance'' results.
In the second one the above two functions have rotation numbers exactly equal to \(n\) and \(n+1\). Then, in order to avoid possible resonance phenomena, they add two Landesman-Lazer conditions, and they prove again the existence of a periodic solution.
The proofs involve delicate analysis in the phase-plane, in order to precisely estimate the rotational properties of the solutions.Periodic solutions for a singular Liénard equation with indefinite weighthttps://zbmath.org/1472.340802021-11-25T18:46:10.358925Z"Lu, Shiping"https://zbmath.org/authors/?q=ai:lu.shiping"Xue, Runyu"https://zbmath.org/authors/?q=ai:xue.runyuIn this paper, the authors study the following singular Liénard equation \[ x''(t)+f(x(t))x'(t)+\frac{\alpha(t)}{x^\mu(t)}= h(t),\tag{1} \] where \(f\in C((0, +\infty), \mathbb{R})\) may have a singularity at \(x=0,\, \mu\in(0, +\infty)\) is a constant, \(\alpha\) and \(h\) are \(T\)-periodic functions with \(\alpha,\, h \in L^1 ([0, T], \mathbb{R}).\) The weight function \(\alpha\) may change sign on \([0, T].\) A new method for estimating a priori bounds of all possible positive \(T\)-periodic solutions is obtained. By using a continuation theorem of Mawhin's coincidence degree theory, some new results on the existence of positive periodic solutions for the equation (1) are established.Existence of \(T/k\)-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvaluehttps://zbmath.org/1472.340812021-11-25T18:46:10.358925Z"Yevstafyeva, V. V."https://zbmath.org/authors/?q=ai:yevstafyeva.victoria-v|yevstafyeva.vistoria-vThe author considers the \(n\)-dimensional system of differential equations of the following form
\[
\dot{Y} = AY + BF(\sigma) + Kf(t).
\]
It is assumed that the matrix \(A\) has real nonzero eigenvalues, among which there are at least one positive and one multiple eigenvalues, the vectors \(B\) and \(K\) are nonzero, and the function \(F(\sigma),\)\,\(\sigma = (C,Y)\) describes a nonideal two-position relay with two output values. The perturbation function \(f(t)\) is \(T\)-periodic with \(T = 2\pi/\omega,\,\omega >0\) of the form
\[
f(t) = f_0 +f_1\sin(\omega t + \varphi_1) + f_2\sin (2\omega t + \varphi_2).
\]
Necessary conditions for the existence of a \(T/k\)-periodic solution of the system with \(k \in \mathbb{N}\) having two switching points in the phase space are studied. An existence theorem for a \(T\)-periodic solution is proved. A numerical example is presented.Determining key parameters in riots using lexicographic directional differentiationhttps://zbmath.org/1472.340822021-11-25T18:46:10.358925Z"Ackley, Matthew"https://zbmath.org/authors/?q=ai:ackley.matthew"Stechlinski, Peter"https://zbmath.org/authors/?q=ai:stechlinski.peter-gSustainable vector/pest control using the permanent sterile insect techniquehttps://zbmath.org/1472.340832021-11-25T18:46:10.358925Z"Anguelov, Roumen"https://zbmath.org/authors/?q=ai:anguelov.roumen"Dumont, Yves"https://zbmath.org/authors/?q=ai:dumont.yves"Djeumen, Ivric Valaire Yatat"https://zbmath.org/authors/?q=ai:yatat-djeumen.ivric-valaireSummary: Vector/pest control is essential to reduce the risk of vector-borne diseases or losses in crops. Among all biological control tools, the sterile insect technique (SIT), which consists of massive releases of sterile insects to reach elimination or to lower a vector/pest population under a certain threshold, is the most promising one. The models presented here are minimalistic with respect to the number of parameters and variables. The first model deals with the dynamics of the vector population, while the second model tackles the interaction between treated males and wild female vectors. For the vector population model, equilibrium \(\mathbf{0}\) is globally asymptotically stable when the basic offspring number, \(\mathcal{R} \leq 1\), whereas \(\mathbf{0}\) becomes unstable and one stable positive equilibrium exists, with well-determined basins of attraction, when \(\mathcal{R} > 1\). For the SIT model, we obtain a threshold number of treated males above which the control of wild population is effective using massive releases. When the amount of treated males is lower than the aforementioned threshold, the SIT model experiences a strong Allee effect, that is, \(\mathbf{0}\) becomes locally asymptotically stable, while a positive equilibrium still exists. Practically, massive releases of sterile males are only possible for a short period. That is why using the Allee effect, we develop a new strategy to maintain the wild population under a certain threshold, for a permanent and sustainable low level of SIT control. We illustrate our theoretical results with numerical simulations. In particular, we study the combination of SIT with other control tools, like mechanical control and adulticide.Mean-field and graph limits for collective dynamics models with time-varying weightshttps://zbmath.org/1472.340842021-11-25T18:46:10.358925Z"Ayi, Nathalie"https://zbmath.org/authors/?q=ai:ayi.nathalie"Pouradier Duteil, Nastassia"https://zbmath.org/authors/?q=ai:duteil.nastassia-pouradierSummary: In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of \textit{indistinguishability}, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. This actually provides an alternative proof for the mean-field limit. We conclude by showing some numerical simulations to illustrate our results.Epidemic threshold in pairwise models for clustered networks: closures and fast correlationshttps://zbmath.org/1472.340852021-11-25T18:46:10.358925Z"Barnard, Rosanna C."https://zbmath.org/authors/?q=ai:barnard.rosanna-c"Berthouze, Luc"https://zbmath.org/authors/?q=ai:berthouze.luc"Simon, Péter L."https://zbmath.org/authors/?q=ai:simon.peter-l"Kiss, István Z."https://zbmath.org/authors/?q=ai:kiss.istvan-zSummary: The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cationshttps://zbmath.org/1472.340862021-11-25T18:46:10.358925Z"Bates, Peter W."https://zbmath.org/authors/?q=ai:bates.peter-w"Wen, Zhenshu"https://zbmath.org/authors/?q=ai:wen.zhenshu"Zhang, Mingji"https://zbmath.org/authors/?q=ai:zhang.mingjiSummary: A quasi-one-dimensional Poisson-Nernst-Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail. Of particular interest is the competition between two cations, which is related to the selectivity phenomena of ion channels. Critical potentials are identified and their roles in characterizing ionic flow properties are studied. Some critical potentials can be estimated experimentally, and this provides an efficient way to adjust/control boundary conditions (electric potential and concentrations) to observe distinct qualitative properties of ionic flows. Mathematical analysis further indicates that to optimize the effect of permanent charges, a short and narrow filter, within which the permanent charge is confined, is expected, which is consistent with the typical structure of an ion channel.Erratum to: ``An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza''https://zbmath.org/1472.340872021-11-25T18:46:10.358925Z"Chong, Nyuk Sian"https://zbmath.org/authors/?q=ai:chong.nyuk-sian"Dionne, Benoit"https://zbmath.org/authors/?q=ai:dionne.benoit"Smith, Robert"https://zbmath.org/authors/?q=ai:smith.robert-jFrom the text: Unfortunately, in the original publication of the article [the authors, ibid. 73, No. 3, 751--784 (2016; Zbl 1353.34053)], the first row of Table 2 was misplaced i.e. the entry \(I_b <g_3\) should be the first row.
The corrected table is given below and the original article has been updated to reflect this change.On the stability of proliferation of kertinocytes in psoriatic skin fractional modelhttps://zbmath.org/1472.340882021-11-25T18:46:10.358925Z"Hassouna, Meryeme"https://zbmath.org/authors/?q=ai:hassouna.meryeme"Ouhadan, Abdelaziz"https://zbmath.org/authors/?q=ai:ouhadan.abdelaziz"El Kinani, El Hassan"https://zbmath.org/authors/?q=ai:el-kinani.el-hassanSummary: In this paper, we introduce a fractional order into a psoriasis model. The system is composed of two fractional equations; we prove the local existence and uniqueness of the solution of the fractional psoriasis system; some stability conditions of the model equilibrium are given; finally, we further present some numerical results that confirm the stability.Optimal tax policy of a one-predator-two-prey system with a marine protected areahttps://zbmath.org/1472.340892021-11-25T18:46:10.358925Z"Huang, Lirong"https://zbmath.org/authors/?q=ai:huang.lirong"Cai, Donghan"https://zbmath.org/authors/?q=ai:cai.donghan"Liu, Weiyi"https://zbmath.org/authors/?q=ai:liu.weiyiSummary: This paper is devoted to handle a dynamic one-predator-two-prey model. In order to protect fish population from over exploitation, we assume that marine protected area (MPA) is established and the fisherman only harvest the prey in the unreserved area, the predator consumes the prey in both the MPA and the unreserved area. And a tax is imposed in the process of harvesting. To begin with, boundeness of the system is discussed. Following this, we studied the existence of the possible equilibrium along with their local stability for both of the unexploited system (8) and exploited system (5). After that, we analyzed the global stability of the positive equilibrium of the exploited system and how the tax \(\tau\) affects the positive equilibrium. Then, the optimal tax policy is obtained by using the Pontryagin's maximum principle. Finally, some numerical simulations are given to support the analytical findings.Codimension-3 bifurcation in the p53 regulatory network modelhttps://zbmath.org/1472.340902021-11-25T18:46:10.358925Z"Jiang, Cuicui"https://zbmath.org/authors/?q=ai:jiang.cuicui"Zhang, Yongxin"https://zbmath.org/authors/?q=ai:zhang.yongxin"Wang, Wendi"https://zbmath.org/authors/?q=ai:wang.wendiIn this paper, the authors study the dynamical behaviors of the p53-Mdm2 model, and show that the model can undergo the saddle-node bifurcation, Hopf bifurcation, codimension-2 Bogdanov-Takens bifurcation, and codimension-3 Bogdanov-Takens bifurcation by rigorous mathematical analysis, and also explain the medical implications of these resluts.Canard-induced complex oscillations in an excitatory networkhttps://zbmath.org/1472.340912021-11-25T18:46:10.358925Z"Köksal Ersöz, Elif"https://zbmath.org/authors/?q=ai:ersoz.elif-koksal"Desroches, Mathieu"https://zbmath.org/authors/?q=ai:desroches.mathieu"Guillamon, Antoni"https://zbmath.org/authors/?q=ai:guillamon.antoni"Rinzel, John"https://zbmath.org/authors/?q=ai:rinzel.john"Tabak, Joël"https://zbmath.org/authors/?q=ai:tabak.joelThis is an excellent article reviewing the canard-induced complex oscillations in an excitatory network. In this article, the authors give a detailed description on complex oscillations including relaxation oscillations, bursting oscillations, mixed-mode oscillations (MMOs) and mixed-mode bursting oscillations (MMBOs) in singular perturbation systems. More importantly, the authors point that the occurrence of these complex oscillations is tightly related to the dimensions of singular perturbation systems under consideration. For example, MMBOs are possible in systems with at least two slow and two fast variables. Based on geometric singular perturbation theory, the complex bifurcation process on the occurrence of these complex oscillations in this excitatory network is revealed in details and a large amount of bifurcation diagrams are given.Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operatorhttps://zbmath.org/1472.340922021-11-25T18:46:10.358925Z"Kopfová, Jana"https://zbmath.org/authors/?q=ai:kopfova.jana"Nábělková, Petra"https://zbmath.org/authors/?q=ai:nabelkova.petra"Rachinskii, Dmitrii"https://zbmath.org/authors/?q=ai:rachinskii.dmitrii-i"Rouf, Samiha C."https://zbmath.org/authors/?q=ai:rouf.samiha-cSummary: We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.Fractional order prey-predator model with infected predators in the presence of competition and toxicityhttps://zbmath.org/1472.340932021-11-25T18:46:10.358925Z"Lemnaouar, M. R."https://zbmath.org/authors/?q=ai:lemnaouar.m-r"Khalfaoui, M."https://zbmath.org/authors/?q=ai:khalfaoui.mehdi|khalfaoui.mohamed"Louartassi, Y."https://zbmath.org/authors/?q=ai:louartassi.younes"Tolaimate, I."https://zbmath.org/authors/?q=ai:tolaimate.iSummary: In this paper, we propose a fractional-order prey-predator model with reserved area in the presence of the toxicity and competition. We prove different mathematical results like existence, uniqueness, non negativity and boundedness of the solution for our model. Further, we discuss the local and global stability of these equilibria. Finally, we perform numerical simulations to prove our results.Simulation-based study of biological systems with threshold policy by a differential linear complementarity systemhttps://zbmath.org/1472.340942021-11-25T18:46:10.358925Z"Luo, Jianfeng"https://zbmath.org/authors/?q=ai:luo.jianfeng"Zhao, Yi"https://zbmath.org/authors/?q=ai:zhao.yi.1Modeling the effect of contaminated objects for the transmission dynamics of COVID-19 pandemic with self protection behavior changeshttps://zbmath.org/1472.340952021-11-25T18:46:10.358925Z"Mekonen, Kassahun Getnet"https://zbmath.org/authors/?q=ai:mekonen.kassahun-getnet"Habtemicheal, Tatek Getachew"https://zbmath.org/authors/?q=ai:habtemicheal.tatek-getachew"Balcha, Shiferaw Feyissa"https://zbmath.org/authors/?q=ai:balcha.shiferaw-feyissaSummary: A mathematical model for the transmission dynamics of Coronavirus diseases (COVID-19) is proposed using a system of nonlinear ordinary differential equations by incorporating self protection behavior changes in the population. The disease free equilibrium point is computed, and both the local and global stability analysis was performed. The basic reproduction number \((R_0)\) of the model is computed using the method of next generation matrix. The disease free equilibrium point is locally asymptotically and globally stable under certain conditions. Based on the available data, the unknown model parameters are estimated using a combination of least square and Bayesian estimation methods for different countries. The forward sensitivity index is applied to determine and identify the key model parameters for the spread of disease dynamics. The sensitive parameters for the spread of the virus vary from country to country. We found out that the reproduction number depends mostly on the infection rates, the threshold value of the force of infection for a population, the recovery rates, and the virus decay rate in the environment. It illustrates that control of the effective transmission rate (recommended human behavioral change towards self-protective measures) is essential to stop the spreading of the virus. Numerical simulations of the model were performed to supplement and verify the effectiveness of the analytical findings.Periodic solutions of a tumor-immune system interaction under a periodic immunotherapyhttps://zbmath.org/1472.340962021-11-25T18:46:10.358925Z"Torres-Espino, Gladis"https://zbmath.org/authors/?q=ai:torres-espino.gladis"Vidal, Claudio"https://zbmath.org/authors/?q=ai:vidal.claudioSummary: In this paper, we consider a mathematical model of a tumor-immune system interaction when a periodic immunotherapy treatment is applied. We give sufficient conditions, using averaging theory, for the existence and stability of periodic solutions in such system as a function of the six parameters associated to this problem. Finally, we provide examples where our results are applied.Analysis of the fractional corona virus pandemic via deterministic modelinghttps://zbmath.org/1472.340972021-11-25T18:46:10.358925Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Tri, Vo Viet"https://zbmath.org/authors/?q=ai:tri.vo-viet"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: With every passing day, one comes to know that cases of the corona virus disease are increasing. This is an alarming situation in many countries of the globe. So far, the virus has attacked as many as 188 countries of the world and 5 549 131 (27 May 2020) human population is affected with 348 224 deaths. In this regard, public and private health authorities are looking for manpower with modeling skills and possible vaccine. In this research paper, keeping in view the fast transmission dynamics of the virus, we have proposed a new mathematical model of eight mutually distinct compartments with the help of memory-possessing operator of Caputo type. The fractional order parameter \(\psi\) of the model has been optimized so that smallest error can be attained while comparing simulations and the real data set which is considered for the country Pakistan. Using Banach fixed point analysis, it has been shown that the model has a unique solution whereas its basic reproduction number \(\mathcal{R}_0\) is approximated to be 6.5894. Disease-free steady state is shown to be locally asymptotically stable for \(\mathcal{R}_0 < 0\), otherwise unstable. Nelder-Mead optimization algorithm under MATLAB Toolbox with daily real cases of the virus in Pakistan is employed to obtain best fitted values of the parameters for the model's validation. Numerical simulations of the model have come into good agreement with the practical observations wherein social distancing, wearing masks, and staying home have proved to be the most effective measures in order to prevent the virus from further spread.Ergodicity and threshold behaviors of a predator-prey model in stochastic chemostat driven by regime switchinghttps://zbmath.org/1472.340982021-11-25T18:46:10.358925Z"Wang, Liang"https://zbmath.org/authors/?q=ai:wang.liang"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqingSummary: This paper deals with a stochastic predator-prey model in chemostat which is driven by Markov regime switching. For the asymptotic behaviors of this stochastic system, we establish the sufficient conditions for the existence of the stationary distribution. Then, we investigate, respectively, the extinction of the prey and predator populations. We explore the new critical numbers between survival and extinction for species of the dual-threshold chemostat model. Numerical simulations are accomplished to confirm our analytical conclusions.Effects on \(I-V\) relations from small permanent charge and channel geometry via classical Poisson-Nernst-Planck equations with multiple cationshttps://zbmath.org/1472.340992021-11-25T18:46:10.358925Z"Wen, Zhenshu"https://zbmath.org/authors/?q=ai:wen.zhenshu"Bates, Peter W."https://zbmath.org/authors/?q=ai:bates.peter-w"Zhang, Mingji"https://zbmath.org/authors/?q=ai:zhang.mingjiExistence and uniqueness of globally attractive positive almost periodic solution in a predator-prey dynamic system with Beddington-DeAngelis functional responsehttps://zbmath.org/1472.341002021-11-25T18:46:10.358925Z"Wu, Wenquan"https://zbmath.org/authors/?q=ai:wu.wenquanSummary: This paper is concerned with a predator-prey system with Beddington-DeAngelis functional response on time scales. By using the theory of exponential dichotomy on time scales and fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive (almost) periodic solution of the above system. Further, by means of Lyapunov functional, the global attractivity of the almost periodic solution for the above continuous system is also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples are given to illustrate the feasibility and effectiveness of the main results.Locally active memristor based oscillators: the dynamic route from period to chaos and hyperchaoshttps://zbmath.org/1472.341012021-11-25T18:46:10.358925Z"Ying, Jiajie"https://zbmath.org/authors/?q=ai:ying.jiajie"Liang, Yan"https://zbmath.org/authors/?q=ai:liang.yan"Wang, Guangyi"https://zbmath.org/authors/?q=ai:wang.guangyi"Iu, Herbert Ho-Ching"https://zbmath.org/authors/?q=ai:iu.herbert-ho-ching"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.1|zhang.jian.5|zhang.jian.3|zhang.jian.4|zhang.jian.6|zhang.jian.2|zhang.jian|zhang.jian.7"Jin, Peipei"https://zbmath.org/authors/?q=ai:jin.peipeiSummary: To explore the complexity of the locally active memristor and its application circuits, a tristable locally active memristor is proposed and applied in periodic, chaotic, and hyperchaotic circuits. The quantitative numerical analysis illustrated the steady-state switching mechanism of the memristor using the power-off plot and dynamic route map. For any pulse amplitude that can achieve a successful switching, there must be a minimum pulse width that enables the state variable to move beyond the attractive region of the equilibrium point. As local activity is the origin of complexity, the locally active memristor can oscillate periodically around a locally active operating point when connected in series with a linear inductor. A chaotic oscillation evolves from periodic oscillation by adding a capacitor in the periodic oscillation circuit, and a hyperchaotic oscillation occurs by further putting an extra inductor into the chaotic circuit. Finally, the dynamic behaviors and complexity mechanism are analyzed by utilizing coexisting attractors, dynamic route map, bifurcation diagram, Lyapunov exponent spectrum, and the basin of attraction.
{\copyright 2021 American Institute of Physics}The dynamics and synchronization of a fractional-order system with complex variableshttps://zbmath.org/1472.341022021-11-25T18:46:10.358925Z"Yang, Xiaoya"https://zbmath.org/authors/?q=ai:yang.xiaoya"Liu, Xiaojun"https://zbmath.org/authors/?q=ai:liu.xiao-jun.2"Dang, Honggang"https://zbmath.org/authors/?q=ai:dang.honggang"He, Wansheng"https://zbmath.org/authors/?q=ai:he.wanshengSummary: A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.Hyers-Ulam stability for a class of perturbed Hill's equationshttps://zbmath.org/1472.341032021-11-25T18:46:10.358925Z"Dragičević, Davor"https://zbmath.org/authors/?q=ai:dragicevic.davorSummary: In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers-Ulam stable. This class consists of equations obtained by perturbing Hill's equation of the form \(x''=(\lambda^2(t)-\lambda '(t))x\).Exponential stability of fast driven systems, with an application to celestial mechanicshttps://zbmath.org/1472.341042021-11-25T18:46:10.358925Z"Chen, Qinbo"https://zbmath.org/authors/?q=ai:chen.qinbo"Pinzari, Gabriella"https://zbmath.org/authors/?q=ai:pinzari.gabriellaThe authors consider a \((n+1+m)\)-dimensional vector-filed \(N\) which, expressed in local coordinates \((I,y,\psi)\in \mathbb P=\mathbb I\times \mathbb Y\times \mathbb T^m\), (where \(\mathbb I\subset \mathbb R^n\), \(\mathbb Y\subset \mathbb R^n\) are open and connected; \(\mathbb T=\mathbb R/(2\pi \mathbb Z)\) is the standard torus), has the form \[N(I,y)=v(I,y)\partial_y+\omega(I,y)\partial_{\psi}.\] Such systems have been extensively investigated in the absence of the coordinate \(y\) and it is known that, after a small perturbing term is switched on, the normalized actions \(I\) turn to have exponential small variations compared to the size of the perturbation. The authors obtain the same result as for the classical situation. In addition, they observe that no trapping argument is needed, as no small denominator arises. They use the result to prove that the level sets of certain function called \textit{Euler integral} have exponential small variations in a short time, closely to collisions.Stability domains for quadratic-bilinear reduced-order modelshttps://zbmath.org/1472.341052021-11-25T18:46:10.358925Z"Kramer, Boris"https://zbmath.org/authors/?q=ai:kramer.borisWild pseudohyperbolic attractor in a four-dimensional Lorenz systemhttps://zbmath.org/1472.341062021-11-25T18:46:10.358925Z"Gonchenko, Sergey"https://zbmath.org/authors/?q=ai:gonchenko.sergey-v"Kazakov, Alexey"https://zbmath.org/authors/?q=ai:kazakov.alexey-o"Turaev, Dmitry"https://zbmath.org/authors/?q=ai:turaev.dmitry-vThe authors consider the 4D extension of the Lorenz system
\begin{align*}
& \dot{x}=\sigma \left( y-x \right), \\
& \dot{y}=x\left( r-z \right)-y, \\
& \dot{z}=xy-bz+\mu w, \\
& \dot{w}=-bw-\mu z, \\
\end{align*}
where \(\sigma ,r,b,\mu \) are system parameters. With the help of numerical experiments, the authors show that the system under consideration has a so-called wild pseudohyperbolic spiral attractor [\textit{D. V. Turaev} and \textit{L. P. Shil'nikov}, Sb. Math. 189, No. 2, 137--160 (1998; Zbl 0927.37017); translation from Mat. Sb. 189, No. 2, 291--314 (1998)], that is, pseudohyperbolic, spiral (contains a saddle-focus equilibrium), and wild (contains a hyperbolic set with homoclinic tangencies).Some remarks on the renormalization group and Chapman-Enskog type methods in singularly perturbed problemshttps://zbmath.org/1472.341072021-11-25T18:46:10.358925Z"Banasiak, Jacek"https://zbmath.org/authors/?q=ai:banasiak.jacekSummary: In this paper, we show that for a large class of singularly perturbed problems, the classical Chapman-Enskog asymptotic procedure leads, in a shorter way, to the same asymptotic expansion as the renormalization group (RG) approach. We also prove that the Chapman-Enskog expansion gives the expected error estimates uniformly on \([0, \infty)\).Singularly perturbed partially dissipative systems of equationshttps://zbmath.org/1472.341082021-11-25T18:46:10.358925Z"Butuzov, V. F."https://zbmath.org/authors/?q=ai:butuzov.valentin-fedorovichIn this paper, the author considers the stationary partially dissipative system consisting of the two equations
\[
\varepsilon^2 \left(\frac{d^2u}{dx^2}-w(x)\frac{du}{dx}\right)=F(u,v,x,\varepsilon),
\]
\[
\varepsilon^2 \frac{dv}{dx}=f(u,v,x,\varepsilon), \ x\in(0;1)
\]
and with the boundary conditions
\[
u(0,\varepsilon)=u^0,\ v(0,\varepsilon)=v^0,\ u(1,\varepsilon)=u^1,
\]
where \(\varepsilon>0\) is a small parameter and \(w, F,\) and \(f\) are given sufficiently smooth functions, with \(F(u,v,x,0)=0\) and \(f(u,v,x,0)=0.\)
Other key assumptions are as follows:
\begin{itemize}
\item The function \(f\) has the form
\[
f(u,v,x,\varepsilon) = -(v - \varphi(u,x))^3 + \varepsilon f_1(u,v,x,\varepsilon).
\]
\item The equation
\[
F(u,\varphi(u,x),x,0) =: g(u,x) = 0
\]
has a root \(u=\bar u_0(x),\) \(x\in[0;1],\) and
\[
\frac{\partial g}{\partial u}(\bar u_0(x),x) > 0, \ x\in[0;1].
\]
\end{itemize}
The purpose of the paper is to obtain conditions under which the boundary value problem has a multizonal boundary layer solution \(u(x,\varepsilon),\) \(v(x,\varepsilon)\) for sufficiently small \(\varepsilon\), that is, a solution that tends to the solution of degenerate system (\(\varepsilon=0\)) as \(\varepsilon\to 0^+\) on the interval \(0 < x < 1\) and to construct an asymptotic approximation of this solution in the parameter \(\varepsilon\) on the whole interval \(0 \leq x \leq 1\) including the boundary layers, that is, small neighborhoods of the boundary points \(x = 0\) and \(x = 1,\) where the solution \(u(x,\varepsilon),\) \(v(x,\varepsilon)\) significantly differs from the solution of the degenerate system.Singularly perturbed boundary-focus bifurcationshttps://zbmath.org/1472.341092021-11-25T18:46:10.358925Z"Jelbart, Samuel"https://zbmath.org/authors/?q=ai:jelbart.samuel"Kristiansen, Kristian Uldall"https://zbmath.org/authors/?q=ai:kristiansen.kristian-uldall"Wechselberger, Martin"https://zbmath.org/authors/?q=ai:wechselberger.martinSummary: We consider smooth systems limiting as \(\epsilon \to 0\) to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the \textit{smooth} system with sufficiently small but non-zero \(\epsilon \), using a combination of \textit{geometric singular perturbation theory} and \textit{blow-up}. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an \(\epsilon\)-dependent domain which shrinks to zero as \(\epsilon \to 0\), identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation oscillations in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation oscillations to regular cycles within the \(\epsilon\)-dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.Asymptotic integration of singularly perturbed differential algebraic equations with turning points. Ihttps://zbmath.org/1472.341102021-11-25T18:46:10.358925Z"Samoilenko, A. M."https://zbmath.org/authors/?q=ai:samoilenko.anatolii-mikhailovich"Samusenko, P. F."https://zbmath.org/authors/?q=ai:samusenko.petro-fIn this paper, the authors study linear differential-algebraic singularly perturbed systems of the form
\[
\varepsilon B(t,\varepsilon)\frac{dx}{dt}=A(t,\varepsilon)x,\quad t\in[0;T],
\]
where
\[
A(t,\varepsilon)=\sum\limits_{k\geq0}\varepsilon^k A_k(t), \quad B(t,\varepsilon)=\sum\limits_{k\geq0}\varepsilon^k B_k(t)
\]
satisfy that
\begin{itemize}
\item[(i)] \(A(0,0) = \mathrm{diag}\,\{E_q,J_p\},\) \(B(0,0) = \mathrm{diag}\,\{J_q,E_p\},\) and \(p + q = n,\) where \(E_q\) is the identity matrix of order \(q,\) \(J_q\) is a square matrix of order \(q,\) the elements of the upper superdiagonal of the matrix are equal to \(1,\) the other elements are equal to zero; and the matrices \(E_p\) and \(J_p\) are defined analogously;
\item[(ii)] \(\frac{d}{dt}(\mathrm{det}\, A(t,0))\vert_{t=0} \neq 0\) and \(\frac{d}{dt}(\mathrm{det}\, B(t,0))\vert_{t=0} \neq 0.\)
\end{itemize}
The authors separately construct the asymptotic solutions of the system under consideration (for small \(\varepsilon > 0\)) on two intervals that do not contain turning point (external decomposition) and on a interval containing the turning point (internal decomposition - this is discussed in the second part of the present paper). Then these solutions are joined to construct asymptotic decomposition and to determine the fundamental matrix of the system on the whole interval \([0;T].\)Stochastic approaches to Lagrangian coherent structureshttps://zbmath.org/1472.341112021-11-25T18:46:10.358925Z"Balasuriya, Sanjeeva"https://zbmath.org/authors/?q=ai:balasuriya.sanjeevaSummary: This note discusses a connection between deterministic Lagrangian coherent structures (robust fluid parcels which move coherently in unsteady fluid flows according to a deterministic ordinary differential equation), and the incorporation of noise or stochasticity which leads to the Fokker-Planck equation (a partial differential equation governing a probability density function). The link between these is via a stochastic ordinary differential equation. It is argued that a closer investigation of the stochastic differential equation offers additional insights to both the other approaches, and in particular to uncertainty quantification in Lagrangian coherent structures.
For the entire collection see [Zbl 1462.35005].Corrigendum to: ``Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion''https://zbmath.org/1472.341122021-11-25T18:46:10.358925Z"Blouhi, T."https://zbmath.org/authors/?q=ai:blouhi.tayeb"Caraballo, T."https://zbmath.org/authors/?q=ai:caraballo.tomas"Ouahab, A."https://zbmath.org/authors/?q=ai:ouahab.abdelghaniSummary: In this paper we correct an error made in our paper [ibid. 34, No. 5, 792--834 (2016; Zbl 1380.34091)]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.Transient probability in basins of noise influenced responses of mono and coupled Duffing oscillatorshttps://zbmath.org/1472.341132021-11-25T18:46:10.358925Z"Cilenti, Lautaro"https://zbmath.org/authors/?q=ai:cilenti.lautaro"Balachandran, Balakumar"https://zbmath.org/authors/?q=ai:balachandran.balakumarThis paper concerns Duffing oscillators perturbed by random noise and modeled using Ito stochastic differential equations (SDE). An improvement of the path integration method is introduced that reduces the number of points used and therefore facilitates less costly computation. The improved method is used in numerical experiments to study the relationship between intensity of random noise in the forcing function and destruction of the high-amplitude mode in the multistability region of hardened Duffing oscillators and also to study the level of intensity of noise that destroys an energy localized mode in arrays of two coupled Duffing oscillators. The paper notes that the new way of selecting a reduced number of points can be applied to lessen Monte-Carlo Euler-Maruyama method simulation costs for the SDEs.Analysis of limit set for trajectories of nonlinear system with random actions for almost all initial conditionshttps://zbmath.org/1472.341142021-11-25T18:46:10.358925Z"Vasylieva, I. G."https://zbmath.org/authors/?q=ai:vasylieva.i-g"Zuyev, A. L."https://zbmath.org/authors/?q=ai:zuev.a-l|zuyev.alexanderSummary: We consider a class of nonlinear differential equations with random actions that admit invariant manifolds of an arbitrary dimension. We study the problem of stability for such manifolds for almost all initial values of the phase space. Sufficient conditions for the attraction to the invariant set in terms of the density function of a measure that has the property of monotonicity on the phase flow are proved. As an illustration, we consider an example of a nonlinear system for which the density function is constructed explicitly.Asymptotic solution of the Cauchy problem for a first-order differential equation with a small parameter in a Banach spacehttps://zbmath.org/1472.341152021-11-25T18:46:10.358925Z"Uskov, V. I."https://zbmath.org/authors/?q=ai:uskov.vladimir-igorevich|uskov.v-i.1Summary: The Cauchy problem for a first-order differential equation with a small parameter multiplying the derivative in a Banach space is considered. The right-hand side of the equation contains the Fredholm operator perturbed by an additional operator term containing a small parameter. The asymptotic expansion of the solution in powers of the small parameter is constructed by the Vasil'yeva-Vishik-Lyusternik method. To calculate the components of the regular part of the expansion, the cascade decomposition method is used, which consists in the step-by-step splitting of the equation into equations in subspaces of decreasing dimensions. The conditions under which the boundary layer phenomenon occurs in the problem are determined.Pseudo \(S\)-asymptotically Bloch type periodicity with applications to some evolution equationshttps://zbmath.org/1472.341162021-11-25T18:46:10.358925Z"Chang, Yong-Kui"https://zbmath.org/authors/?q=ai:chang.yong-kui"Wei, Yanyan"https://zbmath.org/authors/?q=ai:wei.yanyanSummary: This paper is mainly focused upon the pseudo \(S\)-asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo \(S\)-asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo \(S\)-asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo \(S\)-asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo \(S\)-asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.Semilinear fractional-order evolution equations of Sobolev type in the sectorial casehttps://zbmath.org/1472.341172021-11-25T18:46:10.358925Z"Fedorov, Vladimir E."https://zbmath.org/authors/?q=ai:fedorov.v-e"Avilovich, Anna S."https://zbmath.org/authors/?q=ai:avilovich.anna-sergeevnaIn this paper, the authors prove the local unique solvability of the Cauchy-type problem to a semilinear fractional differential equation in a Banach space with Riemann-Liouville derivative. A linear unbounded operator at the unknown function in the equation generates in a sector resolving analytic family of operators of the linear homogeneous fractional-order equation. This result is applied to study the initial-boundary value problems for a class of nonlinear partial differential equations and in particular a nonlinear superdiffusion equation. Further, it is used to investigate the local unique solvability of the Showalter-Sidorov type problem to a semilinear Sobolev-type equation in a Banach space with a sectorial pair of operators and with Riemann-Liouville derivatives. For this the authors use two types of condition on the nonlinear operator: (i) the condition on the image of this operator (ii) the condition of its independence of the elements of the degeneration subspace. Examples are provided to illustrate the abstract results.Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operatorshttps://zbmath.org/1472.341182021-11-25T18:46:10.358925Z"Obukhovskii, Valeri"https://zbmath.org/authors/?q=ai:obukhovskii.valeri"Zecca, Pietro"https://zbmath.org/authors/?q=ai:zecca.pietro"Afanasova, Maria"https://zbmath.org/authors/?q=ai:afanasova.mariaA general nonlocal boundary value problem for a fractional-order semilinear differential inclusion in a separable Banach space \(E\) of a fractional order \(0<q<1,\)
\[
\begin{cases}
^{C}D^{q}x(t)\in Ax(t)+F(t,x(t)), ~~ t\in [0,T],\\
\mathcal{Q}(x)\in \mathcal{S}(x)
\end{cases}
\]
is considered, where \(A: D(A)\subset E\to E\) is a Hille-Yosida operator generating a locally Lipschitz integrated semigroup, \(F: [0,T]\times E\to Kv(E),\) is a nonlinear multivalued map, \(\mathcal{Q}: C([0,T]; \overline{D(A)})\to \overline{D(A)}\) is a bounded linear operator and \(\mathcal{S}: C([0,T]; \overline{D(A)})\to K(\overline{D(A)})\) is a completely upper semicontinuous quasi-\(R_{\delta}\)-multioperator. Here \(Kv(E), K(E)\) denote the collections of all nonempty compact convex and, respectively, compact subsets of \(E.\) By using the theory of integrated semigroups, fractional calculus and the fixed point theory of condensing multivalued maps, the existence of mild solutions is proved. Some important particular cases including a nonlocal Cauchy problem, periodic and anti-periodic boundary value problems are presented.Partial approximate controllability of fractional systems with Riemann-Liouville derivatives and nonlocal conditionshttps://zbmath.org/1472.341192021-11-25T18:46:10.358925Z"Haq, Abdul"https://zbmath.org/authors/?q=ai:haq.abdul"Sukavanam, N."https://zbmath.org/authors/?q=ai:sukavanam.nagarajanSummary: In this work, we investigate the partial approximate controllability of nonlocal Riemann-Liouville fractional systems with integral initial conditions in Hilbert spaces without assuming the Lipschitz continuity of nonlinear function. We also exclude the conditions of Lipschitz continuity and compactness for the nonlocal function. The existence results are derived using Schauder fixed point theorem, then the partial approximate controllability result is proved by assuming that the associated linear system is partial approximately controllable for \(\varphi =0\), where \(\varphi\) is nonlocal function. Lastly, an example is provided to apply our results.Twin semigroups and delay equationshttps://zbmath.org/1472.341202021-11-25T18:46:10.358925Z"Diekmann, O."https://zbmath.org/authors/?q=ai:diekmann.odo"Verduyn Lunel, S. M."https://zbmath.org/authors/?q=ai:verduyn-lunel.sjoerd-mAuthors' abstract: In the standard theory of delay equations, the fundamental solution does not `live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with ``retarded'' equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.Application of algebraic systems in the theory of functional differential equationshttps://zbmath.org/1472.341212021-11-25T18:46:10.358925Z"Rodionov, V."https://zbmath.org/authors/?q=ai:rodionov.vasiliy-n|rodionov.v-a|rodionov.vitalii-ivanovich|rodionov.v-b|rodionov.v-vSummary: The paper provides an overview of the author's results in the study of linear functional differential equations of first order \(\dot x(t)-(\mathcal{F}x)(t)=b(t)\) generated by five families of linear operators \(x\to\mathcal{F}x\). In each of five problems (for each \(\mathcal{F})\) by immersion of equations from algebra with traditional (pointwise) multiplication into an algebraic system with special multiplication explicit epresentations for the solutions of the equations (in the Cauchy form) are obtained.On sufficient conditions of exponential stability for systems of linear autonomous differential equations with aftereffecthttps://zbmath.org/1472.341222021-11-25T18:46:10.358925Z"Sabatulina, T."https://zbmath.org/authors/?q=ai:sabatulina.tatyana-leonidovnaSummary: We consider systems of linear autonomous functional differential equation with aftereffect and propose an approach to obtain effective sufficient conditions of exponential stability for these systems. In the approach we use the positiveness of the fundamental matrix of an auxiliary system (a comparison system) with concentrated and
distributed delays.Some properties of functional-differential operators with involution \(\nu(x)=1-x\) and their applicationshttps://zbmath.org/1472.341232021-11-25T18:46:10.358925Z"Burlutskaya, M. Sh."https://zbmath.org/authors/?q=ai:burlutskaya.m-shSummary: Functional-differential operators with involution \(\nu(x)=1-x\), related to integral operators whose kernels can have points of discontinuity on the lines \(t=x\) and \(t=1-x\) and to Dirac and Sturm-Liouville operators, are used in the study of these operators and various applications. This paper provides a survey on the spectral properties of such operators with involution and their applications in problems on geometric graphs, in the study of Dirac systems, and in the justification of the Fourier method in mixed problems for partial differential equations.Boundary value problems associated with singular strongly nonlinear equations with functional termshttps://zbmath.org/1472.341242021-11-25T18:46:10.358925Z"Biagi, Stefano"https://zbmath.org/authors/?q=ai:biagi.stefano"Calamai, Alessandro"https://zbmath.org/authors/?q=ai:calamai.alessandro"Marcelli, Cristina"https://zbmath.org/authors/?q=ai:marcelli.cristina"Papalini, Francesca"https://zbmath.org/authors/?q=ai:papalini.francescaThe authors study the existence of a solution to the non-local boundary value problem for the functional differential equation
\begin{gather*}
(\Phi(k(t)x'(t)))'+f(t,G(x)(t))\rho(t,x'(t))=0, \\
x(a)=H_a(x),\ \ x(b)=H_b(x)
\end{gather*}
with the \(\Phi\)-Laplacian operator and the Carathéodory functions \(f\), \(\rho\). Under the assumption of the existence of a well-ordered pair of lower and upper functions, a quite general existence result is proved by means of fixed point arguments.Existence of positive solutions of second-order delayed differential system on infinite intervalhttps://zbmath.org/1472.341252021-11-25T18:46:10.358925Z"Ding, Ran"https://zbmath.org/authors/?q=ai:ding.ran"Wang, Fanglei"https://zbmath.org/authors/?q=ai:wang.fanglei"Yang, Nannan"https://zbmath.org/authors/?q=ai:yang.nannan"Ru, Yuanfang"https://zbmath.org/authors/?q=ai:ru.yuanfangSummary: The present paper is focused on the analysis on the existence of positive solutions of a second-order differential system with two delays
\[
\begin{cases}x_1^{\prime\prime}(t)-a_1(t)x_1(t)+m_1(t)f_1(t,x(t),x_\tau(t))=0,t > 0, \\ x_2^{\prime\prime}(t)-a_2(t)x_2(t)+m_2(t)f_2(t,x(t),x_\tau(t))=0,t > 0,\\ x_1(t)=0,-\tau_1\leq t\leq 0,\text{ and }\lim_{t\to\infty}x_1(t)=0, \\ x_2(t)=0,-\tau_2\leq t\leq 0,\text{ and }\lim_{t\to\infty}x_2(t)=0\end{cases}.
\] by using two well-known fixed point theorems.Conditional optimal controls in enclosing solutions to boundary value problems with an uncertaintyhttps://zbmath.org/1472.341262021-11-25T18:46:10.358925Z"Maksimov, V."https://zbmath.org/authors/?q=ai:maksimov.vladimir-p|maksimov.valeri-m|maksimov.v-s|maksimov.vyacheslav-i|maksimov.v-n|maksimov.vyacheslav-i.1|maksimov.v-a.1|maksimov.valerii-vladimirovich|maksimov.v-fSummary: We consider the general linear boundary value problem for a functional differential system with an uncertainty in the right-hand side (the free term). The problem of two-sided point-wise estimating solutions of the boundary value problem is considered. The consideration is based on the point of view accepted in the Control Theory while studying the so called attainability sets (or reachable sets). The information about the free term is constrained by a polyhedron of its admissible values. The problem under consideration is reduced to the generalized moment problem which opens a way to estimations of the solutions with the use of a sequence of linear programming problems. Some illustrative examples are presented.Oscillations for nonlinear neutral delay differential equations with variable coefficientshttps://zbmath.org/1472.341272021-11-25T18:46:10.358925Z"Ahmed, Fatima N."https://zbmath.org/authors/?q=ai:ahmed.fatima-n"Ahmad, Rokiah R."https://zbmath.org/authors/?q=ai:ahmad.rokiah-rozita"Din, Ummul K. S."https://zbmath.org/authors/?q=ai:din.ummul-khair-salma"Noorani, Mohd S. M."https://zbmath.org/authors/?q=ai:noorani.mohd-salmi-mohdSummary: A class of nonlinear neutral delay differential equations is considered. Some new oscillation criteria of all solutions are derived. The obtained results generalize and extend some of well known previous results in the literature.Third-order nonlinear differential equations with nonlinear neutral termshttps://zbmath.org/1472.341282021-11-25T18:46:10.358925Z"Chatzarakis, G."https://zbmath.org/authors/?q=ai:chatzarakis.george-e"Grace, S."https://zbmath.org/authors/?q=ai:grace.said-rIn this paper, the third order nonlinear neutral-delay differential equation
\[
(a(t){(y''(t))}^{\alpha})' + q(t)x^{\gamma}(\tau(t)) = 0
\]
with \(y(t)=x(t)\pm p(t) x^{\beta}(\sigma(t))\) considered. Two new criteria are established for all solutions to be either oscillatory or tend to zero. This work is novel and interesting since it initiates the study of the above oscillation problem via comparison with first order delay differential equations. The results improve, extend, and simplify existing ones in the literature. Two examples are given to demonstrate the significance of this work.Green's function for periodic solutions in alternately advanced and delayed differential systemshttps://zbmath.org/1472.341292021-11-25T18:46:10.358925Z"Chiu, Kuo-Shou"https://zbmath.org/authors/?q=ai:chiu.kuo-shouIn the paper, the differential system
\[
x'(t)=A(t)x(t)+f(t,x(t),x(\gamma(t)))+g(t,x(t),x(\gamma(t)))
\]
is considered with a piecewise continuous argument deviation \(\gamma\). The existence of an \(\omega\)-periodic solution is studied under the assumption that the corresponding linear system has only the trivial \(\omega\)-periodic solution. By using Krasnoselskii's fixed point theorem, the author proves the existence (resp. the existence and uniqueness) of an \(\omega\)-periodic solution to the given system. Applications and illustrative examples are discussed as well.New result of existence of periodic solutions for a generalized \(p\)-Laplacian Liénard type differential equation with a variable delayhttps://zbmath.org/1472.341302021-11-25T18:46:10.358925Z"Eswari, R."https://zbmath.org/authors/?q=ai:eswari.rajendiran"Piramanantham, V."https://zbmath.org/authors/?q=ai:piramanantham.veeraraghavanIn this paper, the authors study the following generalized \(p\)-Laplacian Liénard type differential equation with a variable delay:
\[
(\phi_p(u'(t)))' = f(t, u(t), u'(t))u'(t) + \beta(t)g(t, u(t), u(t-\tau(t))) +s(t),
\]
where \(p\) is a constant, \(\phi_p: \mathbb{R}\to\mathbb{R}\) is defined by \(\phi_p(x) = |x|^{p-2}x\) for \(x\not =0\) and \(\phi_p(0) = 0\), \(\tau, \beta, s\) are periodic continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) with period \(T>0\), and \(f, g\) are continuous and \(T\)-periodic in the first argument. There are many results available for periodic solutions of Duffing type, Rayleigh type and Liénard equations. Motivated by some of the results in literature, the authors consider this generalized equation under the assumptions that \(p\geq q >2\) with \(\frac{1}{p} + \frac{1}{q} = 1\) and \(\beta(t) > 0\) for \(t\in [0, T]\). By applying the Mawhin continuation theorem, a set of sufficient conditions is established for existence of at least one \(T\)-periodic solution. This improves and generalizes some existing results.The existence of periodic solutions for three-order neutral differential equationshttps://zbmath.org/1472.341312021-11-25T18:46:10.358925Z"Huang, Manna"https://zbmath.org/authors/?q=ai:huang.manna"Guo, Chengjun"https://zbmath.org/authors/?q=ai:guo.chengjun"Liu, Junming"https://zbmath.org/authors/?q=ai:liu.junmingIn this paper, the following third-order neutral type delay differential equation is considered:
\begin{eqnarray*}
p(t) &=& x'''(t) +c x'''(t-\tau) +a_2(t) x''(t) + a_1(t)x'(t) + a_0(t)x(t)\\
&& + \sum^n_{i=1}\beta_i(t)g_i(x(t-\tau_i(t))),
\end{eqnarray*}
where \(c, \tau\) are constants satisfying \(|c|<1\) and \(\tau >0\), \(a_0, a_1, a_2, \beta_i, \tau_i (i = 1, \ldots, n)\) and \(p\) are continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) with period \(T> 0\), and the \(g_i\) are continuous from \(\mathbb{R}\) to \(\mathbb{R}\). Under suitable assumptions, sufficient conditions are provided for existence of nontrivial \(T\)-periodic solutions.Existence of \(P\)-mean almost periodic mild solution for fractional stochastic neutral functional differential equationhttps://zbmath.org/1472.341322021-11-25T18:46:10.358925Z"Sun, Xiao-ke"https://zbmath.org/authors/?q=ai:sun.xiaoke"He, Ping"https://zbmath.org/authors/?q=ai:he.pingSummary: A class of fractional stochastic neutral functional differential equation is analyzed in this paper. With the utilization of the fractional calculations, semigroup theory, fixed point technique and stochastic analysis theory, a sufficient condition of the existence for \(p\)-mean almost periodic solution is obtained, which are supported by two examples.Bifurcations in a delay logistic equation under small perturbationshttps://zbmath.org/1472.341332021-11-25T18:46:10.358925Z"Kashchenko, S. A."https://zbmath.org/authors/?q=ai:kashchenko.sergey-aleksandrovichConsider the following equation, known as the delay logistic equation, \[ \frac{du}{dt}=\lambda \big[ 1-u(t-T)\big] u \quad (\lambda >0, \ T>0), \] which is encountered in mathematical ecology, physics, etc. In the first part of the paper, the local behavior of solutions to the above equation is analyzed by using bifurcation methods. In particular, the author studies the dependence of dynamic properties of solutions on small perturbations of the above equation that contain a large delay \(h>0\). The second part of the paper is devoted to the delay logistic equation with periodic perturbations of parameters. Using standard asymptotic methods the parametric resonance is deduced under a two-frequency perturbation.Estimation of the region of global stability of the equilibrium state of the logistic equation with delayhttps://zbmath.org/1472.341342021-11-25T18:46:10.358925Z"Kashchenko, S. A."https://zbmath.org/authors/?q=ai:kashchenko.sergey-aleksandrovich"Loginov, D. O."https://zbmath.org/authors/?q=ai:loginov.d-oThe aim of this work is to study the global stability of the zero equilibrium for the Wright logistic delay differential equation using a parameter. The authors provide several results on the global stability with respect to the parameter. Some satisfactory answers are provided for the Wright conjecture.Fixed-time synchronization of quaternion-valued neural networks with time-varying delayhttps://zbmath.org/1472.341352021-11-25T18:46:10.358925Z"Kumar, Umesh"https://zbmath.org/authors/?q=ai:kumar.umesh"Das, Subir"https://zbmath.org/authors/?q=ai:das.subir-k"Huang, Chuangxia"https://zbmath.org/authors/?q=ai:huang.chuangxia"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jindeSummary: In this article, sufficient conditions for fixed-time synchronization of time-delayed quaternion-valued neural networks (QVNNs) are derived. Firstly, QVNNs are decomposed into four real-valued systems. Then using the available lemmas and by constructing the Lyapunov function, the synchronization criterion for the neural networks is proposed. Activation functions satisfy the Lipschitz condition. A suitable controller has been designed to synchronize the master-slave systems. The effectiveness of the proposed result is validated through a comparison of the settling time obtained by applying two different existing lemmas to a particular problem of synchronization of two identical QVNNs with time-varying delay with the help of suitable controllers.On the asymptotic behavior of solutions to linear autonomous neutral functional differential equationshttps://zbmath.org/1472.341362021-11-25T18:46:10.358925Z"Malygina, V."https://zbmath.org/authors/?q=ai:malygina.v-v|malygina.vera"Chudinov, K."https://zbmath.org/authors/?q=ai:chudinov.kirill-mSummary: We investigate stability, with respect to initial data, of a linear autonomous functional differential equation of neutral type, on the basis of the well-known solution representation formula. We consider stability as a property depending on a functional space which initial functions belong to, and show that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic
stability. We obtain that for initial data from the Lebesgue space \(L_1\) the strong asymptotic stability of the equation under study is equivalent to an exponential estimate of the Cauchy function, which is the kernel of the integral operator in the solution representation formula. Moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from \(L_p\) for all \(p\) from 1 to infinity inclusive. However, strong asymptotic stability with respect to initial data from \(L_p\) for \(p\) greater than one may not coincide with exponential stability.Existence and uniqueness of solutions for fuzzy mixed type of delay differential equationshttps://zbmath.org/1472.341372021-11-25T18:46:10.358925Z"Prasantha Bharathi, D."https://zbmath.org/authors/?q=ai:bharathi.d-prasantha"Jayakumar, T."https://zbmath.org/authors/?q=ai:jayakumar.t"Muthukumar, T."https://zbmath.org/authors/?q=ai:muthukumar.thirumalai-nambi"Vinoth, S."https://zbmath.org/authors/?q=ai:vinoth.sSummary: In this paper we are defining the fuzzy type of mixed delay differential equations. The necessity of studying the mixed delay differential equation in terms of fuzzy is that the single real valued solution can be ordered as the set of fuzzy valued solution. So it is very important in establishing the existence of solution for the fuzzy mixed type of delay differential equations using necessary theorems and lemmas. In addition we are also proving that the existing solution is unique.Existence of global mild solutions for a class of fractional partial functional differential equationshttps://zbmath.org/1472.341382021-11-25T18:46:10.358925Z"Xi, Xuan-Xuan"https://zbmath.org/authors/?q=ai:xi.xuanxuan"Hou, Mimi"https://zbmath.org/authors/?q=ai:hou.mimi"Zhou, Xian-Feng"https://zbmath.org/authors/?q=ai:zhou.xianfengIn this paper, the authors establish sufficient conditions for the local and global existence of mild solutions of a class of fractional partial functional differential equations in Banach spaces. The results are obtained by using standard fixed point theorems. A specific nonlinear function is provided to verify the assumptions. It should be noted that the solution representation (22) for the abstract equation (19) is not true in all cases of the operator \(A\). As claimed in the abstract, explicit nonlocal conditions are not discussed.Stability and approximation of almost automorphic solutions on time scales for the stochastic Nicholson's blowflies modelhttps://zbmath.org/1472.341392021-11-25T18:46:10.358925Z"Dhama, Soniya"https://zbmath.org/authors/?q=ai:dhama.soniya"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed"Sakthivel, Rathinasmay"https://zbmath.org/authors/?q=ai:sakthivel.rathinasmaySummary: The stability and approximation of solutions of Nicholson's blowflies model for the stochastic case on time scales is discussed. Using various tools of analysis, sufficient conditions for the existence of a square mean almost automorphic solution are derived. The randomness and time scales make the model a hybrid model, which is more realistic and useful. The analysis works for both discrete and continuous cases, as well as for several other cases such as quantum and Cantor sets. We establish appropriate conditions and results to explore the Ulam-Hyers-Rassias stability. Furthermore, the model with piecewise constant argument is analyzed. Then the approximate solution and a nicer bound of this model using the discretization method is established. We conclude with an example to demonstrate our analytical results.Interval oscillation of damped second-order mixed nonlinear differential equation with variable delay under impulse effectshttps://zbmath.org/1472.341402021-11-25T18:46:10.358925Z"Muthulakshmi, V."https://zbmath.org/authors/?q=ai:muthulakshmi.velu"Manjuram, R."https://zbmath.org/authors/?q=ai:manjuram.rSummary: In this paper, we study the oscillatory behavior of damped second-order mixed nonlinear differential equation with variable delay under impulse effects. By using Riccati transformation technique, integral averaging method and some inequalities, we obtain sufficient conditions for oscillation of all solutions. Finally, two examples are presented to illustrate the theoretical results.On multi-step estimation of delay for SDEhttps://zbmath.org/1472.341412021-11-25T18:46:10.358925Z"Kutoyants, Yury A."https://zbmath.org/authors/?q=ai:kutoyants.yury-aSummary: We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLEs for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. the parameter, and therefore the numerical calculation of the MLEs encounter certain difficulties. We propose One-step and Two-step MLEs, whose calculation has no such problems and provide an estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First, we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and a modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.Dynamics analysis of a Filippov pest control model with time delayhttps://zbmath.org/1472.341422021-11-25T18:46:10.358925Z"Arafa, Ayman A."https://zbmath.org/authors/?q=ai:arafa.ayman-a"Hamdallah, Soliman A. A."https://zbmath.org/authors/?q=ai:hamdallah.soliman-a-a"Tang, Sanyi"https://zbmath.org/authors/?q=ai:tang.sanyi"Xu, Yong"https://zbmath.org/authors/?q=ai:xu.yong.3"Mahmoud, Gamal M."https://zbmath.org/authors/?q=ai:mahmoud.gamal-mSummary: The most critical factor for increasing crop production is the successful resistance of pests and pathogens which has massive impacts on global food security. Therefore, Filippov systems have been used to model and grasp control strategies for limited resources in Integrated Pest Management (IPM). Extensive studies have been done on these systems where the evolution is governed by a smooth set of ordinary differential equations (ODEs). As far as we know the time delay has not been considered in these systems, which we mean that a set of delay differential equations (DDEs). With this motivation, a Filippov prey-predator (pest-natural enemy) model with time delay is introduced in this paper, where the time delay represents the change of growth rate of the natural enemy before releasing it to feed on pests. The threshold conditions for the stability of the equilibria are derived by using time delay as a bifurcation parameter. It is shown that when the time delay parameter passes through some critical values, a periodic oscillation phenomenon appears through Hopf bifurcation. Further, by employing Filippov convex method we obtain the equation of sliding motion and address the sliding mode dynamics. Numerically, we demonstrate that the time delay plays a substantial role in discontinuity-induced bifurcation. More precisely, one can get boundary focus bifurcation from boundary node bifurcation through variation of the value of the time delay. Moreover, the time delay is used as a bifurcation parameter to obtain sliding-switching and sliding-grazing bifurcations. In conclusion, a Filippov system with time delay can give new insights into pest control models.Collision-free flocking for a time-delay systemhttps://zbmath.org/1472.341432021-11-25T18:46:10.358925Z"Chen, Maoli"https://zbmath.org/authors/?q=ai:chen.maoli"Wang, Xiao"https://zbmath.org/authors/?q=ai:wang.xiao"Liu, Yicheng"https://zbmath.org/authors/?q=ai:liu.yichengThe authors studies a Cucker-Smale type model with two additional mechanism. One is time delay effect, and the other one is the repulsion force when two particles are close. Therefore, the inertial effect is stronger and oscillations appear in the system, which draws difficulty in analysis of large time behavior. Moreover, the mean velocity is no longer conservative, thus the flocking is described via vanishing of relative velocity.
The authors first prove global existence of the solution by showing collision avoidance in finite time. Then, when delay is sufficiently small, the authors construct a proper Lyapunov functional to describe the decay structure, and show the emergence of asymptotical flocking based on Barbalat's lemma.Hopf bifurcation analysis of a density predator-prey model with Crowley-Martin functional response and two time delayshttps://zbmath.org/1472.341442021-11-25T18:46:10.358925Z"Liu, Chunxia"https://zbmath.org/authors/?q=ai:liu.chunxia"Li, Shumin"https://zbmath.org/authors/?q=ai:li.shumin"Yan, Yan"https://zbmath.org/authors/?q=ai:yan.yanSummary: In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.A model for pandemic control through isolation policyhttps://zbmath.org/1472.341452021-11-25T18:46:10.358925Z"Moyo, Sibusiso"https://zbmath.org/authors/?q=ai:moyo.sibusiso"Cruz, Luis Gustavo Zelaya"https://zbmath.org/authors/?q=ai:cruz.luis-gustavo-zelaya"de Carvalho, Rafael Lima"https://zbmath.org/authors/?q=ai:de-carvalho.rafael-lima"Faye, Roger Marcelin"https://zbmath.org/authors/?q=ai:faye.roger-marcelin"Tabakov, Pavel Yaroslav"https://zbmath.org/authors/?q=ai:tabakov.pavel-yaroslav"Mora-Camino, Felix"https://zbmath.org/authors/?q=ai:mora-camino.felixSummary: In this paper we model the dynamics of a spreading pandemic over a country using a new dynamical and decentralised differential model with the main objective of studying the effect of different policies of social isolation (social distancing) over the population to control the spread of the pandemic. A probabilistic infection process with time lags is introduced in the dynamics with the main contribution being the proposed model to explicitly look at levels of interaction between towns and regions within the considered country. We believe the strategies and findings here will help practitioners, planners and Governments to put in place better strategies to control the spread of pandemics, thus saving lives and minimizing the impact of pandemia on socio-economic development and the populations livelihood.Stability and bifurcation analysis of hepatitis B-type virus infection modelhttps://zbmath.org/1472.341462021-11-25T18:46:10.358925Z"Prakash, Mani"https://zbmath.org/authors/?q=ai:prakash.mani"Rakkiyappan, Rajan"https://zbmath.org/authors/?q=ai:rakkiyappan.rajan"Manivannan, Annamalai"https://zbmath.org/authors/?q=ai:manivannan.annamalai"Zhu, Haitao"https://zbmath.org/authors/?q=ai:zhu.haitao"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jindeSummary: The main aim of this study is to analyze the dynamical properties of hepatitis B-type virus (HBV) infection in terms of mathematical model. The presented mathematical model on HBV involves the various factors such as immune impairment, total carrying capacity, logistic growth term, and antiretroviral therapies. In addition, the effect of time delays is also considered into the model, which are inevitable during the activation of immune response and time taken to infect the healthy cells. Mathematically, the qualitative analyses such as stability, bifurcation, and stabilization analysis are performed to explore the dynamical characteristics of HBV over the period of time. The significance of the model parameters is revealed through Hopf-type bifurcation analysis and the global stability analysis of the proposed model. With the help of data set values that are extracted from the literature, the efficiency of the derived theoretical results is explored.Stability analysis of a fractional-order delay dynamical model on oncolytic virotherapyhttps://zbmath.org/1472.341472021-11-25T18:46:10.358925Z"Singh, Hitesh K."https://zbmath.org/authors/?q=ai:singh.hitesh-k"Pandey, Dwijendra N."https://zbmath.org/authors/?q=ai:pandey.dwijendra-narainSummary: In this manuscript, the main objective is to introduce the derivatives of fractional-order into a delayed dynamical model of oncolytic virotherapy. The system consists of populations of infected cells, uninfected cells, and virus particles. The local asymptotic stability of all the equilibrium points is discussed by analyzing the corresponding characteristic polynomials. The existence of Hopf bifurcation is shown due to the effect of delay. The fractional-order dynamical system is compared with the integer order counterpart. Numerical simulations are also carried out to verify how the fractional-order model is more stable than its integer-order counterpart and fractional-order parameter can be used to pacify the effect of delay.Complex dynamics in a quasi-periodic plasma perturbations modelhttps://zbmath.org/1472.341482021-11-25T18:46:10.358925Z"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin|zhang.xin.4|zhang.xin.1|zhang.xin.3"Yang, Shuangling"https://zbmath.org/authors/?q=ai:yang.shuanglingSummary: In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [\textit{D. Constantinescu} et al., ``A low-dimensional model system for quasi-periodic plasma perturbations'', Phys. Plasmas 18, No. 6, Article ID 062307, 7 p. (2011; \url{doi:10.1063/1.3600209})]. In particular, assuming that there exists a finite time lag \(\tau\) between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power \(h\) and the relaxation of the instability \(\delta\) do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient \(\eta\) just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.A completeness theorem for the system of eigenfunctions of the complex Schrödinger operator with potential \(q(x)=cx^\alpha \)https://zbmath.org/1472.341492021-11-25T18:46:10.358925Z"Tumanov, S. N."https://zbmath.org/authors/?q=ai:tumanov.s-n(no abstract)Direct and inverse scattering problems for a first-order system with energy-dependent potentialshttps://zbmath.org/1472.341502021-11-25T18:46:10.358925Z"Aktosun, T."https://zbmath.org/authors/?q=ai:aktosun.tuncay"Ercan, R."https://zbmath.org/authors/?q=ai:ercan.rIn the paper, the direct and inverse scattering problems on the full line are analyzed for a first-order system of ordinary linear differential equations associated with energy-dependent potentials. Using the two potentials as input, the direct problem is solved by determining the scattering coefficients and the bound-state information consisting of bound-state energies, their multiplicities, and the corresponding norming constants. Using two different methods, the corresponding inverse problem is solved by determining the two potentials when the scattering data set is used as input. The first method involves the transformation of the energydependent system into two distinct energy-independent systems. The second method involves the establishment of the so-called alternate Marchenko system of linear integral equations and the recovery of the energy-dependent potentials from the solution to the alternate Marchenko system.Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the linehttps://zbmath.org/1472.341512021-11-25T18:46:10.358925Z"Tyni, Teemu"https://zbmath.org/authors/?q=ai:tyni.teemu"Serov, Valery"https://zbmath.org/authors/?q=ai:serov.valery-sSummary: We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.An investigation of solving third-order nonlinear ordinary differential equation in complex domain by generalising Prelle-Singer methodhttps://zbmath.org/1472.341522021-11-25T18:46:10.358925Z"Joohy, Ali K."https://zbmath.org/authors/?q=ai:joohy.ali-k"Al-Juaifri, Ghassan A."https://zbmath.org/authors/?q=ai:al-juaifri.ghassan-a"Mechee, Mohammed S."https://zbmath.org/authors/?q=ai:mechee.mohammed-sSummary: A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) -- nonlinear ODEs in the complex plane -- by generalizing Prelle-Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.Meromorphic solutions of certain types of non-linear differential equationshttps://zbmath.org/1472.341532021-11-25T18:46:10.358925Z"Liu, Huifang"https://zbmath.org/authors/?q=ai:liu.huifang"Mao, Zhiqiang"https://zbmath.org/authors/?q=ai:mao.zhiqiangIn this paper under review, the authors study the following non-linear differential equation
\[
f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z},
\]
where \(n\geq 2\) is an integer, \(p_{1},\) \(p_{2}\) and \(\alpha _{1},\) \(\alpha _{2}\) are non-zero constants and \(P_{d}\left( z,f\right) \) is a differential polynomial in \(f\) of degree \(d\). The authors find the forms of meromorphic solutions with few poles of the above equation when \(d=n-1\) under some restrictions on \(\alpha _{1},\) \(\alpha _{2},\) The Theorems 1.1 and 1.2 obtained extend the result established by \textit{P. Li} [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; Zbl 1206.30046)] provided \(\alpha _{1}\neq \) \(\alpha _{2}\) and \(d\leq n-2\). Some examples are given to illustrate the results.On Petrenko's deviations and second order differential equationshttps://zbmath.org/1472.341542021-11-25T18:46:10.358925Z"Heittokangas, Janne"https://zbmath.org/authors/?q=ai:heittokangas.janne-m"Zemirni, Mohamed Amine"https://zbmath.org/authors/?q=ai:zemirni.mohamed-amineThe authors ``consider the oscillation of solutions of
\[
f''+A(z)f=0\tag{1.1}
\]
and the growth of solutions of
\[
f''+A(z)f'+B(z)f=0\tag{1.2}
\]
were \(A\) and \(B\) are entire functions.'' For the first equation, an improved (see [\textit{I. Laine} and \textit{P. Wu}, Rev. Roum. Math. Pures Appl. 44, No. 4, 609--615 (1999; Zbl 1004.34079)]) estimate of the value of \(\Lambda (E)\) is obtained, where \(E\) is a product of two linearly independent solutions of the equation (1.1) and \(\Lambda (E)\) denotes the exponent of convergence of zeros of \(E\). For the equation (1.2), conditions are considered under which all its nontrivial solutions have infinite order. A new result is obtained in this direction and, as a consequence, the following statement
Corollary. Let \(A\) and \(B\) be entire functions. Suppose there exists a sector where \(\log^+|A(z)|=O(\log(|z|))\), and suppose that \(B\) is transcedental with Fabry gaps. Then every non-trivial solution of (1.2) is of infinite order.Order and hyper-order of solutions of second order linear differential equationshttps://zbmath.org/1472.341552021-11-25T18:46:10.358925Z"Kumar, Sanjay"https://zbmath.org/authors/?q=ai:kumar.sanjay-v|kumar.sanjay.2|kumar.sanjay.1"Saini, Manisha"https://zbmath.org/authors/?q=ai:saini.manishaIn this paper, the authors investigate the growth of solutions of second order linear differentilal equations. They determine new conditions on the coefficients in order that every non-trivial solution has an infinite order by introducing the notions of function extremal to Yang's inequality and extremal to Denjoy's conjecture. They also study the hyper-order of these solutions. Moreover, they extend these results to higher order linear differential equations. So this work is interesting.Infinite growth of solutions of second order complex differential equations with entire coefficient having dynamical propertyhttps://zbmath.org/1472.341562021-11-25T18:46:10.358925Z"Zhang, Guowei"https://zbmath.org/authors/?q=ai:zhang.guowei"Yang, Lianzhong"https://zbmath.org/authors/?q=ai:yang.lianzhongThis paper is concerned with the classical problem on the growth of solutions of second order linear complex differential equations \(f''+A(z)f'+B(z)f=0\), where \(A(z)\) and \(B(z)\) are two entire functions as coefficients of this equation. The authors give some properties for these two coefficients to obtain the solutions of the equation having infinite growth order. This topic is a research hotspot and many literatures focused on this research areas. The authors introduce some new methods and technology to get the infinity order of the solutions.
In the main results of this paper the authors suppose that the coefficient \(B(z)\) has a dynamical property, that is, has a multiple Fatou component. This property ensures that the maximum and minimum modules of \(B(z)\) satisfy an important inequality as \(|z|=r\) is in an infinitely logarithmic measure set. The coefficient \(A(z)\) satisfies the extreme of Yang's inequality, which says that for an entire function with nonzero finite lower order the number of its Borel direction is the double of the number of its finite deficient values, or \(A(z)\) is a nontrivial solution of another differential equation \(w''+P(z)w=0\), where \(P(z)\) is a polynomial. The author also discusses some other properties to get the infinity growth of the solutions.
In my opinion, this paper is of great significance for future research.Application of resurgent analysis to the construction of asymptotics of linear differential equations with degeneration In coefficientshttps://zbmath.org/1472.341572021-11-25T18:46:10.358925Z"Korovina, M. V."https://zbmath.org/authors/?q=ai:korovina.mariya-victorovna|korovina.margarita-vladimirovnaSummary: This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations.On the Borel summability of WKB solutions of certain Schrödinger-type differential equationshttps://zbmath.org/1472.341582021-11-25T18:46:10.358925Z"Nemes, Gergő"https://zbmath.org/authors/?q=ai:nemes.gergoA linear equation of the Schrödinger type
\[
\frac{d^2W}{d\xi^2}=(u^2+u\phi(\xi)+\psi(\xi))W \tag{1}
\]
admits formal solutions of the form
\[
W^{\pm}(u,\xi)=\exp\left(\pm u\xi \pm \frac 12 \int^{\xi} \phi(t)dt\big) \big(1+\sum_{n=1}^{\infty}\frac{A^\pm_n(\xi)}{u^n} \right),
\]
called \textit{WKB solutions}, where \(u\) is a large parameter, \(\phi(\xi)\) and \(\psi(\xi)\) are holomorphic in a domain \(G\) \((\ni \xi)\), and \(A^{\pm}_{n}(\xi)\) are recursively given by
\[
A^{\pm}_{n+1}(\xi)=-\frac 12\phi(\xi) A^{\pm}_n(\xi)\mp \frac 12 \frac{dA^\pm_n(\xi)} {d\xi} \mp \frac 12 \int^{\xi} \Big(\frac 14 \phi(t)^2 \mp \frac 12 \phi'(t) -\psi(t) \Big)A^{\pm}_n(t)dt
\]
with \(A^{\pm}_0(\xi)=1.\) This paper shows that, under certain conditions on \(\phi(\xi)\) and \(\psi(\xi)\), the expansions in the WKB solutions are Borel summable in well-defined subdomains \(\Gamma^{\pm}\) of \(G\), that is,
\[
\exp\bigg(\pm u\xi \pm \frac 12 \int^{\xi}_{\alpha^{\pm}} \phi(t) dt\bigg) (1+\eta^{\pm}(u,\xi)) \,\,\,{\text with}\,\,\, \eta^{\pm}(u,\xi)=\int^{+\infty}_0e^{-ut} F^{\pm}(t,\xi)dt
\]
solve equation (1). Here \(F^{\pm}(t,\xi)=\sum_{n=0}^{\infty}A^{\pm}_{n+1}(\xi)(n!)^{-1}t^n\) are the convergent series called the Borel transformation; \(A^{\pm}_{n+1}(\xi)\) are determined by choosing the integration constants suitably; and for each \(\sigma>0\), \(\eta^{\pm}(u,\xi)\) admit uniform asymptotic expansions \(\sum_{n=1}^{\infty}A^{\pm}_n(\xi) u^{-n}\) as \(|u|\to +\infty,\) \(\text{Re} u>\sigma.\) Furthermore the error bounds of these asymptotic expansions are evaluated, and \(\eta^{\pm}(u,\xi)\) are also represented by factorial series expansions in \(u\). Finally to a radial Schrödinger equation and the Bessel equation, applications of the results are illustrated.The sigma form of the second Painlevé hierarchyhttps://zbmath.org/1472.341592021-11-25T18:46:10.358925Z"Bobrova, Irina"https://zbmath.org/authors/?q=ai:bobrova.irina"Mazzocco, Marta"https://zbmath.org/authors/?q=ai:mazzocco.martaThe paper discusses the Painlevé II (\(w''(z)=2w(z)^{3}+zw(z)+\alpha_{1}\)) hierarchy consisting of an infinite sequence of nonlinear differential equations of form
\[P_{II}^{(n)}:: \left(\frac{d}{dz}+2w\right)L_{n}\left[w'-w^{2}\right]+\mathop{\sum_{l=1}^{n-1}}t_{l}\left(\frac{d}{dz}+2w\right)L_{1}\left[w'-w^{2}\right]=zw(z)+\alpha_{n},n\geq1\]
where \(t_1 , \dots, t_{n-1}\) and \(\alpha_n\) are parameters, \(L_n\) is the operator defined by the recursion relation \[\frac{d}{dz}L_{n+1}=\left(\frac{d^{3}}{dz^{3}}+4(w'-w^{2})\frac{d}{dz}+2(w'-w^{2})'\right) L_{n}, \quad L_{0}\left[w'-w^{2}\right]=\frac{1}{2},\] with boundary condition \[L_{n}[0]=0,\quad\forall n\geq1.\] As the Hamiltonian form \(H^{(n)}\) of the second Painlevé hierarchy was produced in [\textit{M. Mazzocco} and \textit{M. Y. Mo}, Nonlinearity 20, No. 12, 2845--2882 (2007; Zbl 1156.34079)] the authors were able (and this is the main result) to calculate the sigma function \(\sigma_n (z )\) for this hierarchy \(\sigma_n (z ) := H^{(n)} ( P_1 ( z),\dots, P_n ( z), Q_1 ( z), \dots, Q_n (z )) \).Two variations on \((A_3 \times A_1 \times A_1)^{(1)}\) type discrete Painlevé equationshttps://zbmath.org/1472.341602021-11-25T18:46:10.358925Z"Shi, Yang"https://zbmath.org/authors/?q=ai:shi.yangSummary: By considering the normalizers of reflection subgroups of types \(A^{(1)}_1\) and \(A^{(1)}_3\) in \(\widetilde{W}(D_5^{( 1 )})\), two subgroups: \( \widetilde{W} ( A_3 \times A_1 )^{( 1 )} \ltimes W(A_1^{( 1 )})\) and \(\widetilde{W} ( A_1 \times A_1 )^{( 1 )} \ltimes W(A_3^{( 1 )})\) can be constructed from a \((A_3 \times A_1 \times A_1)^{(1)}\) type subroot system. These two symmetries arose in the studies of discrete Painlevé equations
[\textit{K. Kajiwara} et al., Lett. Math. Phys. 62, No. 3, 259--268 (2002; Zbl 1030.37045);
\textit{T. Takenawa}, Funkc. Ekvacioj, Ser. Int. 46, No. 1, 173--186 (2003; Zbl 1151.34341);
\textit{N. Okubo} and \textit{T. Suzuki}, ``Generalized \(q\)-Painlevé VI systems of type \((A_2n+1 + A_1 + A_1)^{(1)}\) arising from cluster algebra'', Preprint, \url{arXiv:1810.03252}],
where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups [\textit{R. B. Howlett}, J. Lond. Math. Soc., II. Ser. 21, 62--80 (1980; Zbl 0427.20040);
\textit{B. Brink} and \textit{R. B. Howlett}, Invent. Math. 136, No. 2, 323--351 (1999; Zbl 0926.20024)].
This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.On the number of eigenvalues for parameter-dependent diffusion problem on time scaleshttps://zbmath.org/1472.341612021-11-25T18:46:10.358925Z"Gulsen, Tuba"https://zbmath.org/authors/?q=ai:gulsen.tuba"Jadlovská, Irena"https://zbmath.org/authors/?q=ai:jadlovska.irena"Yilmaz, Emrah"https://zbmath.org/authors/?q=ai:yilmaz.emrah-sercanSummary: In this study, we consider parameter-dependent diffusion eigenvalue problem on time scales. An upper bound on the number of eigenvalues for this problem on a finite time scale is given.On recovering the Sturm-Liouville differential operators on time scaleshttps://zbmath.org/1472.341622021-11-25T18:46:10.358925Z"Kuznetsova, M. A."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-andreevnaSummary: We study Sturm-Liouville differential operators on the time scales consisting of finitely many isolated points and closed intervals. In the author's previous paper, it was established that such operators are uniquely determined by the spectral characteristics of all classical types. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm-Liouville boundary-value problems on time scales with one common boundary condition alternate.On inverse spectral problems for Sturm-Liouville differential operators on closed setshttps://zbmath.org/1472.341632021-11-25T18:46:10.358925Z"Kuznetsova, M. A."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-andreevna"Buterin, S. A."https://zbmath.org/authors/?q=ai:buterin.sergey-alexandrovich"Yurko, V. A."https://zbmath.org/authors/?q=ai:yurko.vjacheslav-anatoljevichSummary: We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred to as time scales and often appear in modelling various real-world processes. Depending on the set structure, such operators unify both differential and difference operators. The time scales under consideration consist of a finite number of non-intersecting segments. We obtain properties of the spectral characteristics and prove uniqueness theorems for inverse problems of recovering the operator from two types of spectral data: the Weyl function, as well as the spectra of two boundary value problems for one and the same Sturm-Liouville equation on the time scale with one common boundary condition.Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scaleshttps://zbmath.org/1472.341642021-11-25T18:46:10.358925Z"Li, Zhien"https://zbmath.org/authors/?q=ai:li.zhien"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also a corresponding existence and uniqueness theorem is proved. In addition, the commutativity of \(n \times n\) quaternion-matrix-valued functions is investigated and some sufficient and necessary conditions of commutativity and noncommutativity are established on time scales. Also the fundamental solution matrices of some basic quaternion matrix dynamic equations are obtained. Examples are provided to illustrate the results, which are completely new on hybrid domains particularly when the time scales are the quantum case \(\mathbb{T}=\overline{q^{\mathbb{Z}}}\) and the discrete case \(\mathbb{T}=h\mathbb{Z}; h > 0\), both of which are significant for the study of quaternion q-dynamic equations and quaternion difference dynamic equations. Finally, we present several applications including multidimensional rotations and transformations of the submarine, the gyroscope, and the planet whose dynamical behaviors are depicted by quaternion dynamics on time scales and the corresponding iteration numerical solution for homogeneous quaternion dynamic equations are provided on various time scales.The dispersionless Veselov-Novikov equation: symmetries, exact solutions, and conservation lawshttps://zbmath.org/1472.350202021-11-25T18:46:10.358925Z"Morozov, Oleg I."https://zbmath.org/authors/?q=ai:morozov.oleg-i"Chang, Jen-Hsu"https://zbmath.org/authors/?q=ai:chang.jen-hsuSummary: We study symmetries, invariant solutions, and conservation laws for the dispersionless Veselov-Novikov equation. The emphasis is placed on cases when the odes involved in description of the invariant solutions are integrable by quadratures. Then we find some non-invariant solutions, in particular, solutions that are polynomials of an arbitrary degree \(N \ge 3\) with respect to the spatial variables. Finally we compute all conservation laws that are associated to cosymmetries of second order.Blow-up phenomenon for a reaction-diffusion equation with weighted nonlocal gradient absorption termshttps://zbmath.org/1472.350652021-11-25T18:46:10.358925Z"Liang, Mengyang"https://zbmath.org/authors/?q=ai:liang.mengyang"Fang, Zhong Bo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: This paper deals with the blow-up phenomenon of solutions to a reaction-diffusion equation with weighted nonlocal gradient absorption terms in a bounded domain. Based on the method of auxiliary function and the technique of modified differential inequality, we establish appropriate conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, upper and lower bounds for blow-up time are derived under appropriate measure in higher dimensional spaces.Correction to: ``Traveling wave solutions for degenerate nonlinear parabolic equations''https://zbmath.org/1472.350862021-11-25T18:46:10.358925Z"Ichida, Yu"https://zbmath.org/authors/?q=ai:ichida.yu"Sakamoto, Takashi Okuda"https://zbmath.org/authors/?q=ai:sakamoto.takashi-okudaCorrection to the authors' paper [ibid. 6, No. 2, 795--832 (2020; Zbl 1451.35045)].Oscillations of the string with singuliaritieshttps://zbmath.org/1472.352372021-11-25T18:46:10.358925Z"Kamenskii, Mikhail"https://zbmath.org/authors/?q=ai:kamenskii.mikhail-igorevich"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfeng"Zvereva, Margarita"https://zbmath.org/authors/?q=ai:zvereva.margarita-borisovnaSummary: In the present paper we consider the initial boundary value problem describing discontinuous Stieltjes string oscillations. The possibility of Fourier series expansion of the solution is investigated. The analysis is based on a refined Stieltjes integral.Dynamics of traveling waves for the perturbed generalized KdV equationhttps://zbmath.org/1472.352962021-11-25T18:46:10.358925Z"Ge, Jianjiang"https://zbmath.org/authors/?q=ai:ge.jianjiang"Wu, Ranchao"https://zbmath.org/authors/?q=ai:wu.ranchao"Du, Zengji"https://zbmath.org/authors/?q=ai:du.zengjiSummary: This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.A heat flow for the mean field equation on a finite graphhttps://zbmath.org/1472.354192021-11-25T18:46:10.358925Z"Lin, Yong"https://zbmath.org/authors/?q=ai:lin.yong"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyanSummary: Inspired by works of \textit{J.-B. Castéras} [Pac. J. Math. 276, No. 2, 321--345 (2015; Zbl 1331.53097)], \textit{J. Li} and \textit{C. Zhu} [Calc. Var. Partial Differ. Equ. 58, No. 2, Paper No. 60, 18 p. (2019; Zbl 1415.58014)], \textit{L. Sun} and \textit{J. Zhu} [Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 42, 26 p. (2021; Zbl 1458.35437)], we propose a heat flow for the mean field equation on a connected finite graph \(G=(V,E)\). Namely
\[
\begin{cases}
\partial_t\phi (u)=\Delta u-Q+\rho \frac{e^u}{\int_V e^u d\mu}\\
u(\cdot ,0)=u_0,
\end{cases}
\]
where \(\Delta\) is the standard graph Laplacian, \(\rho\) is a real number, \(Q:V\rightarrow\mathbb{R}\) is a function satisfying \(\int_VQd\mu =\rho\), and \(\phi :\mathbb{R}\rightarrow\mathbb{R}\) is one of certain smooth functions including \(\phi (s)=e^s\). We prove that for any initial data \(u_0\) and any \(\rho \in\mathbb{R} \), there exists a unique solution \(u:V\times [0,+\infty)\rightarrow\mathbb{R}\) of the above heat flow; moreover, \(u(x, t)\) converges to some function \(u_\infty :V\rightarrow\mathbb{R}\) uniformly in \(x\in V\) as \(t\rightarrow +\infty \), and \(u_\infty\) is a solution of the mean field equation
\[
\Delta u_{\infty}-Q+\rho \frac{e^{u_\infty}}{\int_V e^{u_\infty} d\mu}=0.
\]
Though \(G\) is a finite graph, this result is still unexpected, even in the special case \(Q\equiv 0\). Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow.Mild solutions are weak solutions in a class of (non)linear measure-valued evolution equations on a bounded domainhttps://zbmath.org/1472.354212021-11-25T18:46:10.358925Z"Evers, Joep H. M."https://zbmath.org/authors/?q=ai:evers.joep-h-mSummary: We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain \([0,1]\). Mass moves, driven by a velocity field that is either a function of the spatial variable only, \(v=v(x0\), or depends on the solution \(\mu\) itself: \(v=v[\mu](x)\). The flow is stopped at the boundaries of \([0,1]\), while mass is gated away by a certain right-hand side. In previous works \textit{J. H. M. Evers} et al. [J. Differ. Equations 259, No. 3, 1068--1097 (2015; Zbl 1315.35057); SIAM J. Math. Anal. 48, No. 3, 1929--1953 (2016; Zbl 1342.28004)], we showed the existence and uniqueness of appropriately defined mild solutions for \(v=v(x)\) and \(v=v[\mu](x0\), respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when \(v=v(x)\) and when \(v=v[\mu](x)\).Local fractional Moisil-Teodorescu operator in quaternionic setting involving Cantor-type coordinate systemshttps://zbmath.org/1472.354282021-11-25T18:46:10.358925Z"Bory-Reyes, Juan"https://zbmath.org/authors/?q=ai:bory-reyes.juan"Pérez-de la Rosa, Marco Antonio"https://zbmath.org/authors/?q=ai:perez-de-la-rosa.marco-antonioSummary: The Moisil-Teodorescu operator is considered to be a good analogue of the usual Cauchy-Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in \(\mathbb{R}^3\). In the present work, a general quaternionic structure is developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil-Teodorescu operator.Fractional-order model for cooling of a semi-infinite body by radiationhttps://zbmath.org/1472.354312021-11-25T18:46:10.358925Z"Esmaeili, Shahrokh"https://zbmath.org/authors/?q=ai:esmaeili.shahrokhSummary: In this paper, the fractional-order model for cooling of a semi-infinite body by radiation is considered.
In the supposed semi-infinite body, the equation of heat along with an initial condition and an asymptotic boundary condition form an equivalent equation in which the order of derivatives is halved.
This equation and a boundary condition introduced by the radiation heat transfer give rise to an initial value problem, whose differential equation is nonlinear and fractional order.
The semi-analytical solution to this nonlinear model was determined asymptotically at small and large times.
Moreover, two numerical methods including Grünwald-Letnikov approximation and Müntz-Legendre approximation yield numerical solutions to the problem.Application of the fractional Sturm-Liouville theory to a fractional Sturm-Liouville telegraph equationhttps://zbmath.org/1472.354332021-11-25T18:46:10.358925Z"Ferreira, M."https://zbmath.org/authors/?q=ai:ferreira.michel|ferreira.max|ferreira.miguel-h|ferreira.marta|ferreira.m-n|ferreira.m-d-c|ferreira.mario-f-s|ferreira.maria-c-f|ferreira.maria-joao.2|ferreira.m-p|ferreira.maria-teodora|ferreira.mauricio-a|ferreira.miguel-jorge-bernabe|ferreira.maria-joao.1|ferreira.marco-s|ferreira.marina-a|ferreira.marcelo-rodrigo-portela|ferreira.marisa|ferreira.mauro-s|ferreira.marco-a-r|ferreira.marizete-a-c|ferreira.marcio-j-r|ferreira.mardson|ferreira.marcos-r-s|ferreira.manoel-m-jun|ferreira.mariana|ferreira.m-margarida-a|ferreira.m-luisa|ferreira.marcio-v|ferreira.milton|ferreira.maria-margarida|ferreira.marcelo-c"Rodrigues, M. M."https://zbmath.org/authors/?q=ai:rodrigues.maria-manuela|rodrigues.maikol-m"Vieira, N."https://zbmath.org/authors/?q=ai:vieira.newton-j|vieira.nelsonSummary: In this paper, we consider a non-homogeneous time-space-fractional telegraph equation in \(n\)-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm-Liouville operator defined in terms of right and left fractional Riemann-Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.Conformal boundary operators, \(T\)-curvatures, and conformal fractional Laplacians of odd orderhttps://zbmath.org/1472.354352021-11-25T18:46:10.358925Z"Gover, A. Rod"https://zbmath.org/authors/?q=ai:gover.ashwin-rod"Peterson, Lawrence J."https://zbmath.org/authors/?q=ai:peterson.lawrence-jSummary: We construct continuously parametrised families of conformally invariant boundary operators on densities. These generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with conformal Laplacian power operators, the boundary operators yield conformally invariant fractional Laplacian pseudodifferential operators on the boundary of a conformal manifold with boundary. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson \(Q\)-curvature and include higher-order generalisations of the mean curvature and the \(T\)-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We discuss applications of our results and techniques in the setting of Poincaré-Einstein manifolds and also use our constructions to shed light on some conjectures of Juhl.Existence of three nontrivial solutions of asymptotically linear second order operator equationshttps://zbmath.org/1472.354472021-11-25T18:46:10.358925Z"Chen, Yingying"https://zbmath.org/authors/?q=ai:chen.yingyingSummary: In this article, we prove the existence of three nontrivial solutions for some second order operator equations, especially the asymptotically linear ones. The main methods are the Leray-Schauder degree theory and mountain pass theorem.On the existence, uniqueness, and new analytic approximate solution of the modified error function in two-phase Stefan problemshttps://zbmath.org/1472.354632021-11-25T18:46:10.358925Z"Bougoffa, Lazhar"https://zbmath.org/authors/?q=ai:bougoffa.lazhar"Rach, Randolph C."https://zbmath.org/authors/?q=ai:rach.randolph-c"Mennouni, Abdelaziz"https://zbmath.org/authors/?q=ai:mennouni.abdelazizSummary: This paper provides a new proof of the existence and uniqueness of the solution for a nonlinear boundary value problem
\[
\begin{cases}
[(1+\delta y) y^\prime]^\prime &+ \, 2x(1+ \gamma y)y^\prime = 0, 0 < x < \infty, \\
&y(0) = 0, y(\infty) = 1,
\end{cases}
\] which describes the study of two-phase Stefan problems on the semi-infinite line \([0, \infty)\). This result considerably extends the analysis of a recent work. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.Geodesic Loewner paths with varying boundary conditionshttps://zbmath.org/1472.354672021-11-25T18:46:10.358925Z"Mcdonald, Robb"https://zbmath.org/authors/?q=ai:mcdonald.n-robbSummary: Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped `mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of \(\pi /5\).Some remarks on global analytic planar vector fields possessing an invariant analytic sethttps://zbmath.org/1472.370222021-11-25T18:46:10.358925Z"García, Isaac A."https://zbmath.org/authors/?q=ai:garcia.isaac-aSummary: We study the problem of determining the canonical form that a planar analytic vector field in all the real plane can have to possess a given invariant analytic set. We determine some conditions that guarantee the only solution to this inverse problem is the trivial one.A new fractional-order chaotic complex system and its antisynchronizationhttps://zbmath.org/1472.370452021-11-25T18:46:10.358925Z"Jiang, Cuimei"https://zbmath.org/authors/?q=ai:jiang.cuimei"Liu, Shutang"https://zbmath.org/authors/?q=ai:liu.shutang"Luo, Chao"https://zbmath.org/authors/?q=ai:luo.chaoSummary: We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.A novel four-wing hyperchaotic complex system and its complex modified hybrid projective synchronization with different dimensionshttps://zbmath.org/1472.370462021-11-25T18:46:10.358925Z"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.6|liu.jian.5|liu.jian"Liu, Shutang"https://zbmath.org/authors/?q=ai:liu.shutang"Zhang, Fangfang"https://zbmath.org/authors/?q=ai:zhang.fangfangSummary: We introduce a new Dadras system with complex variables which can exhibit both four-wing hyperchaotic and chaotic attractors. Some dynamic properties of the system have been described including Lyapunov exponents, fractal dimensions, and Poincaré maps. More importantly, we focus on a new type of synchronization method of modified hybrid project synchronization with complex transformation matrix (CMHPS) for different dimensional hyperchaotic and chaotic complex systems with complex parameters, where the drive and response systems can be asymptotically synchronized up to a desired complex transformation matrix, not a diagonal matrix. Furthermore, CMHPS between the novel hyperchaotic Dadras complex system and other two different dimensional complex chaotic systems is provided as an example to discuss increased order synchronization and reduced order synchronization, respectively. Numerical results verify the feasibility and effectiveness of the presented schemes.Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuitieshttps://zbmath.org/1472.370622021-11-25T18:46:10.358925Z"Zhu, Mengkun"https://zbmath.org/authors/?q=ai:zhu.mengkun"Wang, Dan"https://zbmath.org/authors/?q=ai:wang.dan"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1|chen.yang.2Summary: In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles
\[
w(z,t)=A\theta (z-t)e^{-z^2+tz},
\]
here \(\theta(x)\) is the Heaviside function, where \(A> 0\), \(t>0\), and \(z\in [0,\infty)\). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities \(R_n(t)\) and \(r_n(t)\) satisfy the coupled Riccati equations, from which we also prove that \(R_n(t)\) satisfies a particular Painlevé IV equation. Even more, \(\sigma_n(t)\), allied to \(R_n(t)\), satisfies both the discrete and continuous Jimbo-Miwa-Okamoto \(\sigma\)-form of the Painlevé IV equation. Finally, we consider the situation when \(n\) gets large, the second order linear differential equation satisfied by the polynomials \(P_n(x)\) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.Complete classification of rational solutions of \(A_{2n}\)-Painlevé systemshttps://zbmath.org/1472.370672021-11-25T18:46:10.358925Z"Gómez-Ullate, David"https://zbmath.org/authors/?q=ai:gomez-ullate.david"Grandati, Yves"https://zbmath.org/authors/?q=ai:grandati.yves"Milson, Robert"https://zbmath.org/authors/?q=ai:milson.robertThe authors provide a complete classification of the rational solutions of the Painlevé IV equation and its higher-order hierarchy, known as the $A_{2n}$-Painlevé or Noumi-Yamada system. First, they recall the equivalence between the Noumi-Yamada system and a cyclic dressing chain of Schrödinger operators. Then, they show by a careful investigation of the local expansions of the rational solutions around their poles, that the solutions have trivial monodromy, and therefore they must be expressible in terms of Wronskian determinants whose entries are Hermite polynomials.
Next, they use Maya diagrams to classify all the \( (2n + 1)\)-cyclic dressing chains and therefore achieve a complete classification. Finally, they connect their results with the geometric approach mastered by the Japanese school, showing a representation for the action of the symmetry group of Bäcklund transformations in terms of Maya cycles and oddly colored integer sequences. The natural extension of this work is to tackle the full classification of the rational solutions to the $A_{2n+1}$-Painlevé systems, which include the Painlevé V and its higher order extensions.
This is a very significative work in the study of the solutions of Painlevé IV equation.A fractional mathematical model for COVID-19 transmission with Atangana-Baleanu derivativehttps://zbmath.org/1472.370892021-11-25T18:46:10.358925Z"Mohammadi, Hakimeh"https://zbmath.org/authors/?q=ai:mohammadi.hakimehSummary: In this paper, we study the COVID-19 transmission model with the Atangana-Baleanu fractional derivative in Caputo sense. The approximate solution of the model is presented by two-step Adams-Bashforth scheme. The existence and uniqueness of the solution are discussed by employing of fixed point theory. Finally, numerical simulation of COVID-19 transmission in Iran is presented.Analytic invariant curves for an iterative equation related to Ricker-type second-order equationhttps://zbmath.org/1472.390052021-11-25T18:46:10.358925Z"Zhao, Hou Yu"https://zbmath.org/authors/?q=ai:zhao.houyu"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michalThe authors show the existence of analytic invariant curves of the difference equation
\[
x_{n+1}=x_{n-1}e^{a-x_{n-1}-x_{n}},
\]
or equivalently of the mapping \(T(x,y)=(y,xe^{a-x-y})\). They seek for analytic invariant curves of \(T\) in the form \(y=f(x)\). The considered system can be written in the form of the iterative equation
\[
f(f(x))=xe^{a-x-f(x)},
\]
where \(x\in\mathbb{C}\) and \(a\in\mathbb{R}\) is a fixed number. This equation is reduced, with \(f(x)=g(\alpha g^{-1}(x) )\), to the auxiliary equation
\[
g(\alpha^{2}x)=g(x)e^{a-g(x)-g(\alpha x)}.
\]
Thus, by proving the existence of analytic solutions for this last equation, the analytic invariant curves of original equation can be determined.
Set \(\alpha = \pm e^{a/2}\). The authors distinguish three different cases for \(\alpha\):
\begin{itemize}
\item[(1)] \(0< |\alpha| < 1\);
\item[(2)] \(\alpha = e^{2\pi i\theta}, \theta\in \mathbb{R}\setminus\mathbb{Q}\) and \(\theta\) defines a Brjuno number, i.e.,
\[
\sum_{n=0}^{\infty}\frac{\log q_{n+1}}{q_{n}}<\infty,
\]
where \(\{p_{n}/q_{n}\}\) denotes the sequence of partial fractions of the continued fraction expansion of \(\theta\);
\item[(3)] \(\alpha = e^{2\pi iq/p}\) for some integer \(p\in \mathbb{N}\) with \(p\geq 2\) and \(q\in \mathbb{Z}\setminus\{ 0 \}\) and \(\alpha \neq e^{2\pi i\xi/\upsilon}\) for all \(1\leq \upsilon\leq p-1\) and \(\xi\in \mathbb{Z}\setminus\{ 0 \}\).
\end{itemize}New estimations for discrete Sturm-Liouville problemshttps://zbmath.org/1472.390072021-11-25T18:46:10.358925Z"Bas, Erdal"https://zbmath.org/authors/?q=ai:bas.erdal"Ozarslan, Ramazan"https://zbmath.org/authors/?q=ai:ozarslan.ramazanThe authors consider the second-order Sturm-Liouville difference equation
\[
\Delta^2x(n-1)+q(n)x(n)=-\lambda x(n), \quad n\in[a,b]_{\mathbb{Z}}, \tag{1}
\]
where \([a,b]_{\mathbb{Z}}:=[a,b]\cap\mathbb{Z}\) is a discrete interval, with the initial conditions
\[
x(a-1)=-h, \quad x(a)=1.
\]
Equation (1) is uniquely solvable by recursion, given the initial values at \(x(a-1)\) and \(x(a)\). Note that the authors use \(x(0)\) and \(x(1)\) at this place without any warning. As a main result the authors present a formula for the solution of (1), which is obtained in the following way. Consider first the homogeneous part of equation (1), i.e., \[ \Delta^2x(n-1)+\lambda x(n)=0, \quad n\in[a,b]_{\mathbb{Z}}, \] whose solution \(x_h(n)=c_1x_1(n)+c_2x_2(n)\) can be obtained explicitly in terms of \(\lambda\) and \(n\). Then they consider the method of variation of constants to incorporate the nonhomogeneous term in the right-hand side of \[ \Delta^2x(n-1)+\lambda x(n)=-q(n)x(n), \quad n\in[a,b]_{\mathbb{Z}}. \] In this way they obtain a formula for \(x(n)\), which of course contains \(x(n)\) on both sides (!) (Equation (13) in Theorem 6). A similar construction (see Equation (33) in Theorem 7) is derived for equation (1) with the initial conditions
\[
x(a-1)=1, \quad x(a)=0.
\]
In addition, certain asymptotic formulas of the form \(x(n)=O(e^{|t|n})\) for \(n\in\mathbb{Z}^+\) are given in Theorems 8 and 9. Here the role of the parameter \(t\) is not specified. The authors also make a comparison of the values of \(x(n)\) with some sample values of \(n\) for the potentials \(q(n)=1/(n+1)\) and \(q(n)=1/\sqrt{n+1}\) and the spectral parameters \(\lambda=1\) and \(\lambda=2\).
Reviewer's remark: The present reviewer finds difficult to understand the usefulness of the above approach. Furthermore, the paper contains inaccuracies in the formulations of the theorems and their proofs.Oscillation criteria for higher-order neutral type difference equationshttps://zbmath.org/1472.390182021-11-25T18:46:10.358925Z"Köprübaşi, Turhan"https://zbmath.org/authors/?q=ai:koprubasi.turhan"Ünal, Zafer"https://zbmath.org/authors/?q=ai:unal.zafer"Bolat, Yaşar"https://zbmath.org/authors/?q=ai:bolat.yasarThe authors consider higher-order neutral-type difference equations of the form \(\Delta(r_n\Delta^{k-1}(y_n+p_ny_{\tau_n}))+q_nf(y_{\sigma_n})=0\), where \(\tau_n\ge n\) and \(\sigma_n\le n\). They establish conditions that guarantee that either the equation is oscillatory for even \(k\) or every nonoscillatory solution tends to zero for odd \(k\).On properties of meromorphic solutions of certain difference Painlevé III equationshttps://zbmath.org/1472.390312021-11-25T18:46:10.358925Z"Lan, Shuang-Ting"https://zbmath.org/authors/?q=ai:lan.shuangting"Chen, Zong-Xuan"https://zbmath.org/authors/?q=ai:chen.zongxuanSummary: We mainly study the exponents of convergence of zeros and poles of difference and divided difference of transcendental meromorphic solutions for certain difference Painlevé III equations.A \(\beta\)-Sturm-Liouville problem associated with the general quantum operatorhttps://zbmath.org/1472.390362021-11-25T18:46:10.358925Z"Cardoso, J. L."https://zbmath.org/authors/?q=ai:cardoso.joao-lopes|cardoso.jose-luis|cardoso-cortes.jose-luisSummary: Let \(I \subseteq \mathbb{R}\) be an interval and \(\beta : I \to I\) a strictly increasing and continuous function with a unique fixed point \(s_0 \in I\) that satisfies \((s_0 - t)(\beta(t)-t)\geq 0\) for all \(t \in I\), where the equality holds only when \(t = s_0\). The general quantum operator defined by \textit{A. E. Hamza} et al. [Adv. Difference Equ. 2015, Paper No. 182, 19 p. (2015; Zbl 1422.39010)], \(D_{\beta}[f](t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t}\) if \(t \neq s_0\) and \(D_{\beta}[f](s_0):=f'(s_0)\) if \(t=s_0\) generalizes the Jackson \(q\)-operator \(D_q\) and also the Hahn (quantum derivative) operator, \(D_{q,\omega}\). Regarding a \(\beta\)-Sturm-Liouville eigenvalue problem associated with the above operator \(D_{\beta}\), we construct the \(\beta\)-Lagrange's identity, show that it is self-adjoint in \(\mathscr{L}_{\beta}^2([a,b])\) and exhibit some properties for the corresponding eigenvalues and eigenfunctions.Linear operators associated with differential and difference systems: what is different?https://zbmath.org/1472.390402021-11-25T18:46:10.358925Z"Zemánek, Petr"https://zbmath.org/authors/?q=ai:zemanek.petrSummary: The existence of a densely defined operator associated with (time-reversed) discrete symplectic systems is discussed and the necessity of the development of the spectral theory for these systems by using linear relations instead of operators is shown. An explanation of this phenomenon is provided by using the time scale calculus. In addition, the density of the domain of the maximal linear relation associated with the system is also investigated.
For the entire collection see [Zbl 1467.39001].Erratum to: ``A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of \(-y'' + qy = \lambda y\), with boundary conditions of general form''https://zbmath.org/1472.410012021-11-25T18:46:10.358925Z"Hormozi, Mahdi"https://zbmath.org/authors/?q=ai:hormozi.mahdiErratum to the author's paper [ibid. 2012, Paper No. 40, 7 p. (2012; Zbl 1278.41002)].An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernelhttps://zbmath.org/1472.450072021-11-25T18:46:10.358925Z"Bondarenko, Natalia Pavlovna"https://zbmath.org/authors/?q=ai:bondarenko.natalia-pThe author obtains uniqueness results for an inverse problem associated to a Dirac system of linear integro-differential equations with convolution kernel. For this purpose, the system of the direct equations is given and the partial inverse problem is formulated. The uniqueness theorem is proved in terms of the completeness of some system of vector functions associated with the given subspectrum. Moreover, a constructive algorithm for the solution of the specific partial inverse problem is provided. Necessary and sufficient conditions for the unique solvability of the inverse problem in this special case are obtained.Impulsive-integral inequalities for attracting and quasi-invariant sets of neutral stochastic integrodifferential equations with impulsive effectshttps://zbmath.org/1472.450082021-11-25T18:46:10.358925Z"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Anguraj, A."https://zbmath.org/authors/?q=ai:anguraj.annamalaiSummary: In this article, we investigate a class of neutral stochastic integrodifferential equations with impulsive effects. The results are obtained by using the new integral inequalities, the attracting and quasi-invariant sets combined with theories of resolvent operators. Moreover, exponential stability of the mild solution is established with sufficient conditions. An example is provided to illustrate the results of this work.Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functionshttps://zbmath.org/1472.460452021-11-25T18:46:10.358925Z"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.On the characteristic operator of an integral equation with a Nevanlinna measure in the infinite-dimensional casehttps://zbmath.org/1472.470032021-11-25T18:46:10.358925Z"Bruk, V. M."https://zbmath.org/authors/?q=ai:bruk.vladislav-moiseevichSummary: We define the families of maximal and minimal relations generated by an integral equation with a Nevanlinna operator measure in the infinite-dimensional case and prove their holomorphic property. We show that if the restrictions of maximal relations are continuously invertible, then the operators inverse to these restrictions are integral. By using these results, we prove the existence of the characteristic operator and describe the families of linear relations generating the characteristic operator.Spectral enclosures for non-self-adjoint extensions of symmetric operatorshttps://zbmath.org/1472.470182021-11-25T18:46:10.358925Z"Behrndt, Jussi"https://zbmath.org/authors/?q=ai:behrndt.jussi"Langer, Matthias"https://zbmath.org/authors/?q=ai:langer.matthias"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Rohleder, Jonathan"https://zbmath.org/authors/?q=ai:rohleder.jonathanIn the description of many quantum mechanical systems, operators appear as a consequence of heuristic arguments which suggest in a first step a formal expression for the Hamiltonian or Schrödinger operator describing the model. These operators \(S\) are typically unbounded and symmetric on a domain \(\operatorname{dom}{S}\) which is a dense subspace of a Hilbert space \(\mathcal{H}\). In a second crucial step for the description of the quantum mechanical system, one has to choose a closed (in many cases self-adjoint) extension of \(S\) in order to start the analysis of the model. Typically, fixing an extension means to specify the relevant boundary conditions for the system. The paper under review focuses on the description of closed non-selfadjoint extensions of \(S\) which appear as restrictions of the adjoint operator \(S^*\) and on the analysis of some of their spectral properties. The article also presents in its final part several applications of their results to elliptic operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with \(\delta\)-interactions, and to quantum graphs with non-self-adjoint vertex couplings.
In the first part of the article, the authors use an abstract and systematic approach to the description of extensions in terms of so-called boundary triples \((\Gamma_0,\Gamma_1,\mathcal{G})\), where \(\Gamma_{0,1}: \operatorname{dom}S^*\to \mathcal{G}\) satisfy a Green identity on the auxiliary Hilbert space \(\mathcal{G}\). Boundary triples (or their generalizations called quasi boundary triples) provide a useful technique to describe extensions encoding abstractly the boundary data of the problem. To formulate their main results in this part (e.g., Theorem 3.1), the authors use the Weyl function, which is an operator-valued function on the auxiliary Hilbert space defined in terms of the boundary triples. In addition, they introduce also a boundary operator \(B\) (in general non-symmetric) which serves to label the different extensions.
The article has an informative and well-written introduction to the topic describing their methods and results, but also introducing the reader to the literature and alternative approaches in this very active field. The bibliography list contains more than 130 references.Convoluted fractional \(C\)-semigroups and fractional abstract Cauchy problemshttps://zbmath.org/1472.470322021-11-25T18:46:10.358925Z"Mei, Zhan-Dong"https://zbmath.org/authors/?q=ai:mei.zhandong"Peng, Ji-Gen"https://zbmath.org/authors/?q=ai:peng.jigen"Gao, Jing-Huai"https://zbmath.org/authors/?q=ai:gao.jinghuaiSummary: We present the notion of convoluted fractional \(C\)-semigroup, which is the generalization of convoluted \(C\)-semigroup in the Banach space setting. We present two equivalent functional equations associated with convoluted fractional \(C\)-semigroup. Moreover, the well-posedness of the corresponding fractional abstract Cauchy problems is studied.Spectrums of solvable pantograph differential-operators for first orderhttps://zbmath.org/1472.470342021-11-25T18:46:10.358925Z"Ismailov, Z. I."https://zbmath.org/authors/?q=ai:ismailov.zameddin-ismailovich"Ipek, P."https://zbmath.org/authors/?q=ai:ipek.pembeSummary: By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.On the existence of solutions to one-dimensional fourth-order equationshttps://zbmath.org/1472.490042021-11-25T18:46:10.358925Z"Shokooh, S."https://zbmath.org/authors/?q=ai:shokooh.saeid"Afrouzi, G. A."https://zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Hadjian, A."https://zbmath.org/authors/?q=ai:hadjian.arminIn this paper, the authors consider the following fourth-order boundary-value problem: \begin{align*} & u^{(4)}h(x,u')-u''=[\lambda f(x,u)+g(u)]h(x,u')\ in\ ]0, 1[\\
& u(0)=u(1)=0=u''(0)=u''(1), \tag{1}\end{align*} where \(\lambda\) is a positive parameter, \(f:[0, 1]\times\mathbb{R}\longrightarrow\mathbb{R}\) is an \(L^{1}-\)Caratheodory function, \(g:\mathbb{R}\longrightarrow\mathbb{R}\) is a Lipschitz continuous function and \(h:[0, 1]\times\mathbb{R}\longrightarrow [0,\infty[\) is a bounded and continuous function. Using variational methods and Ricceri's variational principle, they prove under suitable conditions on \(\lambda,f,g,h\) that problem (1) admits at least one nontrivial weak solution.Optimization-constrained differential equations with active set changeshttps://zbmath.org/1472.490432021-11-25T18:46:10.358925Z"Stechlinski, Peter"https://zbmath.org/authors/?q=ai:stechlinski.peter-gThis paper deals with parameterized DAEs featuring algebraic inequalities (not only exact constraints), for which solutions should realize an optimization objective. Upon topological and metric conditions, the authors build the theory of such problems: solution existence, uniqueness and dependency to the parameters. The theory gives regularities conditions for such a system to have a generalized index one (Theorem 3.1) and adapt them to the parameterized optimization problem (Theorem 4.1). The content is quite notational though comprehensive, and well organized. It is claimed to be computationally relevant, in that the use of lexigographic directional derivative helps to solve the optimization condition, locally. Really useful for experts of the field, this appealing results need additional examples from real control systems (as mentioned in the introduction) to help right understanding.Optimal control of ODEs with state supremahttps://zbmath.org/1472.490482021-11-25T18:46:10.358925Z"Geiger, Tobias"https://zbmath.org/authors/?q=ai:geiger.tobias"Wachsmuth, Daniel"https://zbmath.org/authors/?q=ai:wachsmuth.daniel"Wachsmuth, Gerd"https://zbmath.org/authors/?q=ai:wachsmuth.gerdSummary: We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.Approximation of solutions to the optimal control problems for systems with maximumhttps://zbmath.org/1472.490572021-11-25T18:46:10.358925Z"Dashkovskiy, S."https://zbmath.org/authors/?q=ai:dashkovskiy.sergey-n"Kichmarenko, O."https://zbmath.org/authors/?q=ai:kichmarenko.olga-d"Sapozhnikova, K."https://zbmath.org/authors/?q=ai:sapozhnikova.kateryna-yuThis paper describes a method for the approximate solution of optimal control problems, where the value of the input depends on the maximal values attained by the control parameter and the phase vector:
\[
\dot{x}(t) = f(t,x(t),x_m(t)) + A(x)\xi(t,u(t),u_m(t)),
\]
where \(u\) is the control parameter, and \(x_m\) and \(u_m\) represent the supremum values attained by the functions \(x\) and \(u\) in an interval \([g(t),\gamma(t)]\).
The solution proposed is to use an averaging method, namely to solve the problem
\[
\dot{y}(t) = \bar{f}(y(t),y_m(t)) + A(y(t))v(t),
\]
where \(v\) is the new control parameter and the function \(\bar{f}\) is defined as
\[
\frac{1}{T}\int_0^T f(t,x)dt.
\]
The authors propose an algorithm to find a correspondence between the averaged control \(v\) and the control \(u\), based on discretising the interval \([0,T]\) into subintervals of predefined length \(T_0\) and show that under suitable assumptions of continuity and the existence of relevant limits, there is a correspondence between the controls \(u\) and \(v\). Furthermore, the objective values of the averaged system and the original system can be made arbitrarily close.The smallest eigenvalue distribution of the Jacobi unitary ensembleshttps://zbmath.org/1472.600112021-11-25T18:46:10.358925Z"Lyu, Shulin"https://zbmath.org/authors/?q=ai:lyu.shulin"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight \(x^{\alpha}(1 - x)^{\beta}, x \in [0, 1], \alpha, \beta > -1\), the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval \([t, 1]\) is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval \((- a, a), a > 0\) is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight \((1 - x^2)^{\beta}, x \in [- 1, 1]\).Oscillatory Breuer-Major theorem with application to the random corrector problemhttps://zbmath.org/1472.600652021-11-25T18:46:10.358925Z"Nualart, David"https://zbmath.org/authors/?q=ai:nualart.david"Zheng, Guangqu"https://zbmath.org/authors/?q=ai:zheng.guangquSummary: In this paper, we present an oscillatory version of the celebrated Breuer-Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures.Stability of stochastic dynamic equations with time-varying delay on time scaleshttps://zbmath.org/1472.600942021-11-25T18:46:10.358925Z"Du, Nguyen Huu"https://zbmath.org/authors/?q=ai:nguyen-huu-du."Tuan, Le Anh"https://zbmath.org/authors/?q=ai:tuan.le-anh"Dieu, Nguyen Thanh"https://zbmath.org/authors/?q=ai:dieu.nguyen-thanhSummary: The aim of this article is to consider the existence, uniqueness and uniformly exponential \(p\)-stability of the solution for \(\nabla\)-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral typehttps://zbmath.org/1472.601012021-11-25T18:46:10.358925Z"Hussain, R. Jahir"https://zbmath.org/authors/?q=ai:hussain.r-jahir"Hussain, S. Satham"https://zbmath.org/authors/?q=ai:hussain.s-sathamThe well-posedness and continuous dependence are presented for the mild solution to a class of stochastic neutral integral-differential equations with infinite delay driven by Poisson jumps on a separable Hilbert space \(\mathbb H\):
\[
\begin{aligned}
d P(t,x_t)= &\,\int_0^t \frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)} A P(s, x_s) d s d t\\
&+ f(t,x_t)d t+ \sigma(t,x_t) d W(t)\\
&+ \int_{\mathbb Z} h(t,x_t,y) \tilde N(d t, dy),\\
&\ x_0\in \mathcal B:=C((-\infty,0];\mathbb H), t\in [0,T],
\end{aligned}
\]
where \(T>0\) is a fixed constant, \((A,\mathcal{A})\) is a densely defined linear operator on \(\mathbb H\) of sectorial type, \(W(t)\) is a Wiener process on \(\mathbb H\) with finite trace nuclear covariance, \(\tilde N\) is a compensated Poisson martingale measure over a reference space \(\mathbb Z\), \(x_t\in \mathcal B\) with \(x_t(\theta):= x(t+\theta)\) for \(\theta\in (-\infty,0]\) is the segment of \(x(\cdot)\) up to time \(t\), and \(P, f, \sigma, h\) are proper defined functionals. The main results are illustrated with specific examples.Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memoryhttps://zbmath.org/1472.601382021-11-25T18:46:10.358925Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.10|li.jun.13|li.jun.1|li.jun.3|li.jun.14|li.jun.11|li.jun.12|li.jun.7|li.jun.8|li.jun|li.jun.6|li.jun.2"Xi, Fubao"https://zbmath.org/authors/?q=ai:xi.fubaoSummary: We study a class of diffusion processes, which are determined by solutions \(X(t)\) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain \(\Lambda (t)\). Under suitable conditions, we investigate convergence and boundedness of both the solutions \(X(t)\) and the functional solutions \(X_t\). We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process \((X_t, \Lambda (t))\), and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.The role of differential equations in applied statisticshttps://zbmath.org/1472.621262021-11-25T18:46:10.358925Z"Kitsos, Christos P."https://zbmath.org/authors/?q=ai:kitsos.christos-p"Nisiotis, C. S. A."https://zbmath.org/authors/?q=ai:nisiotis.c-s-aSummary: The target of this paper is to discuss, investigate and present how the differential equations are applied in Statistics. The stochastic orientation of Statistics creates problems to adopt the differential equations as an individual tool, but Applied Statistics is using the differential equations either through applications from other fields, like Chemistry or as a tool to explain ``variation'' in stochastic processes.
For the entire collection see [Zbl 1471.34005].Hereditary Riccati equation with fractional derivative of variable orderhttps://zbmath.org/1472.650052021-11-25T18:46:10.358925Z"Tvyordyj, D. A."https://zbmath.org/authors/?q=ai:tvyordyj.d-aSummary: The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called \textit{Numerical Solution of a Fractional-Differential Riccati Equation} (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.A method for the discretization of linear systems of ordinary fractional differential equations with constant coefficientshttps://zbmath.org/1472.650792021-11-25T18:46:10.358925Z"Aliev, F. A."https://zbmath.org/authors/?q=ai:aliev.fikrat-ahmadali|aliev.fikret-akhmedalioglu"Aliev, N. A."https://zbmath.org/authors/?q=ai:aliev.nihan-a"Velieva, N. I."https://zbmath.org/authors/?q=ai:velieva.naila-i"Gasimova, K. G."https://zbmath.org/authors/?q=ai:gasimova.k-gSummary: We develop an exact discretization method for the solution of linear systems of ordinary fractional differential equations with constant matrix coefficients. It is shown that, in this case, the obtained linear discrete system does not have constant matrix coefficients. The proposed method is compared with the well-known approximate method. The presented scheme is developed for any linear systems with piecewise constant perturbations. The obtained results are used for the discretization of linear controlled systems and are illustrated by numerical examples.A stable minimal search method for solving multi-order fractional differential equations based on reproducing kernel spacehttps://zbmath.org/1472.650802021-11-25T18:46:10.358925Z"Wu, Longbin"https://zbmath.org/authors/?q=ai:wu.longbin"Chen, Zhong"https://zbmath.org/authors/?q=ai:chen.zhong"Ding, Xiaohua"https://zbmath.org/authors/?q=ai:ding.xiaohuaSummary: In this paper, a stable minimal search method based on reproducing kernel space is proposed for solving multi-order fractional differential equations. The existence and uniqueness of solution of the considered equation is proved and the smoothness of the solution is studied. Based on orthonormal bases, we give smooth transformation and a method for obtaining the \(\epsilon \)-approximate solution by searching the minimum value. Subsequently, error estimation and stability analysis of the method are obtained. The final several numerical experiments are presented to illustrate the correctness of the theory and the effectiveness of the method.Accelerated exponentially fitted operator method for singularly perturbed problems with integral boundary conditionhttps://zbmath.org/1472.650912021-11-25T18:46:10.358925Z"Debela, Habtamu Garoma"https://zbmath.org/authors/?q=ai:debela.habtamu-garoma"Duressa, Gemechis File"https://zbmath.org/authors/?q=ai:duressa.gemechis-fileSummary: In this paper, we consider a class of singularly perturbed differential equations of convection diffusion type with integral boundary condition. An accelerated uniformly convergent numerical method is constructed via exponentially fitted operator method using Richardson extrapolation techniques and numerical integration methods to solve the problem. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be \(\varepsilon \)-uniformly convergent.A novel Petrov-Galerkin method for a class of linear systems of fractional differential equationshttps://zbmath.org/1472.650932021-11-25T18:46:10.358925Z"Faghih, A."https://zbmath.org/authors/?q=ai:faghih.a-k"Mokhtary, P."https://zbmath.org/authors/?q=ai:mokhtary.payamSummary: This paper presents a novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. New fractional-order generalized Jacobi functions are introduced, and their approximation properties are investigated. We show that these functions satisfy the given supplementary conditions and have the same asymptotic behavior as the exact solution, which are essential properties to design a high-order spectral Petrov-Galerkin method. For implementing our scheme, we represent the approximate solution by a linear combination of fractional-order generalized Jacobi functions and minimize the residual using shifted fractional Jacobi functions. The numerical solvability of the resultant algebraic system is justified as well as convergence and stability properties of the proposed scheme are explored. Finally, the reliability of the method is evaluated using various analytical and realistic problems.An \textit{hp}-version Legendre spectral collocation method for multi-order fractional differential equationshttps://zbmath.org/1472.650942021-11-25T18:46:10.358925Z"Guo, Yuling"https://zbmath.org/authors/?q=ai:guo.yuling"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqingSummary: In this paper, we consider the multi-order fractional differential equation and recast it into an integral equation. Based on the integral equation, we develop an \textit{hp}-version Legendre spectral collocation method and the integral terms with the weakly singular kernels are calculated precisely according to the properties of Legendre and Jacobi polynomials. The \textit{hp}-version error bounds under the \(L^2\)-norm and the \(L^{\infty}\)-norm are derived rigorously. Numerical experiments are included to illustrate the efficiency of the proposed method and the rationality of the theoretical results.Erratum to: ``A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body''https://zbmath.org/1472.650972021-11-25T18:46:10.358925Z"Müller, Andreas"https://zbmath.org/authors/?q=ai:muller.andreas.1Some typesetting errors in the author's paper [ibid. 56, No. 3, 995--1015 (2016; Zbl 1398.65198)] are corrected.Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite systemhttps://zbmath.org/1472.651162021-11-25T18:46:10.358925Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Khoa, Vo Anh"https://zbmath.org/authors/?q=ai:khoa.vo-anh"Truong, Mai Thanh Nhat"https://zbmath.org/authors/?q=ai:truong.mai-thanh-nhat"Hung, Tran The"https://zbmath.org/authors/?q=ai:hung.tran-the"Minh, Mach Nguyet"https://zbmath.org/authors/?q=ai:mach-nguyet-minh.Summary: The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in \(L^2\)-norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.Control of chaotic systems by deep reinforcement learninghttps://zbmath.org/1472.681712021-11-25T18:46:10.358925Z"Bucci, M. A."https://zbmath.org/authors/?q=ai:bucci.michele-alessandro"Semeraro, O."https://zbmath.org/authors/?q=ai:semeraro.onofrio"Allauzen, A."https://zbmath.org/authors/?q=ai:allauzen.alexandre"Wisniewski, G."https://zbmath.org/authors/?q=ai:wisniewski.grzegorz"Cordier, L."https://zbmath.org/authors/?q=ai:cordier.laurent"Mathelin, L."https://zbmath.org/authors/?q=ai:mathelin.lionelSummary: Deep reinforcement learning (DRL) is applied to control a nonlinear, chaotic system governed by the one-dimensional Kuramoto-Sivashinsky (KS) equation. DRL uses reinforcement learning principles for the determination of optimal control solutions and deep neural networks for approximating the value function and the control policy. Recent applications have shown that DRL may achieve superhuman performance in complex cognitive tasks. In this work, we show that using restricted localized actuation, partial knowledge of the state based on limited sensor measurements and model-free DRL controllers, it is possible to stabilize the dynamics of the KS system around its unstable fixed solutions, here considered as target states. The robustness of the controllers is tested by considering several trajectories in the phase space emanating from different initial conditions; we show that DRL is always capable of driving and stabilizing the dynamics around target states. The possibility of controlling the KS system in the chaotic regime by using a DRL strategy solely relying on local measurements suggests the extension of the application of RL methods to the control of more complex systems such as drag reduction in bluff-body wakes or the enhancement/diminution of turbulent mixing.Doubly-symmetric periodic orbits in the spatial Hill's lunar problem with oblate secondary primaryhttps://zbmath.org/1472.700272021-11-25T18:46:10.358925Z"Xu, Xingbo"https://zbmath.org/authors/?q=ai:xu.xingboSummary: In this article we consider the existence of a family of doubly-symmetric periodic orbits in the spatial circular Hill's lunar problem, in which the secondary primary at the origin is oblate. The existence is shown by applying a fixed point theorem to the equations with periodical conditions expressed in Poincaré-Delaunay elements for the double symmetries after eliminating the short periodic effects in the first-order perturbations of the approximated system.Equilibria and stability of four point vortices on a spherehttps://zbmath.org/1472.700292021-11-25T18:46:10.358925Z"Dritschel, David G."https://zbmath.org/authors/?q=ai:dritschel.david-gSummary: This paper discusses the problem of finding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of a sphere. A random search method is developed which uses a modification of the linearized equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable, though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computationshttps://zbmath.org/1472.700312021-11-25T18:46:10.358925Z"Hetebrij, Wouter"https://zbmath.org/authors/?q=ai:hetebrij.wouter"Mireles James, J. D."https://zbmath.org/authors/?q=ai:mireles-james.jason-dIn this paper, the authors study the robustness of certain basic homoclinic motions in an equilateral restricted four-body problem, where the three gravitating bodies are arranged in the equilateral triangle configuration of Lagrange. The circular restricted four-body problem (CRFBP) studies the dynamics of a fourth massless particle in a co-rotating reference frame. The problem can be viewed as a two-parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as, formal series methods for computing normal forms and center stable/unstable manifold parameterizations. A number of special cases have been studied numerically by the authors and they formulate several conjectures about the global bifurcations of the homoclinic families.On resonances in Hamiltonian systems with three degrees of freedomhttps://zbmath.org/1472.700402021-11-25T18:46:10.358925Z"Karabanov, Alexander A."https://zbmath.org/authors/?q=ai:karabanov.alexander-a"Morozov, Albert D."https://zbmath.org/authors/?q=ai:morozov.albert-dmitrievichSummary: We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.Method for controlling vibration by exploiting piecewise-linear nonlinearity in energy harvestershttps://zbmath.org/1472.740942021-11-25T18:46:10.358925Z"Tien, Meng-Hsuan"https://zbmath.org/authors/?q=ai:tien.meng-hsuan"D'Souza, Kiran"https://zbmath.org/authors/?q=ai:dsouza.kiranSummary: Vibration energy is becoming a significant alternative solution for energy generation. Recently, a great deal of research has been conducted on how to harvest energy from vibration sources ranging from ocean waves to human motion to microsystems. In this paper, a theoretical model of a piecewise-linear (PWL) nonlinear vibration harvester that has potential applications in variety of fields is proposed and numerically investigated. This new technique enables automatic frequency tunability in the energy harvester by controlling the gap size in the PWL oscillator so that it is able to adapt to changes in excitations. To optimize the performance of the proposed system, a control method combining the response prediction, signal measurement and gap adjustment mechanism is proposed in this paper. This new energy harvester not only overcomes the limitation of traditional linear energy harvesters that can only provide the maximum power generation efficiency over a narrow frequency range but also improves the performance of current nonlinear energy harvesters that are not as efficient as linear energy harvesters at resonance. The proposed system is demonstrated in several case studies to illustrate its effectiveness for a number of different excitations.Erratum to: ``Asymptotic approximations for pure bending of thin cylindrical shells''https://zbmath.org/1472.741432021-11-25T18:46:10.358925Z"Coman, Ciprian D."https://zbmath.org/authors/?q=ai:coman.ciprian-dFrom the text: Due to a typesetting error, Eq. 5.9 was incorrect in the original publication [the author, ibid. 68, No. 4, Paper No. 82, 20 p. (2017; Zbl 1386.74088)]. The correct equation is given.Dynamical stability of water distribution networkshttps://zbmath.org/1472.760282021-11-25T18:46:10.358925Z"Masuda, Naoki"https://zbmath.org/authors/?q=ai:masuda.naoki"Meng, Fanlin"https://zbmath.org/authors/?q=ai:meng.fanlinSummary: Water distribution networks are hydraulic infrastructures that aim to meet water demands at their various nodes. Water flows through pipes in the network create nonlinear dynamics on networks. A desirable feature of water distribution networks is high resistance to failures and other shocks to the system. Such threats would at least transiently change the flow rate in various pipes, potentially undermining the functionality of the whole water distribution system. Here we carry out a linear stability analysis for a nonlinear dynamical system representing the flow rate through pipes that are interconnected through an arbitrary pipe network with reservoirs and consumer nodes. We show that the steady state is always locally stable and develop a method to calculate the eigenvalue that corresponds to the mode that decays the most slowly towards the equilibrium, which we use as an index for resilience of the system. We show that the proposed index is positively correlated with the recovery rate of the pipe network, which was derived from a realistic and industrially popular simulator. The present analytical framework is expected to be useful for deploying tools from nonlinear dynamics and network analysis in the design, resilience management and scenario testing of water distribution networks.Onset and limiting amplitude of yaw instability of a submerged three-tethered buoyhttps://zbmath.org/1472.760442021-11-25T18:46:10.358925Z"Orszaghova, J."https://zbmath.org/authors/?q=ai:orszaghova.jana"Wolgamot, H."https://zbmath.org/authors/?q=ai:wolgamot.hugh-a"Draper, S."https://zbmath.org/authors/?q=ai:draper.scott"Taylor, P. H."https://zbmath.org/authors/?q=ai:taylor.paul-h"Rafiee, A."https://zbmath.org/authors/?q=ai:rafiee.ali|rafiee.ashkan|rafiee.aysanSummary: In this paper the dynamics of a submerged axi-symmetric wave energy converter are studied, through mathematical models and wave basin experiments. The device is disk-shaped and taut-moored via three inclined tethers which also act as a power take-off. We focus on parasitic yaw motion, which is excited parametrically due to coupling with heave. Assuming linear hydrodynamics throughout, but considering both linear and nonlinear tether geometry, governing equations are derived in 6 degrees of freedom (DOF). From the linearized equations, all motions, apart from yaw, are shown to be contributing to the overall power absorption. At higher orders, the yaw governing equation can be recast into a classical Mathieu equation (linear in yaw), or a nonlinear Mathieu equation with cubic damping and stiffness terms. The well-known stability diagram for the classical Mathieu equation allows prediction of onset/occurrence of yaw instability. From the nonlinear Mathieu equation, we develop an approximate analytical solution for the amplitude of the unstable motions. Comparison with regular wave experiments confirms the utility of both models for making relevant predictions. Additionally, irregular wave tests are analysed whereby yaw instability is successfully correlated to the amount of parametric excitation and linear damping. This study demonstrates the importance of considering all modes of motion in design, not just the power-producing ones. Our simplified 1 DOF yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices.Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviourshttps://zbmath.org/1472.780292021-11-25T18:46:10.358925Z"Wen, Xiao-Yong"https://zbmath.org/authors/?q=ai:wen.xiaoyong"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya"Zhang, Guoqiang"https://zbmath.org/authors/?q=ai:zhang.guoqiangSummary: The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation \((M, N - M)\)-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation \((1, N - 1)\)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.On discrete spectrum of a model graph with loop and small edgeshttps://zbmath.org/1472.810692021-11-25T18:46:10.358925Z"Borisov, D. I."https://zbmath.org/authors/?q=ai:borisov.denis-i"Konyrkulzhaeva, M. N."https://zbmath.org/authors/?q=ai:konyrkulzhaeva.maral-nurlanovna"Mukhametrakhimova, A. I."https://zbmath.org/authors/?q=ai:mukhametrakhimova.a-iSummary: We consider a perturbed graph consisting of two infinite edges, a loop, and a glued arbitrary finite graph \(\gamma \epsilon\) with small edges, where \(\gamma \epsilon\) is obtained by \(\epsilon^{-1}\) times contraction of some fixed graph and \(\epsilon\) is a small parameter. On the perturbed graph, we consider the Schrödinger operator whose potential on small edges can singularly depend on \(\epsilon\) with the Kirchhoff condition at internal vertices and the Dirichlet or Neumann condition at the boundary vertices. For the perturbed eigenvalue and the corresponding eigenfunction we prove the holomorphy with respect to \(\epsilon\) and propose a recurrent algorithm for finding all coefficients of their Taylor series.The thermal properties of the one-dimensional boson particles in Rindler spacetimehttps://zbmath.org/1472.810702021-11-25T18:46:10.358925Z"Boumali, Abdelmalek"https://zbmath.org/authors/?q=ai:boumali.abdelmalek"Rouabhia, Tarek Imad"https://zbmath.org/authors/?q=ai:rouabhia.tarek-imadSummary: In this paper we study the one-dimension Klein-Gordon (KG) equation in the Rindler spacetime. The solutions of the wave equation in an accelerated reference frame are obtained. As a result, (i) we derive a compact expression for the energy spectrum associated in an accelerated reference, and (ii) we show that the non-inertial effect of the accelerated reference frame mimics an external potential in the Klein-Gordon equation and, moreover, allows the formation of bound states. In addition, the thermal properties of the Klein-Gordon from the partition function, have been investigated, and the effect of the accelerated reference frame parameter \(a\) on these properties has been tested. This study is extended to the case of the one-dimensional Dirac equation where the spectrum of energy is well determined and has an exact form. As a result, we will see that the behavior of the thermal quantities of the fermion particles is similar to the case of boson particles.On the derivative nonlinear Schrödinger equation on the half line with Robin boundary conditionhttps://zbmath.org/1472.810882021-11-25T18:46:10.358925Z"Van Tin, Phan"https://zbmath.org/authors/?q=ai:van-tin.phanSummary: We consider the Schrödinger equation with a nonlinear derivative term on [0, +\(\infty)\) under the Robin boundary condition at 0. Using a virial argument, we obtain the existence of blowing up solutions, and using variational techniques, we obtain stability and instability by blow-up results for standing waves.
{\copyright 2021 American Institute of Physics}Stability of kinklike structures in generalized modelshttps://zbmath.org/1472.811072021-11-25T18:46:10.358925Z"Andrade, I."https://zbmath.org/authors/?q=ai:andrade.ivan|andrade.ivo-h-p"Marques, M. A."https://zbmath.org/authors/?q=ai:marques.miguel-a-l"Menezes, R."https://zbmath.org/authors/?q=ai:menezes.robertoSummary: We study the stability of topological structures in generalized models with a single real scalar field. We show that it is driven by a Sturm-Liouville equation and investigate the conditions that lead to the existence of explicit supersymmetric operators that factorize the stability equation and allow us to construct partner potentials. In this context, we discuss the property of shape invariance as a possible manner to calculate the discrete states and their respective eigenvalues.Two anyons on the sphere: nonlinear states and spectrumhttps://zbmath.org/1472.813072021-11-25T18:46:10.358925Z"Polychronakos, Alexios P."https://zbmath.org/authors/?q=ai:polychronakos.alexios-p"Ouvry, Stéphane"https://zbmath.org/authors/?q=ai:ouvry.stephaneSummary: We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systemshttps://zbmath.org/1472.820022021-11-25T18:46:10.358925Z"Lyu, Shulin"https://zbmath.org/authors/?q=ai:lyu.shulin"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of \textit{C. Min} and \textit{Y. Chen} [Math. Methods Appl. Sci. 42, No. 1, 301--321 (2019; Zbl 1409.33018)] where a second-order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in [\textit{X.-B. Wu} and \textit{S.-X. Xu}, Nonlinearity 34, No. 4, 2070--2115 (2021; Zbl 1470.34238)] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as \(n \rightarrow \infty\), the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second-order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.Optimal control model of immunotherapy for autoimmune diseaseshttps://zbmath.org/1472.820312021-11-25T18:46:10.358925Z"Costa, M. Fernanda P."https://zbmath.org/authors/?q=ai:costa.m-fernanda-p"Ramos, M. P."https://zbmath.org/authors/?q=ai:ramos.m-p-machado"Ribeiro, C."https://zbmath.org/authors/?q=ai:ribeiro.cassio-b|ribeiro.conceicao|ribeiro.claudio-d|ribeiro.cristina|ribeiro.c-c-h|ribeiro.carlos-h-c|ribeiro.celso-carneiro|ribeiro.cassilda|ribeiro.carlos-f-m|ribeiro.clovis-a|ribeiro.carlos-augusto-david|ribeiro.claudia"Soares, A. J."https://zbmath.org/authors/?q=ai:soares.ana-jacintaSummary: In this work, we develop a new mathematical model to evaluate the impact of drug therapies on autoimmunity disease. We describe the immune system interactions at the cellular level, using the kinetic theory approach, by considering self-antigen presenting cells, self-reactive T cells, immunosuppressive cells, and Interleukin-2 (IL-2) cytokines. The drug therapy consists of an intake of Interleukin-2 cytokines which boosts the effect of immunosuppressive cells on the autoimmune reaction. We also derive the macroscopic model relative to the kinetic system and study the wellposedness of the Cauchy problem for the corresponding system of equations. We formulate an optimal control problem relative to the model so that the quantity of both the self-reactive T cells that are produced in the body and the Interleukin-2 cytokines that are administrated is simultaneously minimized. Moreover, we perform some numerical tests in view of investigating optimal treatment strategies and the results reveal that the optimal control approach provides good-quality approximate solutions and shows to be a valuable procedure in identifying optimal treatment strategies.Dynamics of the Szekeres systemhttps://zbmath.org/1472.830142021-11-25T18:46:10.358925Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaSummary: The Szekeres model is a differential system in \(\mathbb{R}^4\) that provides the solutions of the Einstein field equations in the presence of irrotational dust. This differential system is integrable with two rational first integrals and one analytic first integral. We characterize the qualitative behavior of all the orbits of the Szekeres system in the function of the values of the two rational first integrals.
{\copyright 2021 American Institute of Physics}High-order accuracy computation of coupling functions for strongly coupled oscillatorshttps://zbmath.org/1472.920512021-11-25T18:46:10.358925Z"Park, Youngmin"https://zbmath.org/authors/?q=ai:park.youngmin"Wilson, Dan D."https://zbmath.org/authors/?q=ai:wilson.dan-dStage-structured hematopoiesis model with delays in an almost periodic environmenthttps://zbmath.org/1472.920612021-11-25T18:46:10.358925Z"Zhou, Hui"https://zbmath.org/authors/?q=ai:zhou.hui"Wang, Wen"https://zbmath.org/authors/?q=ai:wang.wen"Yang, Liu"https://zbmath.org/authors/?q=ai:yang.liuSummary: In this article, we focus on stage-structured hematopoiesis model with time-dependent delays in an almost periodic environment. A threshold dynamic is characterized by basic reproduction ratio \(R_0\). By making use of skew-product semiflow approach, we show that the population is extinct if \(R_0 < 1\). The existence of a unique positive almost periodic solution is derived when \(R_0 > 1\), meanwhile, the global stability of the solution is verified under weaker conditions.Limit cycles in the model of hypothalamic-pituitary-adrenal axis activityhttps://zbmath.org/1472.920662021-11-25T18:46:10.358925Z"Arcet, Barbara"https://zbmath.org/authors/?q=ai:arcet.barbara"Dolićanin Đjekic, D."https://zbmath.org/authors/?q=ai:dolicanin-djekic.dijana|dolicanin-dekic.diana"Maćešić, S."https://zbmath.org/authors/?q=ai:macesic.senka|macesic.stevan"Romanovski, V. G."https://zbmath.org/authors/?q=ai:romanovskij.v-g|romanovski.valery-gSummary: Oscillatory behavior in a three-dimensional system of differential equations which represents a sub-network of the model of hypothalamic-pituirary-adrenal axes activity is analysed. We show that Hopf bifurcations and degenerate Hopf bifurcations (Bautin bifurcations) can occur in the system for chemically relevant values of parameters, so the system can have two limit cycles.Bifurcations in a cancer and immune model with Allee effecthttps://zbmath.org/1472.920812021-11-25T18:46:10.358925Z"Hernández-López, Eymard"https://zbmath.org/authors/?q=ai:hernandez-lopez.eymard"Núñez-López, Mayra"https://zbmath.org/authors/?q=ai:nunez-lopez.mayraWell-posedness of a mathematical model of diabetic atherosclerosishttps://zbmath.org/1472.920892021-11-25T18:46:10.358925Z"Xie, Xuming"https://zbmath.org/authors/?q=ai:xie.xumingSummary: Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients' cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this paper, we propose a mathematical model for diabetic atherosclerosis; the model is given by a system of partial differential equations with a free boundary. We establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.Synchronization in stochastic biochemical oscillators subject to common multiplicative extrinsic noisehttps://zbmath.org/1472.921032021-11-25T18:46:10.358925Z"MacLaurin, James N."https://zbmath.org/authors/?q=ai:maclaurin.james-n"Vilanova, Pedro A."https://zbmath.org/authors/?q=ai:vilanova.pedro-aNeutralizing of nitrogen when the changes of nitrogen content is rapidhttps://zbmath.org/1472.921042021-11-25T18:46:10.358925Z"Rasappan, Suresh"https://zbmath.org/authors/?q=ai:rasappan.suresh"Mohan, Kala Raja"https://zbmath.org/authors/?q=ai:mohan.kala-rajaSummary: Many people believe oxygen to be the life saver. But in fact, nitrogen also plays an equally important role. Right from the formation of fetus, nitrogen is essential. For healthy living, the content of nitrogen in proper quantity is necessary. Maintaining a neutralized nitrogen content results in good health. When nitrogen content rapidly increases or decreases, it becomes the cause of many diseases. This rapid change in nitrogen content can be mathematically compared to exponential growth.
This paper focuses on the study of nitrogen mass cycle model when there is an exponential growth. Framing of the exponential growth mathematical model for nitrogen mass cycle is followed by the analysis for its boundedness, local stability, global stability, and bifurcation. Numerical simulation describing the stability of nitrogen mass cycle with the exponential growth is accomplished.
For the entire collection see [Zbl 1461.92002].Balancing of nitrogen mass cycle for healthy living using mathematical modelhttps://zbmath.org/1472.921052021-11-25T18:46:10.358925Z"Rasappan, Suresh"https://zbmath.org/authors/?q=ai:rasappan.suresh"Mohan, Kala Raja"https://zbmath.org/authors/?q=ai:mohan.kala-rajaSummary: Nowadays, pollution is the major cause of many health issues. It causes changes in the content of nitrogen in the atmosphere. In some places, there is an increase in the content of nitrogen, and in some other places, a decrease in nitrogen content is found. Both the increase and decrease in nitrogen content result in ill-health of all living beings. In order to rectify this issue, balancing of the nitrogen mass cycle is essential. The aim of this chapter is to stabilize the content of nitrogen mass cycle using its mathematical model.
The mathematical model for the nitrogen mass cycle is formulated. Mathematical properties of the model in both deterministic and nondeterministic models are also discussed. Local and global stabilities of the deterministic model are executed. Stability analysis of the positive equilibrium with respect to nondeterministic model is performed. Numerical simulation with regard to both deterministic and nondeterministic models is carried out.
For the entire collection see [Zbl 1461.92002].Stationary distribution of a chemostat model with distributed delay and stochastic perturbationshttps://zbmath.org/1472.921442021-11-25T18:46:10.358925Z"Gao, Miaomiao"https://zbmath.org/authors/?q=ai:gao.miaomiao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqingSummary: In this paper, we consider a Lotka-Volterra food chain chemostat model that incorporates both distributed delay and stochastic perturbations. We obtain sufficient conditions for the existence of stationary distribution by constructing suitable Lyapunov functions. Stationary distribution indicates the two species in the chemostat can coexist in the long term.Auxin transport model for leaf venationhttps://zbmath.org/1472.921472021-11-25T18:46:10.358925Z"Haskovec, Jan"https://zbmath.org/authors/?q=ai:haskovec.jan"Jönsson, Henrik"https://zbmath.org/authors/?q=ai:jonsson.henrik"Kreusser, Lisa Maria"https://zbmath.org/authors/?q=ai:kreusser.lisa-maria"Markowich, Peter"https://zbmath.org/authors/?q=ai:markowich.peter-alexanderSummary: The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organization of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialized membrane-localized proteins. Many venation models have been based on polarly localized efflux-mediator proteins of the PIN family. Here, we investigate a modelling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e. directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not \textit{a priori} oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's Law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a mid-vein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional responsehttps://zbmath.org/1472.921762021-11-25T18:46:10.358925Z"González-Olivares, Eduardo"https://zbmath.org/authors/?q=ai:gonzalez-olivares.eduardo"Arancibia-Ibarra, Claudio"https://zbmath.org/authors/?q=ai:arancibia-ibarra.claudio"Rojas-Palma, Alejandro"https://zbmath.org/authors/?q=ai:rojas-palma.alejandro"González-Yañez, Betsabé"https://zbmath.org/authors/?q=ai:gonzalez-yanez.betsabeThe paper studies the following modified Leslie--Gower type model
\begin{align*} \dot x &=\left(r_1 - b_1x - \frac{a_1 y}{x+k_1}\right)x,\\
\dot y &= \left( r_2 - \frac{a_2 y}{x+ k_2}\right) y, \tag{1}\end{align*}
where \(x\) and \(y\) represent the size of the prey and predator population, respectively, and the parameters \(r_1, b_1, a_1, k_1, r_2, a_2, k_2\) are assumed to be positive.
The authors determine the number and stability of the (biologically relevant) equilibria and show that in certain parameter regions limit cycles arise as a result of either a Hopf or a homoclinic bifurcation. The stability of these limit cycles is also analyzed.
The results illustrate that system (1) has the flexibility to model several different population dynamical phenomena.Erratum to: ``Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator''https://zbmath.org/1472.921772021-11-25T18:46:10.358925Z"Guo, Xiaoxia"https://zbmath.org/authors/?q=ai:guo.xiaoxia"Ruan, Dehao"https://zbmath.org/authors/?q=ai:ruan.dehaoErratum to the authors' paper [ibid. 15, Paper No. 46, 18 p. (2020; Zbl 1470.92241)].Effect of poaching on Tiger-Deer interaction model with ratio-dependent functional response in the sundarbans ecosystemhttps://zbmath.org/1472.921792021-11-25T18:46:10.358925Z"Hasan, Md. Nazmul"https://zbmath.org/authors/?q=ai:hasan.md-nazmul"Uddin, Md. Sharif"https://zbmath.org/authors/?q=ai:uddin.md-sharif"Biswas, Md. Haider Ali"https://zbmath.org/authors/?q=ai:biswas.md-haider-aliSummary: Some of the biological species like tiger, deer and monkeys in Sundarbans, the largest mangrove forest in the world have been driven to extinction due to several external forces such as illegal poaching, over exploitation, predation, environmental pollution and mismanagement of habitat. This paper deals with a ratio-dependent predator-prey model with prey refuge and illegal harvesting of both species. The boundedness of solution, feasibility of interior equilibria have been determined. Optimal control technique has been applied to investigate anti-poaching patrol strategy for controlling the threat in the predator prey system facing in Sundarbans. The system is also examined so that the better control strategy is achieved. Moreover, the control strategy is obtained on the effect of variation of prey refuge.Estimation and identifiability of parameters for generalized Lotka-Volterra biological systems using adaptive controlled combination difference anti-synchronizationhttps://zbmath.org/1472.921812021-11-25T18:46:10.358925Z"Khan, Taqseer"https://zbmath.org/authors/?q=ai:khan.taqseer"Chaudhary, Harindri"https://zbmath.org/authors/?q=ai:chaudhary.harindriThe authors apply an adaptive control method to identify unknown parameters in a predator-prey model whose dynamics exhibits chaotic behavior. The method is based in the construction of two ``master'' systems and one ``slave'' system, which have similar structure to the original system but different parameters. In addition, the ``slave'' system has an additive control \(\nu\) that is dependent of the state variables and the uncertain parameters of master and slave systems.
The first section (Preliminaries) is devoted to describe the underlying theory of master and slave systems, the second section (asymptotic analysis) describes the error dynamics between master and slave systems. The third section introduces the predator-prey system studied in a work of \textit{N. Samardzija} and \textit{L. D. Greller} [Bull. Math. Biol. 50, No. 5, 465--491 (1988; Zbl 0668.92010)]:
\[ \left\{\begin{array}{rcl}
\dot{p}_{11}&=&p_{11}-p_{11}p_{12}+l_{3}p_{11}^{2}-l_{1}p_{11}^{2}p_{13}\\
\\
\dot{p}_{12}&=&-p_{12}+p_{11}p_{12}\\
\\
\dot{p}_{13}&=&-l_{2}p_{13}+l_{1}p_{11}^{2}p_{13},
\end{array}\right.
\]
where \(p_{11}\) is the prey population while\(p_{12}\) and \(p_{13}\) are the predators population. In absence of predators, the population dynamics of the prey is described by a logistic growth and \(l_{3}>0\) is the carrying capacity. On the other hand, in absence of preys, the predators \(p_{12}\) and \(p_{13}\) have an exponential decreasing.
Nevertheless, in this article, in the third above equation, it is written \(l_{2}\) instead of \(-l_{2}\), which will imply that this predator will have an exponential growth, which is impossible. This reviewer believe that this is a typo, which is supported by the numerical simulations and the general treatment presented in the article.
The authors construct the master and slave systems associated to the predator-prey model, deduce the equations describing the error dynamics and design the control inputs. The main result states that the equation describing the error dynamics is uniformly asymptotically stable and the proof is carried out by using the second Lyapunov method. Finally, a set of numerical simulations supporting the main result is described in detail.Dynamic analysis of a model for spruce budworm populations with delayhttps://zbmath.org/1472.921842021-11-25T18:46:10.358925Z"Muhammadhaji, Ahmadjan"https://zbmath.org/authors/?q=ai:muhammadhaji.ahmadjan"Halik, Azhar"https://zbmath.org/authors/?q=ai:halik.azharSummary: A class of delayed spruce budworm population model is considered. Compared with previous studies, both autonomous and nonautonomous delayed spruce budworm population models are considered. By using the inequality techniques, continuation theorem, and the construction of suitable Lyapunov functional, we establish a set of easily verifiable sufficient conditions on the permanence, existence, and global attractivity of positive periodic solutions for the considered system. Finally, an example and its numerical simulation are given to illustrate our main results.Exotic bifurcations in three connected populations with Allee effecthttps://zbmath.org/1472.921852021-11-25T18:46:10.358925Z"Röst, Gergely"https://zbmath.org/authors/?q=ai:rost.gergely"Sadeghimanesh, AmirHosein"https://zbmath.org/authors/?q=ai:sadeghimanesh.amirhoseinModelling of a two prey and one predator system with switching effecthttps://zbmath.org/1472.921872021-11-25T18:46:10.358925Z"Saha, Sangeeta"https://zbmath.org/authors/?q=ai:saha.sangeeta"Samanta, Guruprasad"https://zbmath.org/authors/?q=ai:samanta.guruprasadSummary: Prey switching strategy is adopted by a predator when they are provided with more than one prey and predator prefers to consume one prey over others. Though switching may occur due to various reasons such as scarcity of preferable prey or risk in hunting the abundant prey. In this work, we have proposed a prey-predator system with a particular type of switching functional response where a predator feeds on two types of prey but it switches from one prey to another when a particular prey population becomes lower. The ratio of consumption becomes significantly higher in the presence of prey switching for an increasing ratio of prey population which satisfies Murdoch's condition [\textit{W. W. Murdoch}, ``Switching in general predators: experiments on predator specificity and stability of prey populations'', Ecol. Monographs 39, No. 4, 335--354 (1969; \url{doi:10.2307/19423})]. The analysis reveals that two prey species can coexist as a stable state in absence of predator but a single prey-predator situation cannot be a steady state. Moreover, all the population can coexist only under certain restrictions. We get bistability for a certain range of predation rate for first prey population. Moreover, varying the mortality rate of the predator, an oscillating system can be obtained through Hopf bifurcation. Also, the predation rate for the first prey can turn a steady-state into an oscillating system. Except for Hopf bifurcation, some other local bifurcations also have been studied here. The figures in the numerical simulation have depicted that, if there is a lesser number of one prey present in a system, then with time, switching to the other prey, in fact, increases the predator population significantly.A predator-prey model with genetic differentiation both in the predator and preyhttps://zbmath.org/1472.921902021-11-25T18:46:10.358925Z"Wang, Lisha"https://zbmath.org/authors/?q=ai:wang.lisha"Zhao, Jiandong"https://zbmath.org/authors/?q=ai:zhao.jiandongThe paper treats a particular case of a predator-prey 4-dimensional ODE model that has been treated in [\textit{L. Castellino} et al., in: AIP Conference Proceedings 1776, 020006 (2016), \url{https://doi.org/10.1063/1.4965312}]. The model in the 2016 paper is of the Lotka-Volterra type (but with carrying capacities on the prey) with both predator and prey having two genotypes. The particular case treated here consists in assuming that the hunting ability of each predator genotype does not depend on the prey genotype and allows this paper to go deeper than the previous paper into the qualitative behaviour of the (simpler) model. The study of the model is performed on an a dimensionalized version.
The paper starts by studying the stability of the model in the absence of predators and then reminds the known stability results of the model with just one genotype for both preys and predators.
The full model is then studied. There are two boundary equilibria. One corresponds to the extinction of both species and is unstable. The other corresponds to the extinction of just the predators and, depending on the value of a parameter \(h\) compared to some critical value \(h_c\), is unstable for \(h<h_c\), globally stable for \(h>h_c\) or has a local centre manifold characterized in the paper for \(h=h_c\). A necessary and sufficient condition for the existence of a unique positive equilibrium of the full model is that \(h<h_c\) and this equilibrium is stable. Thus, the model has a transcritical bifurcation. The paper concludes with numerical examples and corresponding figures illustrating its different possible qualitative behaviours.Stability and bifurcation analysis of a commensal model with additive Allee effect and nonlinear growth ratehttps://zbmath.org/1472.921912021-11-25T18:46:10.358925Z"Wei, Zhen"https://zbmath.org/authors/?q=ai:wei.zhen"Xia, Yonghui"https://zbmath.org/authors/?q=ai:xia.yonghui"Zhang, Tonghua"https://zbmath.org/authors/?q=ai:zhang.tonghuaA mathematical fractional model to study the hepatitis B virus infectionhttps://zbmath.org/1472.921932021-11-25T18:46:10.358925Z"Agarwal, Ritu"https://zbmath.org/authors/?q=ai:agarwal.ritu"Kritika"https://zbmath.org/authors/?q=ai:kritika."Purohit, S. D."https://zbmath.org/authors/?q=ai:purohit.sunil-dutt"Mishra, Jyoti"https://zbmath.org/authors/?q=ai:mishra.jyotiSummary: In this paper, our aim is to present a model for hepatitis B virus (HBV) infection. We have formulated the fractionalized model for the epidemic problem. Here, a numerical algorithm based on fractional homotopy analysis transform method (FHATM) is constituted to study the fractional form of the model for HBV infection. The suggested technique is obtained by merging the homotopy analysis, the Laplace transform, and the homotopy polynomials. The discussion of convergence and uniqueness of the solution is also presented.
For the entire collection see [Zbl 1461.92002].Control intervention strategies for within-host, between-host and their efficacy in the treatment, spread of COVID-19 : a multi scale modeling approachhttps://zbmath.org/1472.921972021-11-25T18:46:10.358925Z"Bhanu Prakash, D."https://zbmath.org/authors/?q=ai:bhanu-prakash.d"Vamsi, D. K. K."https://zbmath.org/authors/?q=ai:vamsi.dasu-krishna-kiran"Rajesh, D. Bangaru"https://zbmath.org/authors/?q=ai:rajesh.d-bangaru"Sanjeevi, Carani B."https://zbmath.org/authors/?q=ai:sanjeevi.carani-bSummary: The COVID-19 pandemic has resulted in more than 65.5 million infections and 15,14,695 deaths in 212 countries over the last few months. Different drug intervention acting at multiple stages of pathogenesis of COVID-19 can substantially reduce the infection induced, thereby decreasing the mortality. Also population level control strategies can reduce the spread of the COVID-19 substantially. Motivated by these observations, in this work we propose and study a multi scale model linking both within-host and between-host dynamics of COVID-19. Initially the natural history dealing with the disease dynamics is studied. Later comparative effectiveness is performed to understand the efficacy of both the within-host and population level interventions. Findings of this study suggest that a combined strategy involving treatment with drugs such as Arbidol, remdesivir, Lopinavir/Ritonavir that inhibits viral replication and immunotherapies like monoclonal antibodies, along with environmental hygiene and generalized social distancing proved to be the best and optimal in reducing the basic reproduction number and environmental spread of the virus at the population level.The bifurcation analysis of an SIRS epidemic model with immunity age and constant treatmenthttps://zbmath.org/1472.922012021-11-25T18:46:10.358925Z"Cao, Hui"https://zbmath.org/authors/?q=ai:cao.hui"Gao, Xiaoyan"https://zbmath.org/authors/?q=ai:gao.xiaoyan"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquan"Yan, Dongxue"https://zbmath.org/authors/?q=ai:yan.dongxue"Yue, Zongmin"https://zbmath.org/authors/?q=ai:yue.zongminSummary: In this paper, an SIRS epidemic model with immunity age is investigated, where the constant treatment rate and the loss of the acquired immunity are incorporated. The well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem, and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found. The theoretic analysis showed that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number is less than unity, and the numerical simulation illustrated that it is asymptotically stable as the number is greater than unity. Combining numerical simulations, the instability and the local stability of different endemic equilibrium, and the existence of saddle-node bifurcation, and Hopf bifurcation are analyzed. Again, we think it is possible that the Bogdanov-Takens bifurcation may occur for the model under some conditions. Both non-periodic and periodic behaviors are shown when the disease persists in population, where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease.Modeling COVID-19 in Cape Verde Islands -- an application of SIR modelhttps://zbmath.org/1472.922042021-11-25T18:46:10.358925Z"da Silva, Adilson"https://zbmath.org/authors/?q=ai:da-silva.adilson-jSummary: The rapid and surprised emergence of COVID-19, having infected three million and killed two hundred thousand people worldwide in less than five months, has led many experts to focus on simulating its propagation dynamics in order to have an estimated outlook for the not too distant future and so supporting the local and national governments in making decisions. In this paper, we apply the SIR model to simulating the propagation dynamics of COVID-19 on the Cape Verde Islands. It will be done firstly for Santiago and Boavista Islands, and then for Cape Verde in general. The choice of Santiago rests on the fact that it is the largest island, with more than 50\% of the population of the country, whereas Boavista was chosen because it is the island where the first case of COVID-19 in Cape Verde was diagnosed. Observations made after the date of the simulations were carried out corroborate our projections.Stability analysis of a fractional ordered COVID-19 modelhttps://zbmath.org/1472.922062021-11-25T18:46:10.358925Z"Das, Meghadri"https://zbmath.org/authors/?q=ai:das.meghadri"Samanta, Guruprasad"https://zbmath.org/authors/?q=ai:samanta.guruprasadSummary: The main purpose of this work is to study transmission dynamics of COVID-19 in Italy 2020, where the first case of Coronavirus disease 2019 (COVID-19) in Italy was reported on 31st January 2020. Taking into account the uncertainty due to the limited information about the Coronavirus (COVID-19), we have taken the modified susceptible-asymptomatic-infectious-recovered (SAIR) compartmental model under fractional order framework. We have formulated our model by subdividing infectious compartment into two sub compartments (reported and unreported) and introduced hospitalized class. In this work, we have studied the local and global stability of the system at different equilibrium points (disease free and endemic) and calculated sensitivity index for Italy scenario. The validity of the model is justified by comparing real data with the results obtained from simulations.Modeling the effect of reactive oxygen species and CTL immune response on HIV dynamicshttps://zbmath.org/1472.922072021-11-25T18:46:10.358925Z"Deng, Qi"https://zbmath.org/authors/?q=ai:deng.qi"Guo, Ting"https://zbmath.org/authors/?q=ai:guo.ting"Qiu, Zhipeng"https://zbmath.org/authors/?q=ai:qiu.zhipeng"Rong, Libin"https://zbmath.org/authors/?q=ai:rong.libinDynamical analysis of a fractional-order foot-and-mouth disease modelhttps://zbmath.org/1472.922122021-11-25T18:46:10.358925Z"Gashirai, Tinashe B."https://zbmath.org/authors/?q=ai:gashirai.tinashe-b"Hove-Musekwa, Senelani D."https://zbmath.org/authors/?q=ai:hove-musekwa.senelani-dorothy"Mushayabasa, Steady"https://zbmath.org/authors/?q=ai:mushayabasa.steadySummary: In this paper, a fractional-order model that describes transmission and control of foot-and-mouth disease is presented and analyzed. The proposed model incorporates two foot-and-mouth disease intervention strategies: vaccination of susceptible animals and quarantine of clinically infected animals. Firstly, the global existence, positivity and boundedness of solutions for the proposed model were proved. This was followed by computation of the basic reproduction number. The basic reproduction number was then used to demonstrate the global stability of the model steady states. Theoretical results were validated by solving the proposed model in MATLAB software using the modified Adams-Bashforth-Moulton predictor corrector scheme. Among several other important results, numerical results demonstrate that coupling vaccination and animal isolation could be essential to attain effective management of foot-and-mouth disease for certain disease transmission levels. However, if disease transmission exceeds a certain threshold value, then coupling vaccination and quarantine may not be sufficient to effectively manage the disease. Furthermore, simulation results also demonstrated that the qualitative nature of the solutions of the classical integer model is the same as that of the fractional-order differential model. In addition, it was shown numerically that in the presence of quarantine alone, disease outbreaks may occur.Oscillatory dynamics in the dilemma of social distancinghttps://zbmath.org/1472.922132021-11-25T18:46:10.358925Z"Glaubitz, Alina"https://zbmath.org/authors/?q=ai:glaubitz.alina"Fu, Feng"https://zbmath.org/authors/?q=ai:fu.fengSummary: Social distancing as one of the main non-pharmaceutical interventions can help slow down the spread of diseases, like in the COVID-19 pandemic. Effective social distancing, unless enforced as drastic lockdowns and mandatory cordon sanitaire, requires consistent strict collective adherence. However, it remains unknown what the determinants for the resultant compliance of social distancing and their impact on disease mitigation are. Here, we incorporate into the epidemiological process with an evolutionary game theory model that governs the evolution of social distancing behaviour. In our model, we assume an individual acts in their best interest and their decisions are driven by adaptive social learning of the real-time risk of infection in comparison with the cost of social distancing. We find interesting oscillatory dynamics of social distancing accompanied with waves of infection. Moreover, the oscillatory dynamics are dampened with a non-trivial dependence on model parameters governing decision-makings and gradually cease when the cumulative infections exceed the herd immunity. Compared to the scenario without social distancing, we quantify the degree to which social distancing mitigates the epidemic and its dependence on individuals' responsiveness and rationality in their behaviour changes. Our work offers new insights into leveraging human behaviour in support of pandemic response.Numerical analysis of coupled time-fractional differential equations arising in epidemiological modelshttps://zbmath.org/1472.922142021-11-25T18:46:10.358925Z"Goyal, Manish"https://zbmath.org/authors/?q=ai:goyal.manish"Prakash, Amit"https://zbmath.org/authors/?q=ai:prakash.amit"Gupta, Shivangi"https://zbmath.org/authors/?q=ai:gupta.shivangiSummary: In this paper, the coupled fractional differential equations that arise in various epidemiological models are examined by a hybrid and innovative homotopy perturbation method via the Laplace transform (HPTM) .Important applications of these equations are to model the transmission dynamics of epidemic models and to model the phenomena of electrical activity in heart. The conditions for convergence of the presented scheme are presented. This scheme contributes to the solution in a fast convergent series. The results are compared with other schemes. The results are demonstrated graphically and in a tabular form to show its competency and accuracy. The outcomes reveal that HPTM is accurate, highly effective, user-friendly, and attractive.
For the entire collection see [Zbl 1461.92002].Fractional dynamics of huanglongbing transmission within a citrus treehttps://zbmath.org/1472.922232021-11-25T18:46:10.358925Z"Kumar, Pushpendra"https://zbmath.org/authors/?q=ai:kumar.pushpendra"Ertürk, Vedat Suat"https://zbmath.org/authors/?q=ai:erturk.vedat-suat"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: The citrus epidemic huanglongbing (HLB), allied with an uncultured bacterial pathogen, is blusterous the industry of citrus worldwide. In this research work, we analysed a fractional huanglongbing model to study the transmission dynamics of the disease. We considered Caputo and new generalized form of Caputo type fractional derivatives to solve the proposed HLB model using two different methods. We exemplified the all necessary graphical observations by the call of real numerical data to show the nature of the given model classes. The analysis regarding to existence of the solution are mentioned with the help of theorems. We observed that the given numerical techniques are strong and worked well to show the dynamics of the model against the time variable.Dynamics of a chronic virus infection model with viral stimulation delayhttps://zbmath.org/1472.922242021-11-25T18:46:10.358925Z"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquan"Ma, Xiangxiang"https://zbmath.org/authors/?q=ai:ma.xiangxiang"Li, Jia"https://zbmath.org/authors/?q=ai:li.jia"Zhang, Dian"https://zbmath.org/authors/?q=ai:zhang.dianSummary: We incorporate a viral stimulation delay into a chronic virus infection model and investigate its dynamical behavior by theoretic analysis and numerical simulations. The obtained results show that the viral stimulation delay may result in new specific model phenomena, such as the dependence of the upper bound of the model solutions on the delay, transient oscillations, the eventual instability of the immune control equilibrium, and the appearance of cytokine storms. In addition to Hopf bifurcation, we provide numerical simulations showing the existence of homoclinic bifurcation and singular closed orbits.Hopf bifurcation of an age-structured epidemic model with quarantine and temporary immunity effectshttps://zbmath.org/1472.922252021-11-25T18:46:10.358925Z"Liu, Lili"https://zbmath.org/authors/?q=ai:liu.lili"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian|zhang.jian.7|zhang.jian.5|zhang.jian.4|zhang.jian.3|zhang.jian.6|zhang.jian.1|zhang.jian.2"Zhang, Ran"https://zbmath.org/authors/?q=ai:zhang.ran"Sun, Hongquan"https://zbmath.org/authors/?q=ai:sun.hongquan|sun.hongquan.1A study of within-host dynamics of dengue infection incorporating both humoral and cellular response with a time delay for production of antibodieshttps://zbmath.org/1472.922302021-11-25T18:46:10.358925Z"Murari Kanumoori, Deva Siva Sai"https://zbmath.org/authors/?q=ai:murari-kanumoori.deva-siva-sai"Prakash, D. Bhanu"https://zbmath.org/authors/?q=ai:bhanu-prakash.d"Vamsi, D. K. K."https://zbmath.org/authors/?q=ai:vamsi.dasu-krishna-kiran"Sanjeevi, Carani B."https://zbmath.org/authors/?q=ai:sanjeevi.carani-bSummary: \textbf{a. Background:} Dengue is an acute illness caused by a virus. The complex behaviour of the virus in human body can be captured using mathematical models. These models helps us to enhance our understanding on the dynamics of the virus. \textbf{b. Objectives:} We propose to study the dynamics of within-host epidemic model of dengue infection which incorporates both innate immune response and adaptive immune response (Cellular and Humoral). The proposed model also incorporates the time delay for production of antibodies from B cells. We propose to understand the dynamics of the this model using the dynamical systems approach by performing the stability and sensitivity analysis. \textbf{c. Methods used:} The basic reproduction number \((R_0)\) has been computed using the next generation matrix method. The standard stability analysis and sensitivity analysis were performed on the proposed model. \textbf{d. Results:} The critical level of the antibody recruitment rate \((q)\) was found to be responsible for the existence and stability of various steady states. The stability of endemic state was found to be dependent on time delay \((\tau)\). The sensitivity analysis identified the production rate of antibodies \((q)\) to be highly sensitive parameter. \textbf{e. Conclusions:} The existence and stability conditions for the equilibrium points have been obtained. The threshold value of time delay \((\tau_0)\) has been computed which is critical for change in stability of the endemic state. Sensitivity analysis was performed to identify the crucial and sensitive parameters of the model.Modelling the human papilloma virus transmission in a bisexually active host communityhttps://zbmath.org/1472.922322021-11-25T18:46:10.358925Z"Ogunmiloro, Oluwatayo M."https://zbmath.org/authors/?q=ai:ogunmiloro.oluwatayo-michaelThe author considers a nonlinear system of a five first-order evolution equations modeling the control of the HPV transmission within a population of men and women.The steady-state analysis and the classical stability analysis of the steady-state solutions performed. The paper ends with some numerical simulations.Mathematical modeling, analysis, and simulation of the COVID-19 pandemic with explicit and implicit behavioral changeshttps://zbmath.org/1472.922332021-11-25T18:46:10.358925Z"Ohajunwa, Comfort"https://zbmath.org/authors/?q=ai:ohajunwa.comfort"Kumar, Kirthi"https://zbmath.org/authors/?q=ai:kumar.kirthi"Seshaiyer, Padmanabhan"https://zbmath.org/authors/?q=ai:seshaiyer.padmanabhanSummary: As COVID-19 cases continue to rise globally, many researchers have developed mathematical models to help capture the dynamics of the spread of COVID-19. Specifically, the compartmental SEIR model and its variations have been widely employed. These models differ in the type of compartments included, nature of the transmission rates, seasonality, and several other factors. Yet, while the spread of COVID-19 is largely attributed to a wide range of social behaviors in the population, several of these SEIR models do not account for such behaviors. In this project, we consider novel SEIR-based models that incorporate various behaviors. We created a baseline model and explored incorporating both explicit and implicit behavioral changes. Furthermore, using the next generation matrix method, we derive a basic reproduction number, which indicates the estimated number of secondary cases by a single infected individual. Numerical simulations for the various models we made were performed and user-friendly graphical user interfaces were created. In the future, we plan to expand our project to account for the use of face masks, age-based behaviors and transmission rates, and mixing patterns.A co-infection model for oncogenic human papillomavirus and tuberculosis with optimal control and cost-effectiveness analysishttps://zbmath.org/1472.922342021-11-25T18:46:10.358925Z"Omame, Andrew"https://zbmath.org/authors/?q=ai:omame.andrew"Okuonghae, Daniel"https://zbmath.org/authors/?q=ai:okuonghae.danielSummary: A co-infection model for oncogenic human papillomavirus (HPV) and tuberculosis (TB), with optimal control and cost-effectiveness analysis is studied and analyzed to assess the impact of controls against incident infection and against infection with HPV by TB-infected individuals as well as optimal TB treatment in reducing the burden of the co-infection of the two diseases in a population. The co-infection model exhibits backward bifurcation when the associated reproduction number is less than unity. Furthermore, it is shown that TB and HPV re-infection parameters \((\varphi_p \neq 0\) and \(\sigma_t \neq 0)\) as well as TB exogenous re-infection term \((\epsilon_1 \neq 0)\) induced the phenomenon of backward bifurcation in the oncogenic HPV-TB co-infection model. The global asymptotic stability of the disease-free equilibrium of the co-infection model is \textit{shown not to exist}, when the associated reproduction number is below unity. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established using the Pontryagin's maximum principle. Numerical simulations of the optimal control model reveal that the intervention strategy which combines and implements control against HPV infection by TB infected individuals as well as TB treatment control for dually infected individuals is the most cost-effective of all the control strategies for the control and management of the burden of oncogenic HPV and TB co-infection.Optimal control of a Nipah virus transmission modelhttps://zbmath.org/1472.922362021-11-25T18:46:10.358925Z"Panja, Prabir"https://zbmath.org/authors/?q=ai:panja.prabir"Jana, Ranjan Kumar"https://zbmath.org/authors/?q=ai:jana.ranjan-kumarSummary: In this chapter, a Nipah virus transmission model in human population has been formulated. According to the transmission mechanism of Nipah virus, it is assumed that the Nipah virus is transmitted in human population in two possible ways: (i) through infected pig and (ii) through infected human. Optimal control strategies have been developed to minimize the number of infected humans from Nipah virus as well as to minimize the cost of control strategies. Boundedness of solutions and different possible equilibrium points have been studied. The existence condition of the optimal control problem has been investigated. The numerical simulation results without control and with control (vaccination and treatment) of Nipah virus transmission model have been compared. From the theoretical and numerical results, it can be concluded that in order to reduce Nipah virus transmission from human population, vaccination and treatment controls can be considered together.
For the entire collection see [Zbl 1461.92002].Fractional SIRI model with delay in context of the generalized Liouville-Caputo fractional derivativehttps://zbmath.org/1472.922402021-11-25T18:46:10.358925Z"Sene, Ndolane"https://zbmath.org/authors/?q=ai:sene.ndolaneSummary: This chapter addresses an analysis of an epidemic model in the context of fractional calculus. We consider the fractional SIRI model with the delay using the generalized Liouville-Caputo derivative. The existence and uniqueness of the SIRI epidemic model have been investigated. We determine the reproduction number \(F_0\). We establish the free disease point and the endemic point. We present the stability analysis of the free disease point and the endemic point according to the reproduction number \(F_0\).
For the entire collection see [Zbl 1461.92002].A modelling framework to assess the likely effectiveness of facemasks in combination with `lock-down' in managing the COVID-19 pandemichttps://zbmath.org/1472.922412021-11-25T18:46:10.358925Z"Stutt, Richard O. J. H."https://zbmath.org/authors/?q=ai:stutt.richard-o-j-h"Retkute, Renata"https://zbmath.org/authors/?q=ai:retkute.renata"Bradley, Michael"https://zbmath.org/authors/?q=ai:bradley.michael-g|bradley.michael-t|bradley.michael-j|bradley.michael-d"Gilligan, Christopher A."https://zbmath.org/authors/?q=ai:gilligan.christopher-a"Colvin, John"https://zbmath.org/authors/?q=ai:colvin.johnSummary: COVID-19 is characterized by an infectious pre-symptomatic period, when newly infected individuals can unwittingly infect others. We are interested in what benefits facemasks could offer as a non-pharmaceutical intervention, especially in the settings where high-technology interventions, such as contact tracing using mobile apps or rapid case detection via molecular tests, are not sustainable. Here, we report the results of two mathematical models and show that facemask use by the public could make a major contribution to reducing the impact of the COVID-19 pandemic. Our intention is to provide a simple modelling framework to examine the dynamics of COVID-19 epidemics when facemasks are worn by the public, with or without imposed `lock-down' periods. Our results are illustrated for a number of plausible values for parameter ranges describing epidemiological processes and mechanistic properties of facemasks, in the absence of current measurements for these values. We show that, when facemasks are used by the public all the time (not just from when symptoms first appear), the effective reproduction number, \(R_e\), can be decreased below 1, leading to the mitigation of epidemic spread. Under certain conditions, when lock-down periods are implemented in combination with 100\% facemask use, there is vastly less disease spread, secondary and tertiary waves are flattened and the epidemic is brought under control. The effect occurs even when it is assumed that facemasks are only 50\% effective at capturing exhaled virus inoculum with an equal or lower efficiency on inhalation. Facemask use by the public has been suggested to be ineffective because wearers may touch their faces more often, thus increasing the probability of contracting COVID-19. For completeness, our models show that facemask adoption provides population-level benefits, even in circumstances where wearers are placed at increased risk. At the time of writing, facemask use by the public has not been recommended in many countries, but a recommendation for wearing face-coverings has just been announced for Scotland. Even if facemask use began after the start of the first lock-down period, our results show that benefits could still accrue by reducing the risk of the occurrence of further COVID-19 waves. We examine the effects of different rates of facemask adoption without lock-down periods and show that, even at lower levels of adoption, benefits accrue to the facemask wearers. These analyses may explain why some countries, where adoption of facemask use by the public is around 100\%, have experienced significantly lower rates of COVID-19 spread and associated deaths. We conclude that facemask use by the public, when used in combination with physical distancing or periods of lock-down, may provide an acceptable way of managing the COVID-19 pandemic and re-opening economic activity. These results are relevant to the developed as well as the developing world, where large numbers of people are resource poor, but fabrication of home-made, effective facemasks is possible. A key message from our analyses to aid the widespread adoption of facemasks would be: `my mask protects you, your mask protects me'.Complex patterns of an SIR model with a saturation treatment on complex networks: an edge-compartmental approachhttps://zbmath.org/1472.922442021-11-25T18:46:10.358925Z"Yang, Junyuan"https://zbmath.org/authors/?q=ai:yang.junyuan"Duan, Xinyi"https://zbmath.org/authors/?q=ai:duan.xinyi"Li, Xuezhi"https://zbmath.org/authors/?q=ai:li.xuezhiSummary: In this paper, we adopt an edge-compartmental approach to deeply investigate an SIR model with a saturation treatment on complex networks. We find that the system may exhibit the coexistence of multi-endemic equilibria, whose stabilities are determined by signs of tangent slopes of the epidemic curve. Numerical examples illustrate the theoretical results.Periodic oscillation and tri-stability in mutualism systems with two consumershttps://zbmath.org/1472.922592021-11-25T18:46:10.358925Z"Wang, Yuanshi"https://zbmath.org/authors/?q=ai:wang.yuanshi"Wu, Hong"https://zbmath.org/authors/?q=ai:wu.hong"DeAngelis, Donald L."https://zbmath.org/authors/?q=ai:deangelis.donald-lSummary: This paper considers mutualistic interactions between two consumers, in which one consumer can consume a resource only by exchange of service for service with the other. By rigorous analysis on the one-resource and two-consumer model with Holling-type I response, we show periodic oscillations and tri-stability in the mutualism system: when their initial densities decrease, the consumers' interaction outcomes would change from coexistence in periodic oscillation, to persistence at a steady state, and to extinction. Under certain conditions, we also show two types of bi-stability in the system: the consumers would change from coexisting in periodic oscillation (resp. at a steady state) to going to extinction when their initial densities decrease. Then we analyze a modified system with Holling-type II response. Based on theoretical analysis and numerical computation, we show that there also exist tri-stability and two types of bi-stability in this system. Moreover, it is shown that varying the degree of obligation can lead to transition of interaction outcomes between coexistence in periodic oscillation (resp. at a steady state) and extinction of both consumers. These results are important in understanding complexity in mutualism.Stability, bifurcation analysis and chaos control in chlorine dioxide-iodine-malonic acid reactionhttps://zbmath.org/1472.923522021-11-25T18:46:10.358925Z"Din, Qamar"https://zbmath.org/authors/?q=ai:din.qamar"Donchev, Tzanko"https://zbmath.org/authors/?q=ai:donchev.tzanko-donchev"Kolev, Dimitar"https://zbmath.org/authors/?q=ai:kolev.dimitar-aSummary: In this paper, the qualitative behavior of discrete-time models related to chlorine dioxide-iodine-malonic acid reaction is discussed. The discrete-time models are obtained by implementing forward Euler's scheme followed by nonstandard difference scheme. The parametric conditions for local asymptotic stability of positive steady-state are investigated. Moreover, we discuss the existence and direction for both period-doubling and Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time model due to emergence of period-doubling and Neimark-Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.On controllability for a system governed by a fractional-order semilinear functional differential inclusion in a Banach spacehttps://zbmath.org/1472.930112021-11-25T18:46:10.358925Z"Afanasova, Maria"https://zbmath.org/authors/?q=ai:afanasova.maria"Liou, Yeong-Cheng"https://zbmath.org/authors/?q=ai:liou.yeongcheng"Obukhovskii, Valeri"https://zbmath.org/authors/?q=ai:obukhovskii.valeri"Petrosyan, Garik"https://zbmath.org/authors/?q=ai:petrosyan.garik-garikovich|petrosyan.garikSummary: We obtain some controllability results for a system governed by a semilinear functional differential inclusion of a fractional order in a Banach space assuming that the linear part of inclusion generates a noncompact \(C_0\)-semigroup. We define the multivalued operator whose fixed points are generating solutions of the problem. By applying the methods of fractional analysis and the fixed point theory for condensing multivalued maps we study the properties of this operator, in particular, we prove that under certain conditions it is condensing w.r.t. an appropriate measure of noncompactness. This allows to present the general controllability principle in terms of the topological degree theory and to consider certain important particular cases.Differential-algebraic systems are generically controllable and stabilizablehttps://zbmath.org/1472.930152021-11-25T18:46:10.358925Z"Ilchmann, Achim"https://zbmath.org/authors/?q=ai:ilchmann.achim"Kirchhoff, Jonas"https://zbmath.org/authors/?q=ai:kirchhoff.jonasSummary: We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts
(see the survey article by \textit{T. Berger} and \textit{T. Reis} [in: Surveys in differential-algebraic equations I. Berlin: Springer. 1--61 (2013; Zbl 1266.93001)]),
we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.Ensemble control on Lie groupshttps://zbmath.org/1472.930172021-11-25T18:46:10.358925Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.19"Li, Jr-Shin"https://zbmath.org/authors/?q=ai:li.jr-shinOn systems of fractional differential equations with the \(\psi \)-Caputo derivative and their applicationshttps://zbmath.org/1472.930732021-11-25T18:46:10.358925Z"Almeida, Ricardo"https://zbmath.org/authors/?q=ai:almeida.ricardo"Malinowska, Agnieszka Barbara"https://zbmath.org/authors/?q=ai:malinowska.agnieszka-barbara"Odzijewicz, Tatiana"https://zbmath.org/authors/?q=ai:odzijewicz.tatianaSummary: Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order \(0< \alpha <1\) of the \(\psi \)-Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader-follower consensus for fractional multi-agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples.Periodic dynamics for nonlocal Hopfield neural networks with random initial datahttps://zbmath.org/1472.930742021-11-25T18:46:10.358925Z"Chen, Zhang"https://zbmath.org/authors/?q=ai:chen.zhang"Yang, Dandan"https://zbmath.org/authors/?q=ai:yang.dandan"Zhong, Shitao"https://zbmath.org/authors/?q=ai:zhong.shitaoSummary: In this paper, a class of nonlocal Hopfield neural networks with random initial data is introduced, where the randomness may be of probability uncertainty. Sufficient conditions are derived to ensure the existence and globally exponential convergence of periodic solution for the addressed system in the frame of nonlinear expectation and linear expectation, respectively. Moreover, numerical examples are given to show the effectiveness of the obtained results.Stochastic Nicholson-type delay differential systemhttps://zbmath.org/1472.930792021-11-25T18:46:10.358925Z"Wang, Wentao"https://zbmath.org/authors/?q=ai:wang.wentao"Shi, Ce"https://zbmath.org/authors/?q=ai:shi.ce"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.2|chen.wei.1|chen.wei.3|chen.wei.4|chen.weiSummary: Focusing on Nicholson-type delay differential system in random environments, we introduce the stochastic system to model the dynamics of Nicholson's blowflies population sizes with mortality rates perturbed by white noises. We study the existence and uniqueness of the global positive solution with nonnegative initial conditions. Then the ultimate boundedness in the mean of a solution is derived under the same condition. Moreover, we estimate the sample Lyapunov exponent of the solution, which is less than a positive vector. Finally, an example with its numerical simulations is carried out to support theoretical results.Adaptive single input sliding mode control for hybrid-synchronization of uncertain hyperchaotic Lu systemshttps://zbmath.org/1472.930922021-11-25T18:46:10.358925Z"Singh, Satnesh"https://zbmath.org/authors/?q=ai:singh.satnesh"Han, Seungyong"https://zbmath.org/authors/?q=ai:han.seungyong"Lee, S. M."https://zbmath.org/authors/?q=ai:lee.sunmin|lee.sangmin.1|lee.seong-baek|lee.shueh-mien|lee.sung-mhan|lee.shen-ming|lee.sangman|lee.shing-meng|lee.sung-min|lee.sinmin|lee.sangmin|lee.shi-min|lee.sang-myung|lee.sangmi|lee.sunmi|lee.su-mi|lee.seok-minSummary: This paper addresses the problem of hybrid synchronization for hyperchaotic Lu systems without and with uncertain parameters via a single input sliding mode controller (SMC). Based on the SMC approach, the proposed controller not only minimizes the influence of uncertainty but also enhances the robustness of the system. The uncertain parameters are estimated by using new adaptation laws which ensure the uncertain parameters convergence to their original value. A hybrid synchronization scheme is useful to maintain the vastly secured and secrecy in the area of secure communication by using the control theory approach. The proposed hybrid synchronization results are providing a superiority of forming a chaotic secure communication scheme. Finally, a numerical example is provided to demonstrate the validity of the theoretical analysis.To the regulation of indefinite nonlinear dynamic objects by continuous controlhttps://zbmath.org/1472.930952021-11-25T18:46:10.358925Z"Ulanov, Boris Vasil'evich"https://zbmath.org/authors/?q=ai:ulanov.boris-vasilevichSummary: We propose an algorithm for the synthesis of continuous scalar control that regulates an indefinite nonlinear dynamic object so that all solutions of the system of differential equations describing a closed system are bounded and the state vector of the object asymptotically tends to zero. The object is described by a system of nonlinear ordinary differential equations with indefinite right-hand sides. The original continuous nonlinear control decreases the dimension of the problem and introduces a new control function -- adaptive control, which solves the problem of the regulation for a low-order controlled system. A study of the behavior of the solutions of the closed system is given.Robust reliable \(H_{\infty}\) optimization control for uncertain discrete-time Takagi-Sugeno fuzzy systems with time-varying delayhttps://zbmath.org/1472.931102021-11-25T18:46:10.358925Z"Sun, Shaoxin"https://zbmath.org/authors/?q=ai:sun.shaoxin"Zhang, Huaguang"https://zbmath.org/authors/?q=ai:zhang.huaguang"Su, Hanguang"https://zbmath.org/authors/?q=ai:su.hanguang"Liang, Yuling"https://zbmath.org/authors/?q=ai:liang.yulingSummary: In this article, the problems of robust reliable \(H_{\infty}\) optimization control for discrete-time nonlinear systems with or without time-delay subject to faults are investigated. A static state feedback controller is explored for Takagi-Sugeno fuzzy systems against nonlinear dynamics, exogenous disturbances, and model uncertainties. In contrast to the design of conventional controllers, the proposed controller is easier and more flexible. The Lyapunov function is utilized to assure the stability of the closed-loop systems. Besides, linear matrix inequalities are utilized to derive the sufficient conditions by MATLAB toolbox. Finally, two simulation examples are shown to validate the feasibility of the suggested approach.Composite slack-matrix-based integral inequality and its application to stability analysis of time-delay systemshttps://zbmath.org/1472.931452021-11-25T18:46:10.358925Z"Tian, Yufeng"https://zbmath.org/authors/?q=ai:tian.yufeng"Wang, Zhanshan"https://zbmath.org/authors/?q=ai:wang.zhanshanSummary: This paper focuses on the delay-dependent stability problem of time-varying delay linear systems. A composite slack-matrix-based integral inequality (CSMBII) is presented in delay-product types. It overcomes the limitation of reciprocal convexity in Bessel-Legendre integral inequality such that CSMBII is more convenient to cope with time-varying delay systems. And it also overcomes the shortcoming that the slack matrices in previous works are independent of time-varying delay. Based on CSMBII, a new delay-dependent stability criterion is derived for time delay systems. A numerical example is given to illustrate the effectiveness of the stability criterion.