Feedback Chaotic Synchronization with Disturbances

Based on Lyapunov stability theorem, a method is proposed for feedback synchronization with parameters perturbation and external disturbances. It is proved theoretically that if the perturbation and disturbances are bounded, the synchronization error can be ensured to approach to and stay within the pre-specified bound which can be arbitrarily small. Some typical chaotic systems with different types of nonlinearity, such as Lorenz system and the original Chua’s circuit, are used for detailed description. The simulation results show the feasibility of the method.


Introduction
In 1990, Pecora and Carroll presented the conception of "chaotic synchronization" and introduced a method to synchronize two identical chaotic systems with different initial conditions [1] [2].Since chaos control and synchronization have great potential applications in many areas such as information science, medicine, biology and engineering, they have received a great deal of attention.
synchronized Rössler and Chen systems via active control method [9] and Impulsive control [10].Guo et al. proposed a simple adaptive-feedback controller for chaos synchronization [11].Agrawal et al. realized the synchronization of fractional order chaotic systems using active control method [12].Norelys et al.
presented the adaptive synchronization of fractional Lorenz systems using a reduced number of control signals and parameters [13].Kajbaf et al. used sliding mode controller to obtain chaotic systems [14].Wang et al. proposed a new feedback synchronization criterion based on the largest Lyapunov exponent [15].However, most synchronization criterions were obtained under ideal circumstances.If parameters perturbation and external disturbance exist, this kind of criterions will take no effect.According to this practical problem, some solutions have been presented.For examples, Jiang et al. proposed a LMI criterion [16] for chaotic feedback synchronization.Although the simulations showed that it is robust to a random noise with zero mean, but no rigorous mathematical proof was provided and we can't determine if their method is effective for other kinds of noise.In Ref. [17], parameters perturbation was involved in their scheme.The theoretical proof and numerical simulations were given in their work, but external disturbance didn't receive attention, which made their method unilateral.Above all, these methods are effective, but still lack generality or robustness.
In this paper, we propose a practical synchronization scheme for chaotic synchronization with parameters perturbation and external disturbance.Rigorous mathematical proof is provided, and simulation results show the feasibility and robustness of our scheme.

Theory and Method
In the following scheme, a universal robust synchronization method is proposed.
In the method, synchronization will be achieved with bounded parameter disturbances and noise.
Suppose a class of ideal chaotic systems as ( ) where AX is the linear part, ( ) f X is the nonlinear part, then the system can be described as where is the external disturbance.Choose system (1) as the drive system, the relevant response system can be described as where are the relevant disturbances in the response system.We choose ( ) (n is the dimension of the chaotic system).Let the error vector = − E Y X , then the error is Set a pre-defined bound ε for the synchronization error, suppose , then system (1) and system (2) achieve approximate synchronization, the precision is ε .When ε is very small, we can consider sys- tem (1) and system (2) have been synchronized.

Choose the following Lyapunov function
According to Equation ( 3), the derivative of j e can be described as , , e a e h e g d k e ) e k e e e e e ε = = we can obtain That is to say, when the error is not within the bound ε , it will exponentially converge to zero.Hence system (1) and system (2) will achieve approximate synchronization, the precision is ε at least.

Numerical Simulations
Lorenz system and the original Chua's circuit have different types of nonlinearity.Next we will adopt the two systems for detailed description.

Taking Lorenz System as Example
Lorenz system [18] is described as Choose the following Lorenz system with parameters perturbation and external disturbances as drive system, then the relevant response system is In system (10) and system (11), , , , , , , , , ,  d d d d d )  where ( ) ( ) We have V t e e e e e e = + +     Substitute Equation ( 14) into Equation ( 17), obtain x k e is satisfied, we will obtain ( ) 0 V t <  .According to Lyapunov stability theorem, the error system (13) will converge to zero when the error is not within the bound ε , i.e. system (10) and system (11) will achieve approximate synchronization, the precision is ε at least． When the parameters perturbation and external disturbances are small, we can consider the variables of system (10) and system (11) are bounded as shown in Figure 1.Suppose the upper bounds of these disturbances and perturbation are 0.5, choose , substitute Equation ( 15) into Equation ( 18), after calculating we obtain if   , , , ,  d d d d d d ′ ′ ′ are random from −0.5 to 0.5.A time step of size 0.0001 (sec.) is employed and fourth-order Runge-Kutta method is used to solve Equation (10) and Equation (11).Let 2 shows the history of ( )

Taking the Original Chua's Circuit as Example
The original Chua's circuit [19] is described as ( ) ( ) x a y x f x y x y z z by where ( ) ( )( ) so that system (20) exhibits a chaotic behavior [19].The projections of the original Chua's circuit's attractor are shown in Figure 3. Obviously we have 4, 1, 5.5 Choose the following Chua's circuit with parameters perturbation and external disturbances ) , , , , , when the parameters perturbation and external disturbances are small, we can consider the variables of system (21) and system (22) are bounded as shown in
is satisfied, Equation (18) will be always true.In the simulation, suppose ( )

Figure 2 .
Figure 2. The history of the error (within 0.1 sec.).

Figure 3 .
Figure 3.The projections of the original Chua's circuit's attractor.
The error system is

Figure 4 .
Figure 4.The history of the error (within 0.5 sec.).
we will obtain ( ) 0 V t <  .According to Lyapunov stability theorem, the error system (24) will converge to zero when the error is not within the bound ε , i.e. system (21) and system (22) will achieve approximate synchronization.Suppose the upper bounds of these disturbances and perturbation are 0.2, choose 0.05 ε = , substitute Equation (29) into Equation (32), after calculating