Attractivity of nonlinear impulsive delay differential equations

Zhichun Yang,Daoyi Xu

doi:10.1155/jamsa/2006/83152

Abstract

The attractivity of nonlinear differential equations with time delays and impulsive effects is discussed. We obtain some criteria to determine the attracting set and attracting basin of the impulsive delay system by developing an impulsive delay differential inequality and introducing the concept of nonlinear measure. Examples and their simulations illustrate the effectiveness of the results and different asymptotical behaviors between the impulsive system and the corresponding continuous system.

Highlights

The stability and attractivity of impulsive differential equations have been deeply investigated in the monographs of Baınov and Simeonov [1], Lakshmikantham et al [6], Samoılenko and Perestyuk [16], Borisenko et al [3]
The recent work has provided a full discussion of this subject for impulsive delay differential equations
Most of these results on asymptotic behavior are valid locally in the neighborhood of the equilibrium state, but do not estimate the range of the stable region and domain of attraction

Summary

Introduction

The stability and attractivity of impulsive differential equations have been deeply investigated in the monographs of Baınov and Simeonov [1], Lakshmikantham et al [6], Samoılenko and Perestyuk [16], Borisenko et al [3]. The recent work has provided a full discussion of this subject for impulsive delay differential equations (see, e.g., Yan and Shen [19], Liu and Ballinger [10], Liu et al [12], Yu [20], Zhang and Sun [21], etc.) Most of these results on asymptotic behavior are valid locally in the neighborhood of the equilibrium state, but do not estimate the range of the stable region and domain of attraction (referring to the definition given by Lakshmikantham and Leela [7], Siljak [17], Kolmanovskii and Nosov [5]). The criteria present a feasible and effective approach to estimate the attracting set, attracting basin, and asymptotically stable region of the impulsive systems by solving an algebraic equation Examples and their simulations are given to demonstrate the effectiveness of our results

Preliminaries

Main results

Illustrative examples

Global attracting set S1