#### Abstract

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.

#### 1. Introduction

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 [1–6]. There exists a functional relationship between the states of the master system and those of the slave system. The symplectic chaos synchronization concept [7]: is studied, where , are the state vectors of the master system and of the slave system, respectively, is a given function of time in different form. The may be a regular function or a chaotic function. When and of (1) reduces to the traditional generalized chaos synchronization and the traditional chaos synchronization given in [8–10], respectively.

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 [11]. This system is similar to the celebrated Lorenz equation but more complex than it due to the introduction of a new feedback control and a new state variable and thus is called a Lorenz-Stenflo system after the names of Lorenz and Stenflo [12]. The nonlinear dynamical behaviors of the Lorenz-Stenflo system have been investigated in [13–16].

This paper is organized as follows. In Section 2, by the Lyapunov asymptotic stability theorem, the symplectic chaos synchronization with uncertain chaotic parameters by adaptive control scheme is given. In Section 3, various adaptive controllers and update laws are designed for the symplectic chaos synchronization of the identical Lorenz-Stenflo systems. Numerical simulations are also given in Section 3. Finally, some concluding remarks are given in Section 4.

#### 2. Symplectic Chaos Synchronization with Chaotic Parameters by Adaptive Control Scheme

There are two identical nonlinear dynamical systems, and the partner controls the partner . The partner is given by where is a state vector, is a vector of uncertain coefficients in , and is a vector function.

The partner is given by where is a state vector, is an estimated vector of uncertain coefficients in .

So a controller is added on partner , and the partner becomes where is the control vector function.

Our goal is to design the controller so that the state vector of the partner asymptotically approaches , a given function plus a given vector function which is the regular function or the chaotic function.

To define error vector : is demanded.

From (4), it is obtained that where .

By (2), (3a), and (3b), (6) can be rewritten as

A positive definite Lyapunov function is chosen: where .

Its derivative along any solution of (7) is In (9), the and are designed so that where and are two diagonal negative definite matrices. The is a negative definite function of and . By Lyapunov theorem of asymptotical stability

The symplectic chaos synchronization with uncertain chaotic parameters is obtained [3–7, 11, 17–23].

#### 3. Numerical Results for the Symplectic Chaos Synchronization of Lorenz-Stenflo System with Uncertain Chaotic Parameters via Adaptive Control

The master Lorenz-Stenflo system can be described as [11] And the slave Lorenz-Stenflo system can be described as where , , , , and are called the first Prandtl number, geometric parameter, second Prandtl number, Rayleigh number, and rotation number, respectively. The parameters of Lorenz-Stenflo system are chosen as , , , , and .

The controllers, , , , and , are added to the four equations of (12), respectively:

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

*Case 1. *A time delay symplectic synchronization.

We take , , , and . They are chaotic functions of time, where time delay sec. The are given. By (5) we have
From (14) we have
where .

Equation (15) can be expressed as where

Choose a positive definite Lyapunov function: where , , , , , and , , , , and are estimates of uncertain parameters a, b, c, d, and r, respectively.

Its time derivative along any solution of (16) is

The adaptive controllers are chosen as and the update laws are chosen as The initial values of estimate for uncertain parameters are .

Equation (19) becomes which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The time delay symplectic synchronization of the identical Lorenz-Stenflo systems is achieved. The numerical results of the phase portrait of master system, chaotic system (13), the time series of states, state errors, and parameter differences are shown in Figures 1, 2, and 3, respectively. The symplectic chaos synchronization is accomplished using adaptive control method.

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*Case 2. *A time delay symplectic synchronization with uncertain chaotic parameters.

The Lorenz-Stenflo master system with uncertain chaotic parameters is
where , , , , and are uncertain chaotic parameters. In simulation, we take
where , , , , and are positive constants. Chosen are , , , , and . The system (23) is chaotic dynamic motion, shown in Figure 4.

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The is chaotic system, and the chaotic signal of goal system can be described as where initial conditions of the chaotic signal of system are (0) = 2, (0) = 5, (0) = −4, and (0) = −6. The are given. By (5) we have From (26) we have Equation (27) can be expressed as where

Choose a positive definite Lyapunov function: where , , , , , and , , , , and are estimates of uncertain parameters , , , , and , respectively.

Its time derivative along any solution of (28) is

The adaptive controllers are chosen as and the update laws are chosen as The initial values of estimate for uncertain parameters are .

Equation (31) becomes which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The time delay symplectic synchronization with uncertain chaotic parameters of the identical Lorenz-Stenflo systems is achieved. The numerical results of the phase portrait of chaotic system (13), the time series of states, state errors, parameters, and parameter differences are shown in Figures 5, 6, and 7, respectively. The symplectic chaos synchronization is accomplished using adaptive control method.

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*Case 3. *A multitime delay symplectic synchronization with uncertain chaotic parameters.

We take , , , and . They are chaotic functions of time, where multi time delay , , , and are positive constants, = 0.5 sec, sec, sec, and sec. The are given. By (5) we have
From (35) we have
Equation (36) can be expressed as
where

Choose a positive definite Lyapunov function:
where , , , , , and , , , , and are estimates of uncertain parameters , , , , and , respectively.

Its time derivative along any solution of (37) is

The adaptive controllers are chosen as and the update laws are chosen as The initial values of estimate for uncertain parameters are .

Equation (40) becomes which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The multi time delay symplectic synchronization with uncertain chaotic parameters of the identical Lorenz-Stenflo systems is achieved. The numerical results of the phase portrait of chaotic system (13), the time series of states, state errors, and parameter differences are shown in Figures 8 and 9, respectively. The symplectic chaos synchronization is accomplished using adaptive control method.

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#### 4. Conclusions

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.

#### Acknowledgment

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