The impulsive control and synchronization of unified chaotic system are proposed. By applying impulsive control theory and introducing a piecewise continuous auxiliary function, some novel and useful conditions are provided to guarantee the globally asymptotical stabilization and synchronization of unified chaotic system under impulsive control. Compared with some previous results, our criteria are superior and less conservative. Finally, the effectiveness of theoretical results is shown through numerical simulations.

In 1963, Lorenz presented the first chaotic attractor in a simple three-dimensional autonomous system [

For a long time, people thought that chaos was neither predictable nor controllable. However, in the 1990s, the viewpoint and theory about control of chaotic system introduced by Ott. Grebogi Yorke and Pecora Carroll completely changed the situation [

In the past several years, impulsive control strategy has been widely used to stabilize and synchronize nonlinear dynamical systems. From the control point of view, impulsive control is an effective method based on the theory of impulsive dynamic systems [

In this paper, based on impulsive control theory, the stabilization and synchronization of unified chaotic system are investigated via designing impulsive controller. By introducing a piecewise continuous function, some new and useful criteria are established to guarantee the stabilization and synchronization of unified chaotic system. Compared with the main criteria obtained in [

This paper is organized as follows. In Section

In this section, two central lemmas are provided to support our analysis in later sections.

Assume that

Since matrix

Assume that

For an arbitrary column vector

Similarly, we can prove

The proof is completed.

Consider the unified chaotic system [

It is easy to see that system (

For convenience, denote the above equilibrium points as

In order to stabilize the equilibrium point, we now introduce a control model of system (

Integrating from

For convenience, denote

The following theorem is provided to ensure the globally asymptotical stability of the origin for the impulsive control system (

Let

For all

For all

Firstly, construct an auxiliary function described by the following form:

In the following, we consider the following cases.

From the above analysis, the trivial solution

From Theorem

If

Taking

The origin of the controlled unified chaotic system (

For all

For all

In Corollary

In practice, the equal impulsive interval and the constant gain matrix are often selected for convenience. In Corollary

The origin of system (

From Corollary

In this section, we discuss the impulsive synchronization of unified chaotic system. It is easy to see that system (

Similar to the technique of system (

In this section, we are interested in synchronizing the driving system (

Let

For all

For all

Observe that error system (

Similar to Corollary

The driving system (

For all

For all

The impulsive synchronization of unified chaotic system was investigated in [

Similar to Corollary

The driving system (

In this section, based on the results obtained in the previous sections, some numerical simulations are represented to show the effectiveness of our results.

First, we consider numerical simulations of impulsive control for system (

The chaotic behavior of system (

For convenience, select (

Choosing impulsive control interval

Impulsive stabilization of system (

In the second simulation, we study impulsive synchronization of the driving system (

The chaotic behavior of system (

Choose

Synchronization error curves with

Evidently, it follows from (

In this paper, the issue on the stabilization and synchronization of unified chaotic system under impulsive control is investigated. Some novel and useful criteria are derived by using impulsive control theory. Finally, the effectiveness and feasibility of the developed methods are shown by some numerical simulations. Compared with some previous results given in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Natural Science Foundation of Xinjiang (Grant no. 2013211B06), the Project funded by China Postdoctoral Science Foundation (Grant no. 2013M540782), the Natural Science Foundation of Xinjiang University (Grant no. BS120101), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20136501120001), and the National Natural Science Foundation of China (Grant no. 61164004).