Synchronization Between Two Different Switched Chaotic Systems By Switching Control

Abstract. This paper is concerned with the synchronization problem of two different switched chaotic systems, considering the general case that the master-slave switched chaotic systems have uncertainties. Two basic problems are considered: one is projective synchronization of switched chaotic systems under arbitrary switching; the other is projective synchronization of switched chaotic systems by design of switching when synchronization cannot achieved by using any subsystems alone. For the two problems, common Lyapunov function method and multiple Lyapunov function method are used respectively, an adaptive control scheme has been presented, some sufficient synchronization conditions are attainted, and the switching signal is designed . Finally, the numerical simulation is provide to show the effectiveness of our method.

Most studies are assumed that the master and the slave systems are single accurate mathematical model of the chaotic system with same dynamical structures.However, it is difficult to find two exactly identical chaotic systems.On the other hand, the master and slave chaotic systems may change very quickly-even by jumping or switching.To the best of the author's knowledge, no synchronization criteria have been reported for this kind of studies.Obviously, synchronization between two different switched chaotic systems are more interesting and worth researching.For example, it can improves safety features of chaos synchronization secret communication while applying in secret communication because of the richness of selection in switched chaotic systems which achieves the variety of transmisson signal.This paper studies the projective synchronization problems between different switched chaotic systems with uncertainty.By utilizing switching control theory and adaptive control technology, an adaptive control scheme has been presented, some sufficient synchronization conditions are attainted, and the switching signal is designed based on multiple Lyapunov function method.

Problem Formulation
Consider the master-slave switched chaotic systems is described as follow ( , , ), where is the state variable of master where e y hx is the synchronization error.So the synchronization problem is now replaced by the problem of stabilizing the system (3).
Assumption 1: The nonlinear part of the system ( 1) and : , , It is easy to see that the synchronization problem turn into the stabilization problem of the system (3).

Common Lyapunov function method
In this section, we will discuss the projective synchronization between different switched chaotic systems by common Lyapunov function method.First, we design the controllers as the form of (5). 1 , , , , 2 Where r represents the dynamical estimate of 2 l in assumption (4), and The estimate ( ) r t is updated according to the following algorithm: Under control law (5) and the adaptation algorithm (6), the ith subsystem of (3) can be characterized as follow form: where e y hx , then we discuss the asymptotic stabilization of the system (7).
Theorem 1. consider the error system (7) of master system (1) and the controlled slave system (2) satisfying Assumption 1.If there exist a positive definite matrix P , a matrix i K and a positive constant O such that the following LMI 1 ( ) ( ) 0 , , is satisfied.then the error system ( 7) is global asymptotic stable with the adaptive controller(5) under arbitrary switching , that is to say, the master-slave systems achieve projective synchronization.Proof.Construct a common Lyapunov function in the form of The derivative of Lyapunov function along trajectory of each subsystem in system ( 7) is  V e (( ) ( ) ) 0.
According to the stability theory of switching system, error system ( 7) is asymptotically stable with arbitrary switching conditions, i.e. ( ) ( ) ( ) 0 e t y t hx t o as t o f .It is obviously that that the master-slave systems achieve projective synchronization, which complete the proof.

Multiple Lyapunov function method
In this section, we will study the case that none of subsystems can achieve synchronization.For this kind of condition, the common Lyapunov function could not get, so we discuss how to realize synchronization by the suitable design of switching ruler between sub chaotic systems and give a multiple Lyapunov function methos.
Then, for any i M , let construct Lyapunov function as the form of The derivative of ith Lyapunov function along the trajectory of ( 7) is ( ) From (15), it is easy to see that error system ( 7) is asymptotically stable under switching ruler (13), and the theorem 2 is proved.

Numerical Simulation
In this section we give numerical experiments on the problems of projective synchronization of master system(1) and slave system(2) to illustrate the validity of Theorems.
Let take Lorzen and Genesio system as the subsystems of master switched chaotic system and Liu and Rossler system as the subsystems of slave switched chaotic sysem.The Lorenz system is a very typical chaotic system, it is presented as x a x x x a x x x x x a x xx °® °where x is the state variable of master system, 1 , , a a a are the parameters of system, If we choose Genesio system is described as: Liu system is another typical chaotic system model, which is described as: 8 / 3, the system stands in chaos, as the initial point of system 1 y (0) 5 , 2 y (0) 6 , 3 y (0) 4 , its chaotic attractor is displayed as Figure 3.
Rossler system has plentiful chaotic features and is described by: It is easy to see that the system (17) (18)and system (19) (20)can be rewrited as the form of Eqs. ( 1) and ( 2), if we take 1 -10 10 0 28 0 0 ,

O
then we can get the positive matrix 0.8780 -0.3505 0 -0.0505 0.1812 0 , 0 0 3.5787 which satisfied the inequality (8).If the initial state of system (0) (0.1 0.3 0.5) x , (0) (5 6 4) y , (0) 8 r , and put in control as t 5 ! ,the simulation result is displayed as Figure 5. From Figure 5, we can clearly see that the error state of system reaches zero quickly with adaptive controller under arbitrary switching, which demonstrates the validity of our results.
Remark 1.In the paper, if we take 1, h then the two different switched chaotic systems will get complete synchronization with the adaptive controller under arbitrary switching, and if we take 1, h the masterslave switched chaotic systems will get antisynchronization under arbitrary switching.

CONCLUSIONS
In the paper, we present a novel method to solve the problem of projective synchronization between different switched chaotic systems, By adopting common Lyapunov function method and multiple Lyapunov function method, an adaptive controller is achieved, which realize the projective synchronization between the master-slave system with different switched chaotic systems under arbitrary switching and designed switching.Numerical simulation indicates the accuracy of the results.

Figure 5 .
Figure 5.The evolution of the error vector