This paper considers impulsive control for the synchronization of chaotic systems with time delays. Based on the Lyapunov functions and the Razumikhin technique, some new synchronization criteria with an exponential convergence rate are derived. Our results show that impulses do contribute to globally exponential synchronization of dynamical systems. Besides, the impulsive moments are independent of the upper bound of time delays. Furthermore, a bigger upper bound of impulsive intervals for the synchronization of chaotic systems can be obtained when compared with many previous studies. Hence, our results are less conservative and more effective for the synchronization analysis. A numerical example is given to show the validity and potential of the developed results.
In the last two decades, control and synchronization problems of chaotic systems have been extensively studied, due to their potential applications in many areas. For instance, they are used to understand self-organizational behavior in the brain as well as in ecological systems, and they also have been applied to produce secure message communication between a sender and a receiver. So far, different synchronization techniques have been proposed and implemented in practice, such as PC (Pecora and Carroll) method [
Impulsive phenomena exist in many biological systems and mechanics fields. Impulsive control has been widely studied and gradually become an interesting and useful synchronization approach [
In this paper, we will investigate globally exponential synchronization of coupled chaotic systems with time delay by using impulsive control. The main contributions of this paper include the following. (i) Our results show that impulses do contribute to globally exponential synchronization of dynamical systems and the time delays in systems not to be smaller than the length of impulsive intervals. Therefore, they can be usually used as an effective control strategy to synchronize the underlying delayed dynamical systems in more practical application. (ii) A new approach for the exponential synchronization of impulsive chaotic systems with any finite delay is given. (iii) In order to deal with uncertainty and/or measurement noise effectively in nominal case for identical chaotic and hyperchaotic systems, the impulse distance should increases which in turn decreases the control cost in accordance with [
The rest of this paper is organized as follows. In Section
Denote that
Now we introduce the coupled chaotic systems that will be studied. It usually consists of two chaotic systems at the transmitter and the receiver ends.
At the transmitter end, we have
Then, we can get the following error dynamical system:
Obviously,
For a given
We assume that (
The trivial solution of (
Given a function
In the following, we shall address the exponential stability problem for impulsive delayed nonlinear differential equation (
Let
Then, the trivial solution of system (
Let
and so
As for any
If
In LMI (
Compared with Theorem 3.1 in [
Similarly to Theorem
Let
Then, the trivial solution of system (
Condition (ii) in Theorem
As Corollary
Since the amount of transmitted information decreases leading to reduced control cost, the cost of impulsive synchronization of chaotic systems is closely related to the impulse distances and having a minimum level of synchronization error with the largest impulsive intervals as generally a desired [
In this section, an example is presented here to illustrate our main results.
Consider the Lorenz system in [
Solving (
A comparison of the upper bound of impulsive intervals of (
Corollary |
[ |
[ |
[ |
| |||
0.0396 | 0.0072 | 0.0108 | 0.0178 |
The trajectories of the error Lorenz system under impulsive controller with
In this paper, we have investigated the synchronization of coupled chaotic systems with time delay by using impulsive control. The time delays in systems need not to be smaller than the length of impulsive interval as many works existing in the literature assumed. A numerical example has been given to demonstrate the effectiveness of the theoretical results, and the estimation of the stable region of the impulsive intervals has also been presented. By comparison, the upper bound of impulsive intervals is greater than it was in some existing results. The obtained results can be easily used to control many systems, especially to stabilize and synchronize chaotic systems.
The authors would like to take this opportunity to thank Professor. Jinde Cao and the reviewers for their constructive comments and useful suggestions. This work was partially supported by the NNSF of China (Grants nos. 61175119, 11101373, 11271333, and 61074011), the NSF of Jiangsu Province of China (Grant no. BK2010408), Huo Ying-Dong Education Foundation (Grant no. 132037), and Zhejiang Innovation Project, China (Grant no. T200905).