基于媒体报道的Filippov传染病模型的全局动力学

在经典传染病模型的基础上,通过考虑阈值策略,研究了一类基于媒体报道的不连续的传染病模型.利用Filippov意义下的右端不连续微分方程理论,对阈值策略下传染病模型的动力学行为进行了定性分析,并利用Poincaré映射研究了无病平衡点、地方病平衡点及伪平衡点的全局渐近稳定性.     

基于Allee效应诱导的Filippov生态系统的动力学行为研究

该文建立了一类由Allee效应诱导的非光滑Filippov切换系统.运用Filippov系统的定性分析方法,从理论上研究了系统的滑动区域、滑动模态和各类平衡点的存在性.同时用数值方法研究了系统的滑动模态分支、边界焦点分支及全局动力学行为.研究发现:Allee效应的强度可使种群的动态不稳定,不利于濒危生物种群的管理.     

基于阈值策略带有扩散和不连续控制项的单物种生态模型的全局稳定性分析（英文）

本文通过阈值策略(TP)研究了带有扩散的单物种生态模型的控制问题.利用一致持久性理论和Filippov理论方法,得到了新模型的正平衡点的存在性定理.通过使用耦合系统的图论方法和构建Lyapunov函数思想,得到了新模型的正平衡点唯一且全局渐进稳定的充分条件.推广了文献[6]的相关结果.     

一类具有Allee效应的Filippov害虫治理模型的研究

本文构建一类具有Allee效应的Filippov害虫治理模型.首先研究两个子系统的平衡点的存在性和稳定性,并证明了鞍结点的存在.然后运用非光滑动力学系统理论,讨论了真、假、伪平衡点的存在性和稳定性条件.最后应用理论和数值模拟研究与边界结点分支有关的滑动分支,并讨论相应的生物学意义.     

具有分布时滞的病毒感染模型动力学性质研究

本文研究具有分布时滞的病毒感染模型的动力学性质.构建该模型,基于Lyapunov泛函分析方法,研究该系统平衡点的全局稳定性.分别得到该系统无病平衡点和感染平衡点全局稳定的充分条件.     

基于忆阻的分数阶时滞复值神经网络的全局渐近稳定性

研究了分数阶复值神经网络的稳定性.针对一类基于忆阻的分数阶时滞复值神经网络,利用Caputo分数阶微分意义上Filippov解的概念, 研究其平衡点的存在性和唯一性.采用了将复值神经网络分离成实部和虚部的研究方法, 将实数域上的比较原理、不动点定理应用到稳定性分析中, 得到了模型平衡点存在性、唯一性和全局渐近稳定性的充分判据.数值仿真实例验证了获得结果的有效性.     

具有标准发生率和因病死亡率的离散SIS传染病模型的全局稳定性分析

主要研究了具有标准发生率和因病死亡率的离散SIS传染病模型的动力学性质,利用构造Lyapunov函数,得到模型无病平衡点和地方性平衡点的全局稳定性,即无病平衡点是全局渐近稳定的当且仅当基本再生数R_0≤1,地方病平衡点是全局渐近稳定的当且仅当R_0>1.     

具有治疗控制的传染病模型分析

针对筛查和药物治疗对染病者传染性产生的影响,本文考虑了具有两种不同传染水平染病者的仓室数学模型.分析了模型平衡点的稳定性态,结果表明,当基本再生数小于1时,模型的无病平衡点全局稳定;当基本再生数大于1时,地方病平衡点在一定条件下也是全局稳定的.同时利用控制理论本文也研究了药物治疗的实施对染病者进行干预和影响的最优控制措施,寻找到了使目标函数值最小的治疗控制方法,并用数值模拟显示了模型解的动力学性态及治疗措施对防止疾病蔓延所起的作用.     

无标度网络上带有时滞的SIRS模型的稳定性分析

建立了一个无标度网络上带有时滞的SIRS模型,并分析了在度不相关情况下模型的动力学性态.当基本再生数R_01时,模型只有无病平衡点,运用Jacobi矩阵和Lyapunov泛函得出无病平衡点的全局稳定性;当R_01时,无病平衡点不稳定,存在唯一地方病平衡点且是持续的.     

具有空间效应及一般感染率的时滞病毒模型分析

考虑到时滞效应及空间扩散的影响,建立了一个具有一般传染率的病毒感染仓室模型,分析了模型的动力学性态.定义了模型的基本再生数R_0,讨论了平衡点的存在性,并通过构造Lyapunov函数分析了平衡点的稳定性.结果表明,当R_01时,无病平衡点全局渐近稳定;当R_0 1时,无病平衡点不稳定且地方病平衡点在一定条件下全局渐近稳定.同时,以Beddington-DeAngelis感染率为例的数值模拟进一步验证和扩展了理论结果.     

Global dynamics of a Filippov plant disease model with an economic threshold of infected-susceptible ratio

This paper presents a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio, above which control strategies of replanting or removing are needed to be carried out. Based on the Filippov approach, we study the sliding mode dynamics and further the global dynamics. It is shown that there is a unique equilibrium, which is a disease-free equilibrium, an endemic equilibrium or a pseudo-equilibrium. Moreover, the equilibrium is proved to be globally asymptotically stable. Our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.     

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator–prey model with mutual interference and Beddington–DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set‐valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system. Copyright © 2016 John Wiley & Sons, Ltd.     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.     

Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations

We present two codimension-one bifurcations that occur when an equilibrium collides with a discontinuity in a piecewise smooth dynamical system. These simple cases appear to have escaped recent classifications. We present them here to highlight some of the powerful results from Filippov’s book Differential Equations with Discontinuous Righthand Sides (Kluwer, 1988). Filippov classified the so-called boundary equilibrium collisions without providing their unfolding. We show the complete unfolding here, for the first time, in the particularly interesting case of a node changing its stability as it collides with a discontinuity. We provide a prototypical model that can be used to generate all codimension-one boundary equilibrium collisions, and summarize the elements of Filippov’s work that are important in achieving a full classification.     

Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks

We study some properties of piecewise linear differential systems describing gene regulatory networks, where the dynamics are governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing discontinuous differential equations with differential inclusions. The global existence and some other basic properties of the Filippov solutions such as continuous dependence on parameters are studied. We also study the uniqueness and non-uniqueness of the Filippov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two ways of defining the singular stationary points: one is based on the Filippov theory and the other consists in replacing step functions with steep sigmoids and investigating the smooth systems thus obtained. The results are illustrated by a number of examples.     

一类不连续系统的推广Matrosov稳定性定理

在一般Filippov解意义下给出利用一族满足一定条件的Lyapunov函数的更一般的Matrosov定理.     

Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting

This paper is concerned with a delayed Nicholson's blowflies model with discontinuous harvesting, which is described by an almost periodic nonsmooth dynamical system. Under some reasonable assumptions on the discontinuous harvesting function, by using the Filippov regulation techniques and the theory of dichotomy, together with the Halanay inequality, we establish some new criteria on the existence of positive almost periodic solution and its convergence. An example with numerical simulation is also presented to support the theoretical results.     

一类不连续时滞系统的一致最终有界性

主要讨论不连续的时滞自治系统,在Filippov解意义下的一致最终有界性问题．基于Lyapunov-Krasovskii泛函给出了全局强一致最终有界的Lyapunov定理,并将其应用到一类带有不连续摩擦项的时滞力学系统．     

Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations

This paper investigates the dynamics of a class of recurrent neural networks where the neural activations are modeled by discontinuous functions. Without presuming the boundedness of activation functions, the sufficient conditions to ensure the existence, uniqueness, global exponential stability and global convergence of state equilibrium point and output equilibrium point are derived, respectively. Furthermore, under certain conditions we prove that the system is convergent globally in finite time. The analysis in the paper is based on the properties of M-matrix, Lyapunov-like approach, and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend previous works on global stability of recurrent neural networks with not only Lipschitz continuous but also discontinuous neural activation functions.