A new symplectic chaos synchronization of chaotic systems with uncertain chaotic parameters is studied. The traditional chaos synchronizations are special cases of the symplectic chaos synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics and a parameter difference. The symplectic chaos synchronization with uncertain chaotic parameters may be applied to the design of secure communication systems. Finally, numerical results are studied for symplectic chaos synchronized from two identical Lorenz-Stenflo systems in three different cases.

Chaos has been detected in a large number of nonlinear dynamic systems of physical characteristics. In addition to the control and stabilization of chaos, chaos synchronization systems are a fascinating concept which has received considerable interest among nonlinear scientists in recent times. However, chaos is desirable in some systems, such as convective heat transfer, liquid mixing, encryption, power converters, secure communications, biological systems and chemical reactions. There are a chaotic master system (or driver) and either an identical or a different slave system (or responser). Our goal is the synchronization of the chaotic master system and the chaotic slave system by coupling or by other methods. In practice, some or all of the parameters of chaotic system parameters are uncertain. A lot of works have been preceded to solve this problem using adaptive control concept.

Among many kinds of chaos synchronization, generalized chaos synchronization is investigated [

As numerical examples, in 1996, Stenflo originally used a four-dimensional autonomous chaotic system to describe the low-frequency short-wavelength gravity wave disturbance in the atmosphere [

This paper is organized as follows. In Section

There are two identical nonlinear dynamical systems, and the partner

The partner

So a controller

Our goal is to design the controller

To define error vector

From (

By (

A positive definite Lyapunov function

Its derivative along any solution of (

The symplectic chaos synchronization with uncertain chaotic parameters is obtained [

The master Lorenz-Stenflo system can be described as [

The controllers,

The initial values of the states of the master system and of the slave system are taken as

A time delay symplectic synchronization.

We take

Equation (

Choose a positive definite Lyapunov function:

Its time derivative along any solution of (

The adaptive controllers are chosen as

Equation (

Projections of phase portrait for master system (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors, parameter differences,

A time delay symplectic synchronization with uncertain chaotic parameters.

The Lorenz-Stenflo master system with uncertain chaotic parameters is

Projections of the phase portrait for chaotic system (

The

Choose a positive definite Lyapunov function:

Its time derivative along any solution of (

The adaptive controllers are chosen as

Equation (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors,

Time histories of

A multitime delay symplectic synchronization with uncertain chaotic parameters.

We take

Choose a positive definite Lyapunov function:

Its time derivative along any solution of (

The adaptive controllers are chosen as

Equation (

Projections of the phase portrait for chaotic system (

Time histories of states, state errors, parameter differences,

A novel symplectic synchronization of a Lorenz-Stenflo system with uncertain chaotic parameters is obtained by the Lyapunov asymptotical stability theorem. The simulation results of three cases are shown in corresponding figures which imply that the adaptive controllers and update laws we designed are feasible and effective. The symplectic synchronization of chaotic systems with uncertain chaotic parameters via adaptive control concept can be used to increase the security of secret communication system.

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