We present a method that uses successor functions in ordinary differential systems to address the “center-focus” problem of a class of planar systems that have an impulsive perturbation. By deriving solution formulae for impulsive systems, several interesting criteria for distinguishing between the center and the focus of linear and nonlinear planar systems with state-dependent impulsions are established. The conditions describing the stability of the focus of the considered models are also given. The computing methods presented here are more convenient for determining the center of impulsive systems than those in the literature. Numerical examples are given to show the effectiveness of the theoretical results.

Many differential evolutionary processes that arise in physical, chemical, and biological phenomena are characterized by the fact that they experience an abrupt change of state at certain moments in time. These processes can be modeled using impulsive differential equations. It is known, for example, that many phenomena involve thresholds and exhibit impulsive attributes, including bursting rhythm models (medicine and biology), optimal control models (economics), pharmacokinetics, and frequency modulated systems [

There are many results regarding the bifurcation of impulsive differential systems, for example, [

The rest of this paper is organized as follows. In Section

The following conventions will be used throughout this paper. The transpose of matrix

Consider the following planar differential systems with an impulsive perturbation:

The following assumptions will be imposed throughout the paper.

Set

The Jacobian matrix of

Set

Next, one must make the transformation

Similar to the concepts of the center and focus in the ordinary differential systems [

For any sufficiently small

Consider the following linear impulsive differential system:

Using polar coordinates,

Let

When

Now, suppose that (

According to Definition

Denote

a center if

an asymptotically stable focus if

an unstable focus if

When

When

In this section, successor functions are used to study the center-focus problem of a class of nonlinear impulsive differential systems. We will consider the following nonlinear impulsive differential systems:

Let us subject (

When

Substituting (

The following theorem yields the solution to system (

Let

We define the successor function as

Assume that

In order to obtain the solution to nonlinear impulsive differential systems, we use the impulsive jumps of the series solutions in differential systems. This method is fundamentally different from that in [

In this section, two numerical examples are given to illustrate the theoretical results presented in the previous sections.

Consider the following linear impulsive differential system:

The origin is the center (focus) of (

Numerical simulations are presented in Figure

The center and focus of system (

Consider the following nonlinear impulsive differential system:

Assume that

Numerical simulations are displayed in Figure

The focus of system (

In this paper, we studied the center-focus problem of planar differential systems that have a state-dependent impulsion. A criterion for distinguishing between the center and the focus in linear and nonlinear impulsive differential systems was established using successor functions. Our results are potentially useful if applied to the theory of impulsive differential systems. An extension of these results to three-dimensional impulsive differential systems is an interesting topic that will be considered in future research.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank the editors and reviewers for their helpful suggestions. This research is partially supported by the National Natural Science Foundation of China (61463002) and the Yunnan provincial Natural Science Foundation of China (2012FB175).