A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finite-time stability theory, step-by-step control and nonlinear control method, a suitable controller is designed to achieve finite-time combination-combination synchronization among four hyperchaotic systems. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique.
As a new subject in 1980s, chaos almost covers all the fields of science. It is known that chaos is an interesting nonlinear phenomenon which may lead to irregularity and unpredictability in the dynamic system, and it has been intensively studied in the last three decades. Since Pecora and Carroll proposed the PC method to synchronize two chaotic systems in 1990 [
Most of the aforementioned works are based on the synchronization scheme which consists of one drive system and one response system and can be seen as one-to-one system. However, we found it not secure and flexible enough in many real world applications, for instance, in secure communication. Recently, Runzi et al. presented a new type of synchronization with two drive systems and one response system [
Notice that the mentioned literatures mainly investigated the asymptotic synchronization of chaotic systems. However, in the view of practical application, optimizing the synchronization time is more important than achieving synchronization asymptotically [
Motivated by the previous discussion, this paper aims to study the finite-time synchronization between a combination of two drive systems and a combination of two response systems in drive-response synchronization scheme. We have applied the finite-time stability theory to our analysis to achieve finite-time combination-combination synchronization. The step-by-step control method and nonlinear control technique are adopted to synchronize four different hyperchaotic systems. Numerical simulations are presented to verify the theoretical findings.
Consider the drive systems and response systems as follows:
If there exist four constant matrices
Assume that a continuous, positive-definite function
Thus, for any initial value
Consider two identity hyperchaotic Chen systems as the drive systems [
Consider two identity hyperchaotic Lorenz systems as the response systems [
Without loss of generality, we choose
The objective of the synchronization scheme is to design a suitable controller
Let
Thus, we can get the error system as follows:
Our aim is to design a suitable controller, such that the drive systems (
The design plan and its steps are as follows.
Choose
is a proper rational number, and
Substituting
Choose a candidate Lyapunov function
Thus, the derivative of
According to Lemma
Choose
For
Choose a candidate Lyapunov function
Thus, the derivative of
According to Lemma
Choose
For
Choose a candidate Lyapunov function
Thus, the derivative of
According to Lemma
Choose
For
Choose a candidate Lyapunov function
Thus, the derivative of
According to Lemma
The controller is designed as follows:
According to what we discussed previously, we can obtain this conclusion that the error system (
To verify the effectiveness of the proposed finite-time synchronization method, we consider the hyperchaotic Chen system with the parameters
Hyperchaotic attractors of the hyperchaotic Chen system with
Consider the hyperchaotic Lorenz system with the parameters
Hyperchaotic attractors of the hyperchaotic Lorenz system with
In the following simulation, we assume
The errors between drive systems (
The error vector is achieved to zero which implies that systems (
In this paper, the problem of finite-time combination-combination synchronization with two drive systems and two response systems was investigated. Based on the finite-time stability theory, the step-by-step control and nonlinear control approach, a suitable controller was introduced. The simulation results demonstrated that the proposed controller works well for synchronizing four hyperchaotic systems in finite time.
The authors would like to thank the Editor and the referees for their valuable comments and suggestions that helped to improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 61074012) and the Natural Science Foundation of Fujian Province (Grant no. 2011J01025).