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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.

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 [

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.

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

where

where

Set a pre-defined bound

Choose the following Lyapunov function

According to Equation (3), the derivative of

If

we can obtain

That is to say, when the error is not within the bound

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

Lorenz system [

In the paper choose

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),

Then

Hence

where

Choose Lyapunov function

We have

Substitute Equation (14) into Equation (17), obtain

If

is satisfied, we will obtain

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

is satisfied, Equation (18) will be always true.

In the simulation, suppose

The original Chua’s circuit [

where

Choose the following Chua’s circuit with parameters perturbation and external disturbances

As drive system, where

where

Then

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

Because

we have

Hence

where

and

Choose Lyapunov function

We have

Substitute Equation (28) into Equation (31), obtain

If

is satisfied, we will obtain

Suppose the upper bounds of these disturbances and perturbation are 0.2, choose

is satisfied, Equation (32) will be always true.

In the above simulation, let

In this paper, a practical scheme is proposed for feedback synchronization with parameters perturbation and external disturbances. Lorenz system and the original Chua’s circuit are used for detailed description. The simulation results show the feasibility of the method. According to Ref. [

The work was supported by Natural Science Foundation of Liaoning Province (No. 201602034).

Wang, M.J., Yu, W.B. and Zhao, J. (2017) Feedback Chaotic Synchronization with Disturbances. International Journal of Modern Nonlinear Theo- ry and Application, 6, 1-10. http://dx.doi.org/10.4236/ijmnta.2017.61001