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In this paper, anti-synchronization of two different chaotic systems is investigated. On the basis of Lyapunov theory, adaptive control scheme is proposed when system parameters are unknown or uncertain; sufficient conditions for the stability of the error dynamics are derived, where the controllers are designed by using the sum of the state variables in chaotic systems. Numerical simulations are performed for the Chen system and Genesio system to demonstrate the effectiveness of the proposed control strategy.

Since the pioneering work by Pecora and Carroll [

The layout of the rest of the paper is as follows. Section 2 describes the systems and their mathematical models; in Section 3, adaptive anti-synchronization between Chen and Genesio systems is presented; numerical simulation results are given for illustration and verification. Finally, conclusions are drawn in Section 4.

Consider nonlinear chaotic system as follows.

where

where

The Genesio system, proposed by Genesio and Tesi [

where

Chen system is described by

When

In the next section, we will study chaos anti-synchronization between Chen and Genesio systems with known or unknown parameters using adaptive control.

We assume that Genesio system (2) is the drive system, and the controlled Chen system (4) is the response system.

We add (2) from Equation (4) and yield

Our goal is to find the proper controllers

where

Theorem: If the controllers are chosen as

and the update laws of parameters are chosen as

Then system (4) globally anti-synchronizes system (2) asymptotically, where

Proof: Applying control laws (6) to (5) yields the resulting error dynamics as follows.

where

Consider the following Lyapunov function

The time derivative of V along the solution of error dynamical system (8) gives that

where

Since

where

In simulation, Fourth order Runge-Kutta integration method is used to solve the systems of differential Equations (2) and (4) with the controllers (6) and the parameter update laws (7). We select the parameters of Genesio system as

In this paper, chaos anti-synchronization between two different chaotic systems with different structures using adaptive control is presented. Chen system and Genesio system are taken as an illustrative example to verify the effectiveness of the proposed methods.

Xianyong Wu,Hao Wu,Zhengrong Yan,Yuehua Huang, (2016) Chaos Anti-Synchronization between Chen System and Genesio System. Open Access Library Journal,03,1-5. doi: 10.4236/oalib.1102856