Abstract and Applied Analysis

Volume 2013 (2013), Article ID 193138, 12 pages

http://dx.doi.org/10.1155/2013/193138

Research Article

Enhanced Symplectic Synchronization between Two Different Complex Chaotic Systems with Uncertain Parameters

Cheng-Hsiung Yang

Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, 43 Section 4, Keelung Road, Taipei 106, Taiwan

Received 12 October 2012; Accepted 13 April 2013

Academic Editor: Haydar Akca

Copyright © 2013 Cheng-Hsiung Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1–4]. Thus, it is of considerable interest and potential utility to devise control techniques capable of achieving the desired type of behavior in nonlinear and chaotic systems. Many approaches have been presented for the synchronization of chaotic systems [5–10]. There are a chaotic master system and either an identical or a different slave system. Our goal is the synchronization of the chaotic master and the chaotic slave by coupling or by other methods.

The symplectic chaos synchronization concept [11]

is studied, where , are the state vectors of the master system and of the slave system, respectively, and is a given function of time in different form. The may be a regular motion function or a chaotic motion function. When and , (1) reduces to the generalized chaos synchronization and the traditional chaos synchronization given in [1–3], respectively. In this paper, a new enhance symplectic chaos synchronization:

As numerical examples, we select hyperchaotic Chen system [12] and hyperchaotic Lorenz system [13] as the master system and the slave system, respectively.

This paper is organized as follows. In Section 2, by the Lyapunov asymptotical stability theorem, a symplectic synchronization scheme is given. In Section 3, various feedbacks of nonlinear controllers are designed for the enhanced symplectic synchronization of a hyperchaotic Chen system with uncertain parameters and a hyperchaotic Lorenz system. Numerical simulations are also given in Section 3. Finally, some concluding remarks are given in Section 4.

2. Enhanced Symplectic Synchronization Scheme

There are two different nonlinear chaotic systems. The partner controls the partner partially. 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 a vector of uncertain coefficients in , and is a vector function different from .

After a controller is added, partner becomes

where is the control vector.

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 a regular or a chaotic function. Define error vector :

is demanded.

From (5), it is obtained that

where .

Using (3), (4a), and (4b), (7) can be rewritten as

Proof. A positive definite Lyapunov function is chosen [14, 15] as

Its derivative along any solution of (8) is

In (10), the is designed so that , where is a diagonal negative definite matrix. The is a negative definite function of .

Remark 1. Note that approaches zero when time approaches infinitly, according to Lyapunov theorem of asymptotical stability. The enhanced symplectic synchronization is obtained [12, 13, 16–19].

3. Numerical Results for the Enhanced Symplectic Chaos Synchronization of Chen System with Uncertain Parameters and Hyperchaotic Lorenz System

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

where , and are parameters. The parameters of master system and slave system are chosen as , and .

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

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

Case 1 (a symplectic synchronization). We take ,, and . They are chaotic functions of time. are given. By (6), we have

From (7), we have

Equation (8) can be expressed as

where , , , and .

Choose a positive definite Lyapunov function as

Its time derivative along any solution of (16) is

According to (10), we get the controller

Equation (18) becomes

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of the Chen system and the Lorenz system is achieved. The numerical results are shown in Figures 1, 2, and 3. After 1 second, the motion trajectories enter a chaotic attractor.

(a)

(b)

(c)

(d)

Figure 1: Projections of phase portrait for master system (11) and slave system (12).

(a)

(b)

(c)

(d)

Figure 2: Projections of the phase portrait for chaotic system (13) of Case 1.

(a)

(b)

(c)

(d)

(e)

Figure 3: Time histories of states, state errors, , and for Case 1.

Case 2 (a symplectic synchronization with uncertain parameters). The master Chen system with uncertain variable parameters is

where , and are uncertain parameters. In simulation, we take

where , and are constants. Take ,  ,  ,  ,  ,  ,  ,  , and . So, (21) is chaotic system, shown in Figure 4.

We take ,, and . They are chaotic functions of time. are given. By (6), we have

From (7), we have

Equation (8) can be expressed as

where , , , and .

Choose a positive definite Lyapunov function as

Its time derivative along any solution of (25) is

According to (10), we get the controller

Equation (27) becomes

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of the Chen system with uncertain parameters and the Lorenz system is achieved. The numerical results are shown in Figures 5 and 6. After 1 second, the motion trajectories enter a chaotic attractor.

(a)

(b)

(c)

(d)

Figure 4: Projections of the phase portrait for chaotic system (21).

(a)

(b)

(c)

(d)

Figure 5: Projections of the phase portrait for chaotic system (13) of Case 2.

(a)

(b)

(c)

(d)

(e)

Figure 6: Time histories of states, state errors, , and for Case 2.

Case 3 (an enhanced symplectic synchronization with uncertain parameters). We take ,, and . They are chaotic functions of time. are given. The value is 0.0001. By (6), we have

From (7) we have

Equation (8) can be expressed as

where , , , and .

Choose a positive definite Lyapunov function as

Its time derivative along any solution of (32) is

According to (10), we get the controller

Equation (34) becomes

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The enhanced symplectic synchronization of the Chen system with uncertain parameters and the Lorenz system is achieved. The numerical results are shown in Figures 7 and 8. After 1 second, the motion trajectories enter a chaotic attractor.

(a)

(b)

(c)

(d)

Figure 7: Projections of the phase portrait for chaotic system (13) of Case 3.

(a)

(b)

(c)

(d)

(e)

Figure 8: Time histories of states, state errors, , and for Case 3.

4. Conclusions

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.

Acknowledgment

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

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