An enhanced symplectic synchronization of complex chaotic systems with uncertain parameters is studied. The traditional chaos synchronizations are special cases of the enhanced symplectic synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics. The enhanced symplectic synchronization may be applied to the design of secure communication. Finally, numerical simulations results are performed to verify and illustrate the analytical results.

A synchronized mechanism that enables a system to maintain a desired dynamical behavior (the goal or target) even when intrinsically chaotic has many applications ranging from biology to engineering [

The symplectic chaos synchronization concept [

As numerical examples, we select hyperchaotic Chen system [

This paper is organized as follows. In Section

There are two different nonlinear chaotic systems. The partner

The partner

After a controller

where

Our goal is to design the controller

From (

Using (

A positive definite Lyapunov function

Its derivative along any solution of (

Note that

To further illustrate the effectiveness of the controller, we select hyperchaotic Chen system and hyperchaotic Lorenz system as the master system and the slave system, respectively. Consider

The controllers

The initial values of the states of the Chen system and of the Lorenz system are taken as

We take

Choose a positive definite Lyapunov function as

According to (

Projections of phase portrait for master system (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors,

The master Chen system with uncertain variable parameters is

We take

Equation (

Choose a positive definite Lyapunov function as

Projections of the phase portrait for chaotic system (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors,

We take

Choose a positive definite Lyapunov function as

According to (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors,

We achieve the novel enhanced symplectic synchronization of a Chen system with uncertain parameters, and a Lorenz system is obtained by the Lyapunov asymptotical stability theorem. All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the three cases of proposed synchronization schemes. The enhanced symplectic synchronization of chaotic systems with uncertain parameters can be used to increase the security of secret communication.

This research was supported by the National Science Council, Taiwan, under Grant no. 98-2218-E-011-010.