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

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] 𝑦=𝐻𝑡,𝑥,𝑦+𝐹(𝑡) (1) is studied, where 𝑥, 𝑦 are the state vectors of the master system and of the slave system, respectively, and 𝐹(𝑡) is a given function of time in different form. The 𝐹(𝑡) may be a regular motion function or a chaotic motion function. When 𝐻(𝑡,𝑥,𝑦)+𝐹(𝑡)=𝑥 and 𝐻(𝑡,𝑥,𝑦)=𝑥, (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: 𝑦=𝐻𝑡,𝑥̇,𝑦,̇𝑥,𝑦+𝐹(𝑡). (2)

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 𝐴 controls the partner 𝐵 partially. The partner 𝐴 is given by 𝑥̇=𝑓(𝑡,𝑥,𝐴(𝑡)), (3) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 is a state vector, 𝐴(𝑡)=[𝐴₁(𝑡),𝐴₂(𝑡),…,𝐴_𝑀(𝑡)]^𝑇∈𝑅^𝑀 is a vector of uncertain coefficients in 𝑓, and 𝑓 is a vector function.

The partner 𝐵 is given by𝑦̇=𝑔𝑡,𝑦,𝐵(𝑡), (4a) where 𝑦=[𝑦₁,𝑦₂,…,𝑦_𝑛]^𝑇∈𝑅^𝑛 is a state vector, 𝐵(𝑡)=[𝐵₁(𝑡),𝐵₂(𝑡),…,𝐵_𝑚(𝑡)]^𝑇∈𝑅^𝑚 is a vector of uncertain coefficients in 𝑔, and 𝑔 is a vector function different from 𝑓.

After a controller 𝑢(𝑡) is added, partner 𝐵 becomes 𝑦̇=𝑔𝑡,𝑦,𝐵(𝑡)+𝑢(𝑡), (4b)

where 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the control vector.

Our goal is to design the controller 𝑢(𝑡) so that the state vector 𝑦 of the partner 𝐵 asymptotically approaches 𝐻(𝑡,𝑥̇,𝑦,̇𝑥,𝑦)+𝐹(𝑡), a given function 𝐻(𝑡,𝑥̇,𝑦,̇𝑥,𝑦) plus a given vector function 𝐹(𝑡)=[𝐹₁(𝑡),𝐹₂(𝑡),…,𝐹_𝑛(𝑡)]^𝑇 which is a regular or a chaotic function. Define error vector 𝑒(𝑡)=[𝑒₁,𝑒₂,…,𝑒_𝑛]^𝑇: 𝑒=𝐻𝑡,𝑥̇,𝑦,̇𝑥,𝑦−𝑦+𝐹(𝑡),lim_𝑡→∞𝑒=0 (5)(6) is demanded.

From (5), it is obtained that 𝑒̇=𝜕𝐻𝜕𝑡+∇𝐻^𝑇Ψ̇−𝑦̇+𝐹̇(𝑡), (7) where Ψ̇=[𝑥̈𝑦̈𝑥̇𝑦]̇^𝑇.

Using (3), (4a), and (4b), (7) can be rewritten as 𝑒̇=𝜕𝐻+𝜕𝑡𝜕𝐻𝜕𝑥̇𝑥̈+𝜕𝐻𝜕𝑦̇𝑦̈+𝜕𝐻𝑓((𝑡+𝜕𝑥𝑡,𝑥,𝐴))𝜕𝐻𝑔(𝑡)(𝑡)(𝑡)+𝐹̇(𝑡).𝜕𝑦𝑡,𝑦,𝐵−𝑔𝑡,𝑦,𝐵−𝑢 (8)

Proof. A positive definite Lyapunov function 𝑉(𝑒) is chosen [14, 15] as 𝑉(𝑒)=12𝑒^𝑇𝑒. (9)
Its derivative along any solution of (8) is 𝑉̇(𝑒)=𝑒^𝑇𝜕𝐻+𝜕𝑡𝜕𝐻𝜕𝑥̇𝑥̈+𝜕𝐻𝜕𝑦̇𝑦̈+𝜕𝐻((𝑡+𝜕𝑥×𝑓𝑡,𝑥,𝐴))𝜕𝐻𝑔(𝑡)(𝑡)+𝐹̇(𝑡)(𝑡.𝜕𝑦𝑡,𝑦,𝐵−𝑔𝑡,𝑦,𝐵−𝑢) (10) In (10), the 𝑢(𝑡) is designed so that 𝑉̇(𝑒)=𝑒^𝑇𝐶_𝑛×𝑛𝑒, where 𝐶_𝑛×𝑛 is a diagonal negative definite matrix. The 𝑉̇ is a negative definite function of 𝑒.

Remark 1. Note that 𝑒 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 𝑥̇₁=𝑎𝑥₂−𝑥₁+𝑥₄,𝑥̇₂=𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝑏𝑥₃+𝑥₁𝑥₂,𝑥̇₄=𝑟𝑥₄+𝑥₂𝑥₃,𝑦̇₁=𝑎₁𝑦₂−𝑦₁+𝑦₄,𝑦̇₂=𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃,𝑦̇₃=−𝑐₁𝑦₃+𝑦₁𝑦₂,𝑦̇₄=𝑑₁𝑦₄−𝑦₁𝑦₃, (11)(12) where 𝑎,𝑏,𝑐,𝑑,𝑟,𝑎₁,𝑏₁,𝑐₁, and 𝑑₁ are parameters. The parameters of master system and slave system are chosen as 𝑎=31,𝑏=3.5,𝑐=11,𝑑=7.7,𝑟=0.1,𝑎₁=11,𝑏₁=28,𝑐₁=2.8, and 𝑑₁=1.2.

The controllers 𝑢₁,𝑢₂,𝑢₃, and 𝑢₄ are added to the four equations of (12), respectively as follows: 𝑦̇₁=𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑢₁,𝑦̇₂=𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑢₂,𝑦̇₃=−𝑐₁𝑦₃+𝑦₁𝑦₂+𝑢₃,𝑦̇₄=𝑑₁𝑦₄−𝑦₁𝑦₃+𝑢₄. (13)

The initial values of the states of the Chen system and of the Lorenz system are taken as 𝑥₁(0)=11,  𝑥₂(0)=13,   𝑥₃(0)=12,  𝑥₄(0)=12,  𝑦₁(0)=−11,  𝑦₂(0)=−13,  𝑦₃(0)=−12, and 𝑦₄(0)=−12.

Case 1 (a symplectic synchronization). We take 𝐹₁(𝑡)=𝑥³₄(𝑡),𝐹₂(𝑡)=𝑥³₁(𝑡),𝐹₃(𝑡)=𝑥³₂(𝑡), and 𝐹₄(𝑡)=𝑥³₃(𝑡). They are chaotic functions of time. 𝐻_𝑖(𝑥,𝑦,𝑡)=−𝑥²_𝑖𝑦_𝑖(𝑖=1,2,3,4) are given. By (6), we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞−𝑥²_𝑖𝑦_𝑖−𝑦_𝑖+𝑥³_𝑗=0,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (14) From (7), we have 𝑒_𝑖̇=−2𝑥̇_𝑖𝑥_𝑖𝑦_𝑖−𝑥²_𝑖𝑦̇_𝑖−𝑦̇_𝑖+3𝑥̇_𝑗𝑥²_𝑗,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (15) Equation (8) can be expressed as 𝑒₁̇=−2𝑦₁𝑥₁𝑎𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄−𝑢₁+3𝑥²₄𝑟𝑥₄+𝑥₂𝑥₃,𝑒₂̇=−2𝑦₂𝑥₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃−𝑢₂+3𝑥²₁𝑎𝑥₂−𝑥₁+𝑥₄,𝑒₃̇=−2𝑦₃𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂−𝑢₃+3𝑥²₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃,𝑒₄̇=−2𝑦₄𝑥₄𝑟𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃−𝑢₄+3𝑥²₃−𝑏𝑥₃+𝑥₁𝑥₂, (16) where 𝑒₁=−𝑥²₁𝑦₁−𝑦₁+𝑥³₄, 𝑒₂=−𝑥²₂𝑦₂−𝑦₂+𝑥³₁, 𝑒₃=−𝑥²₃𝑦₃−𝑦₃+𝑥³₂, and 𝑒₄=−𝑥²₄𝑦₄−𝑦₄+𝑥³₃.
Choose a positive definite Lyapunov function as 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄. (17) Its time derivative along any solution of (16) is 𝑉̇=𝑒₁−2𝑦₁𝑥₁𝑎𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄−𝑢₁+3𝑥²₄𝑟𝑥₄+𝑥₂𝑥₃+𝑒₂−2𝑦₂𝑥₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃−𝑢₂+3𝑥²₁𝑎𝑥₂−𝑥₁+𝑥₄+𝑒₃−2𝑦₃𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂−𝑢₃+3𝑥²₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃+𝑒₄−2𝑦₄𝑥₄𝑟𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃−𝑢₄+3𝑥²₃−𝑏𝑥₃+𝑥₁𝑥₂. (18)
According to (10), we get the controller 𝑢₁=−2𝑦₁𝑥₁𝑎𝑥₂−𝑥₁+𝑥₄+3𝑥²₄𝑟𝑥₄+𝑥₂𝑥₃−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑒₁,𝑢₂=−2𝑦₂𝑥₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃+3𝑥²₁𝑎𝑥₂−𝑥₁+𝑥₄−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑒₂,𝑢₃=−2𝑦₃𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂+3𝑥²₂𝑑𝑥₁+𝑐𝑥₂−𝑥₁𝑥₃+1+𝑥²₃×𝑐₁𝑦₃−𝑦₁𝑦₂+𝑒₃,𝑢₄=−2𝑦₄𝑥₄𝑟𝑥₄+𝑥₂𝑥₃+3𝑥²₃−𝑏𝑥₃+𝑥₁𝑥₂−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃+𝑒₄. (19) Equation (18) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄<0, (20) 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, 𝐹1,𝐹2,𝐹3,𝐹4,𝐻1,𝐻2,𝐻3, and 𝐻4 for Case 1.

Case 2 (a symplectic synchronization with uncertain parameters). The master Chen system with uncertain variable parameters is 𝑥̇₁=𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄,𝑥̇₂=𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂,𝑥̇₄=𝑟(𝑡)𝑥₄+𝑥₂𝑥₃, (21) where 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡), and 𝑟(𝑡) are uncertain parameters. In simulation, we take 𝑎(𝑡)=𝑎1+𝑘₁sin𝜔₁𝑡,𝑏(𝑡)=𝑏1+𝑘₂sin𝜔₂𝑡,𝑐(𝑡)=𝑐1+𝑘₃sin𝜔₃𝑡,𝑑(𝑡)=𝑑1+𝑘₄sin𝜔₄𝑡,𝑟(𝑡)=𝑟1+𝑘₅sin𝜔₅𝑡, (22) where 𝑘₁,𝑘₂,𝑘₃,𝑘₄,𝑘₅,𝜔₁,𝜔₂,𝜔₃,𝜔₄, and 𝜔₅ are constants. Take 𝑘₁=0.3,  𝑘₂=0.5,  𝑘₃=0.2,𝑘₄=0.4,  𝑘₅=0.6,  𝜔₁=13,  𝜔₂=17,  𝜔₃=19,  𝜔₄=23, and 𝜔₅=29. So, (21) is chaotic system, shown in Figure 4.
We take 𝐹₁(𝑡)=𝑥³₄(𝑡),𝐹₂(𝑡)=𝑥³₁(𝑡),𝐹₃(𝑡)=𝑥³₂(𝑡), and 𝐹₄(𝑡)=𝑥³₃(𝑡). They are chaotic functions of time. 𝐻_𝑖(𝑥,𝑦,𝑡)=−𝑥²_𝑖𝑦_𝑖(𝑖=1,2,3,4) are given. By (6), we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞−𝑥²_𝑖𝑦_𝑖−𝑦_𝑖+𝑥³_𝑗=0,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (23) From (7), we have 𝑒_𝑖̇=−2𝑥̇_𝑖𝑥_𝑖𝑦_𝑖−𝑥²_𝑖𝑦̇_𝑖−𝑦̇_𝑖+3𝑥̇_𝑗𝑥²_𝑗,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (24)
Equation (8) can be expressed as 𝑒₁̇=−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑢₁+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃,𝑒₂̇=−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑢₂+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄,𝑒₃̇=−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂+𝑢₃+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃,𝑒₄̇=−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃+𝑢₄+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂, (25) where 𝑒₁=−𝑥²₁𝑦₁−𝑦₁+𝑥³₄, 𝑒₂=−𝑥²₂𝑦₂−𝑦₂+𝑥³₁, 𝑒₃=−𝑥²₃𝑦₃−𝑦₃+𝑥³₂, and 𝑒₄=−𝑥²₄𝑦₄−𝑦₄+𝑥³₃.
Choose a positive definite Lyapunov function as 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄. (26) Its time derivative along any solution of (25) is 𝑉̇=𝑒₁−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄−𝑢₁+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃+𝑒₂−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃−𝑢₂+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄+𝑒₃−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂−𝑢₃+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃+𝑒₄−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃−𝑢₄+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂. (27) According to (10), we get the controller 𝑢₁=−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₁𝑎₁𝑦₂−𝑦₁−𝑦₄+𝑒₁,𝑢₂=−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₂𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑒₂,𝑢₃=−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃+1+𝑥²₃𝑐₁𝑦₃−𝑦₁𝑦₂+𝑒₃,𝑢₄=−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₄𝑑₁𝑦₄−𝑦₁𝑦₃+𝑒₄. (28) Equation (27) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄<0, (29) 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, 𝐹1,𝐹2,𝐹3,𝐹4,𝐻1,𝐻2,𝐻3, and 𝐻4 for Case 2.

Case 3 (an enhanced symplectic synchronization with uncertain parameters). We take 𝐹₁(𝑡)=𝑥³₄(𝑡),𝐹₂(𝑡)=𝑥³₁(𝑡),𝐹₃(𝑡)=𝑥³₂(𝑡), and 𝐹₄(𝑡)=𝑥³₃(𝑡). They are chaotic functions of time. 𝐻_𝑖(𝑥̇,𝑦,̇𝑥,𝑦,𝑡)=−𝑥²_𝑖𝑦_𝑖−𝑥̇−𝐾𝑦̇(𝑖=1,2,3,4) are given. The 𝐾 value is 0.0001. By (6), we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞−𝑥²_𝑖𝑦_𝑖−𝑥̇_𝑖−𝐾𝑦̇_𝑖−𝑦_𝑖+𝑥³_𝑗=0,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (30) From (7) we have 𝑒_𝑖̇=−2𝑥_𝑖𝑥̇_𝑖𝑦_𝑖−𝑥²_𝑖𝑦̇_𝑖−𝑥̈_𝑖−𝐾𝑦̈_𝑖−𝑦̇_𝑖+3𝑥̇_𝑗𝑥²_𝑗,𝑖=1,2,3,4;𝑗=4,𝑖=1,𝑖−1,𝑖≠1. (31) Equation (8) can be expressed as 𝑒₁̇=−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑢₁−𝑥̈₁−𝐾𝑦̈₁+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃,𝑒₂̇=−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑢₂−𝑥̈₂−𝐾𝑦̈₂+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄,𝑒₃̇=−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂+𝑢₃−𝑥̈₃−𝐾𝑦̈₃+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃,𝑒₄̇=−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃+𝑢₄−𝑥̈₄−𝐾𝑦̈₄+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂, (32) where 𝑒₁=−𝑥²₁𝑦₁−𝑦₁−𝑥̇₁−𝐾𝑦̇₁+𝑥³₄, 𝑒₂=−𝑥²₂𝑦₂−𝑦₂−𝑥̇₂−𝐾𝑦̇₂+𝑥³₁, 𝑒₃=−𝑥²₃𝑦₃−𝑦₃−𝑥̇₃−𝐾𝑦̇₃+𝑥³₂, and 𝑒₄=−𝑥²₄𝑦₄−𝑦₄−𝑥̇₄−𝐾𝑦̇₄+𝑥³₃.
Choose a positive definite Lyapunov function as 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄. (33) Its time derivative along any solution of (32) is 𝑉̇=𝑒₁−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₁×𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑢₁−𝑥̈₁−𝐾𝑦̈₁+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃+𝑒₂−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃−1+𝑥²₂×𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑢₂−𝑥̈₂−𝐾𝑦̈₂+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄+𝑒₃−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₃×−𝑐₁𝑦₃+𝑦₁𝑦₂+𝑢₃−𝑥̈₃−𝐾𝑦̈₃+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃+𝑒₄−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₄×𝑑₁𝑦₄−𝑦₁𝑦₃+𝑢₄−𝑥̈₄−𝐾𝑦̈₄+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂. (34)
According to (10), we get the controller 𝑢₁=−2𝑦₁𝑥₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−𝑥̈₁−𝐾𝑦̈₁+3𝑥²₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−1+𝑥²₁𝑎₁𝑦₂−𝑦₁+𝑦₄+𝑒₁,𝑢₂=−2𝑦₂𝑥₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃−𝑥̈₂−𝐾𝑦̈₂+3𝑥²₁𝑎(𝑡)𝑥₂−𝑥₁+𝑥₄−1+𝑥²₂𝑏₁𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑒₂,𝑢₃=−2𝑦₃𝑥₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−𝑥̈₃−𝐾𝑦̈₃+3𝑥²₂𝑑(𝑡)𝑥₁+𝑐(𝑡)𝑥₂−𝑥₁𝑥₃+1+𝑥²₃𝑐₁𝑦₃−𝑦₁𝑦₂+𝑒₃,𝑢₄=−2𝑦₄𝑥₄𝑟(𝑡)𝑥₄+𝑥₂𝑥₃−𝑥̈₄−𝐾𝑦̈₄+3𝑥²₃−𝑏(𝑡)𝑥₃+𝑥₁𝑥₂−1+𝑥²₄𝑑₁𝑦₄−𝑦₁𝑦₃+𝑒₄. (35) Equation (34) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄<0, (36) 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, 𝐹1,𝐹2,𝐹3,𝐹4,𝐻1,𝐻2,𝐻3, and 𝐻4 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.