This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for determining attracting sets and attracting basin of the impulsive system are obtained by developing the properties of quasi-invariant sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive systems are unbounded.

Impulsive delay differential equations have attracted increasing interests since time delays and impulsive effects commonly exist in many fields such as population dynamics, automatic control, drug administration, and communication networks [

However, under impulsive perturbation, the solutions may not be attracted to an equilibrium point or periodical trajectory but to some bounded region. In this case, it is interesting to investigate the attracting set and attracting basin, that is, the region attracting the solutions and the range in which initial values vary when remaining the attractivity for impulsive delay differential equations. In [

In this paper, our objective is to mainly discuss the asymptotical behavior on (locally) attracting set and its attracting basin for a class of impulsive delay differential equations. Based on the quasi-invariant properties, we estimate the existence range of attracting set and attracting basin of the impulsive delay systems by solving algebraic equations and employing differential inequality technique. Examples are given to illustrate the effectiveness of our method and show that the asymptotic behavior of the impulsive systems may be different from one of the corresponding continuous systems.

Let

Let

Morever,

We define

For

In this paper, we will consider a impulsive delay differential equations:

A function

In this paper, we need the following definitions involving attracting set, attracting basin, the quasi-invariant set of impulsive systems, and monotonous vector functions.

The set

The set

Let

In this paper, we always make the following assumptions.

There exist nonnegative constants

To obtain attractivity, we first give the quasi-invariant properties of (

Assume that in addition to

Let

Based on the obtained quasi-invariant set, we have the following

Let

From (

From the above theorems, we can obtain sufficient conditions ensuring global attractivity and stability in the following corollaries.

Assume that (

Since

Assume that all conditions in Corollary

The following illustrative examples will demonstrate the effectiveness of our results and also show the different asymptotical behaviors between the impulsive system and the corresponding continuous system.

Consider a scalar nonlinear impulsive delay system

The trajectory of (

Consider a 2-dimensional impulsive delay system

Global attracting set of (

This work is supported partially by the National Natural Science Foundation of China under Grant nos. 10971240, 61263020, and 61004042, the Key Project of Chinese Education Ministry under Grant no. 212138, the Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and the Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ120630.