Hong- Li

Global stability problem for feedback control systems of impulsive fractional differential equations on networks

This paper studies a novel class of feedback control systems of impulsive fractional differential equations on networks (FCSIFDENs). By combining some graph theory and the Lyapunov method, we provide a systematic method for constructing a global Lyapunov function for FCSIFDENs. Consequently, a new global asymptotic stability principle and a new global Mittag-Leffler stability principle, which have a close relation to the topology property of the network, are given. Finally, numerical examples are given to demonstrate the effectiveness of the theoretical results. HighlightsWe provide a systematic method for constructing a Lyapunov function for FCSIFDENs by using graph theory and Lyapunov method.A new global asymptotic stability principle and a new global Mittag-Leffler stability principle are given.Numerical examples are given to demonstrate the effectiveness of the theoretical results.

Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network

Abstract This paper investigates the global Mittag–Leffler stability of coupled system of fractional-order differential equations on network (CSFDEN). By using graph theory and the Lyapunov method, we provide a method for constructing a global Lyapunov function for CSFDEN. Consequently, several sufficient conditions are obtained to ensure the Mittag–Leffler stability of CSFDEN. These criteria have a close relation to the topology property of the network. Finally, a numerical example is presented to demonstrate the validity and feasibility of the theoretical result.

Global Mittag-Leffler stability for a coupled system of fractional-order differential equations on network with feedback controls

This paper investigates a coupled system of fractional-order differential equations on network with feedback controls (CSFDENFCs). By using the contraction mapping principle, Lyapunov method, graph theoretic approach and inequality techniques, some sufficient conditions are derived to ensure the existence, uniqueness and global Mittag-Leffler stability of the equilibrium point of CSFDENFCs. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the theoretical results. HighlightsWe obtain a global Mittag-Leffler stability principle.The methods used are graph theory and Lyapunov method.Numerical simulations are given to illustrate the results.

Impulsive synchronization of time delay bursting neuron systems with unidirectional coupling

In the article, impulsive synchronization of chaotic bursting in Hindmarsh-Rose neuron systems with time delay via partial state signal is investigated. Based on impulsive control theory of dynamical systems, the sufficient conditions on feedback strength and impulsive interval are established to guarantee the synchronization. Numerical simulations show the effectiveness of the proposed scheme. The obtained results may be helpful to understand dynamical mechanism of signal transduction in real neuronal activity.

Synchronization of multiple bursting neurons ring coupled via impulsive variables

Synchronization behavior of bursting neurons is investigated in a neuronal network ring impulsively coupled, in which each neuron exhibits chaotic bursting behavior. Based on the Lyapunov stability theory and impulsive control theory, sufficient conditions for synchronization of the multiple systems coupled with impulsive variables can be obtained. The neurons become synchronous via suitable impulsive strength and resetting period. Furthermore, the result is obtained that synchronization among neurons is weakened with the increasing of the reset period and the number of neurons. Finally, numerical simulations are provided to show the effectiveness of the theoretical results.© 2014 Wiley Periodicals, Inc. Complexity 21: 29-37, 2015

Dynamic analysis of a fractional-order single-species model with diffusion

Abstract. In this paper, we consider a fractional-order single-species model, which is composed of several patches connected by diffusion. We first prove the existence, uniqueness, non-negativity, and boundedness of solutions for the model, as desired in any population dynamics. Moreover, we also obtain some sufficient conditions ensuring the existence and uniform asymptotic stability of the positive equilibrium point for the investigated system. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the theoretical results.

Stability analysis for coupled systems of fractional differential equations on networks

ABSTRACT This paper considers the stability problem for some coupled systems of fractional differential equations on networks (CSFDENs). We provide a systematic method for constructing a Lyapunov function for CSFDENs by using graph theory and the Lyapunov method. Consequently, some sufficient conditions for stability, uniform stability and uniform asymptotic stability of CSFDENs are obtained. Finally, an example and some numerical simulations are presented to verify the effectiveness of the theoretical results.

Global Stability for a Three-Species Food Chain Model in a Patchy Environment

We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse among different patches . By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive equilibrium of this model is unique and globally asymptotically stable if it exists.

Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos.

Global stability of an SI epidemic model with feedback controls in a patchy environment

Abstract In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n ( n  ≥ 2) patches. We derive the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable if R 0  ≤ 1. In the case of R 0  > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results.