Impulsive Control for Synchronization of Lorenz Chaotic System

Chaotic synchronization is the key technology of secure communication. In this paper, an impulsive control method for chaotic synchronization of two coupled Lorenz chaotic system was proposed. The global asymptotic synchronization of two Lorenz systems was realized by using the linear error feedback of the state variables of the drive system and the response system as impulsive control signal. Based on stability theory of impulsive differential equation, conditions were obtained to guarantee the global asymptotic synchronization of two Lorenz systems. The theory analysis and computer simulation results validated its effectiveness.


Introduction
As the key technology of secure communication, chaotic synchronization has been widely development since Pecora and Carroll [1] proposed the principle of chaos synchronization and realized it in the circuit in 1990.Several synchronization methods have been proposed so far, such as drive-response synchronization, coupling synchronization, feedback-perturb synchronization, self-adapt synchronization, impulse synchronization, and so on [2][3][4].Impulse synchronization has been widely appreciated by researcher and made some good progress [5][6][7][8].However, many of the impulsive control methods for synchronization are subject to certain restrictions.Paper [9] studied impulsive synchronization for Rössler chaotic system, and paper [10] researched impulsive control for synchronization of a class of chaotic system.In their papers the drive signal is generated by the impulsive signal and continuous signal of the system variables, so the controller is very complicated.In this paper, we use an impulsive control method, and design the controller for Lorenz chaotic system.Designed controller is simple and easy to be realized.

Problem Formulation
In this section, we study the impulsive control of Lorenz chaotic system [11] described by the following differential equation: where σ , ρ , β are system's positive real number parameters.We choose the parameters 10 σ = , 28 ρ = , 8 3

β =
, the initial condition is given by System (1) can be rewritten into the following form: ( ) where Regarding (2) as a drive system, the response system can be described as: ( , , ) T y y y y = is state variable of response system.Using linear feedback of synchronization error as impulsive signal, we can obtain the following impulsive response system: y Ay g y t t k y B y x t t y t y here

Synchronization of Lorenz Chaotic System
Theorem: Denote k β , A λ and M λ be the largest eigenvalue of ( ) ( ) V

e e e e e Ae M y x e e e Ae M y x e e A A e e M y x M y x e
e e V e λ λ λ λ On the other hand, when , ( 1, 2, ) According to the inequality ( 6) and ( 7), we can get Then the trivial solu- tion of system ( 5) is global asymptotically stable.That is, system (4) and ( 2 , where ⋅ denotes any kinds of norms of a matrix.We can obtain the following corollary: Corollary: if there exists a constant 1 then system (4) and ( 2) are global asymptotic synchronization.

Numerical Simulation
In this section, we present some numerical simulations to demonstrate our results.We choose the parameters of Lorenz system (1) as 10 , then we can calculate that 28.05A λ = . By observing the Lorenz attractor (Figure 1), we can get state variable's value ranges:  2) is (2, 0.1, 0.1) T while initial value of response system ( 4) is (0.1,5, 0.5) , T and 0.5, µ = − 0.007 τ = . We can see that synchronization error converges to zero quickly.
trajectories are shown in Figure1, which is the notable Lorenz attractor.