Symplectic Synchronization of Lorenz-Stenflo System with Uncertain Chaotic Parameters via Adaptive Control

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]: 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, 𝐹(𝑡) is a given function of time in different form. The 𝐹(𝑡) may be a regular function or a chaotic function. When 𝐻(𝑡,𝑥,𝑦)+𝐹(𝑡)=𝑥 and 𝐻(𝑡,𝑥,𝑦)=𝑥 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 𝐴 controls the partner 𝐵. The partner 𝐴 is given by 𝑥̇=𝑓(𝑡,𝑥,𝐴(𝑡)), (2) 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=𝑓𝑡,𝑦,𝐴(𝑡), (3a) where 𝑦=[𝑦₁,𝑦₂,…,𝑦_𝑛]^𝑇∈𝑅^𝑛 is a state vector, 𝐴(𝑡)=[𝐴₁(𝑡),𝐴₂(𝑡),…,𝐴_𝑚(𝑡)]^𝑇∈𝑅^𝑚 is an estimated vector of uncertain coefficients in 𝑓.

So a controller 𝑢(𝑡) is added on partner 𝐵, and the partner 𝐵 becomes 𝑦̇=𝑓𝑡,𝑦,𝐴(𝑡)+𝑢(𝑡), (3b)where 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the control vector function.

Our goal is to design the controller 𝑢(𝑡) so that the state vector 𝑦 of the partner 𝐵 asymptotically approaches 𝐻(𝑡,𝑥,𝑦)+𝐹(𝑡), a given function 𝐻(𝑡,𝑥,𝑦) plus a given vector function 𝐹(𝑡)=[𝐹₁(𝑡),𝐹₂(𝑡),…,𝐹_𝑛(𝑡)]^𝑇 which is the regular function or the chaotic function.

To define error vector 𝑒(𝑡)=[𝑒₁,𝑒₂,…,𝑒_𝑛]^𝑇: 𝑒=𝐻𝑡,𝑥,𝑦−𝑦+𝐹(𝑡),lim_𝑡→∞𝑒=0 (4)(5) is demanded.

From (4), it is obtained that 𝑒̇=𝜕𝐻𝜕𝑡+∇𝐻^𝑇Ψ̇−𝑦̇+𝐹̇(𝑡), (6) where Ψ̇^𝑇=[𝑥̇𝑦]̇.

By (2), (3a), and (3b), (6) can be rewritten as 𝑒̇=𝜕𝐻+𝜕𝑡𝜕𝐻𝑓((𝑡+𝜕𝑥𝑡,𝑥,𝐴))𝜕𝐻𝑓𝐴(𝑡)𝐴(𝑡(𝑡)+𝐹̇(𝑡).𝜕𝑦𝑡,𝑦,−𝑓𝑡,𝑦,)−𝑢 (7)

A positive definite Lyapunov function 𝑉(𝑒,𝐴(𝑡)) is chosen: 𝑉𝐴(𝑡=1𝑒,)2𝑒^𝑇1𝑒+2𝐴^𝑇(𝑡)𝐴(𝑡), (8) where 𝐴(𝑡)=𝐴(𝑡)−𝐴(𝑡).

Its derivative along any solution of (7) is 𝑉̇𝑒,𝐴(𝑡)=𝑒^𝑇𝜕𝐻+𝜕𝑡𝜕𝐻𝑓((𝑡+𝜕𝑥𝑡,𝑥,𝐴))𝜕𝐻𝑓𝐴(𝑡)𝐴(𝑡+𝐹̇(𝑡)(𝑡+𝐴𝜕𝑦𝑡,𝑦,−𝑓𝑡,𝑦,)−𝑢)_𝑇(𝑡)𝐴̇(𝑡). (9) In (9), the 𝑢(𝑡) and 𝐴̇(𝑡) are designed so that 𝑉̇=𝑒^𝑇𝐶_𝑛×𝑛𝑒+𝐴^𝑇(𝑡)𝐷_𝑚×𝑚𝐴(𝑡) where 𝐶_𝑛×𝑛 and 𝐷_𝑚×𝑚 are two diagonal negative definite matrices. The 𝑉̇ is a negative definite function of 𝑒 and 𝐴(𝑡). By Lyapunov theorem of asymptotical stability lim_𝑡→∞𝑒=0,lim_𝑡→∞𝐴(𝑡)=0. (10)

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] 𝑥̇₁=𝑎𝑥₂−𝑥₁+𝑟𝑥₄,𝑥̇₂=𝑑𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝑏𝑥₃+𝑥₁𝑥₂,𝑥̇₄=−𝑥₄−𝑐𝑥₁. (11) And the slave Lorenz-Stenflo system can be described as 𝑦̇₁=𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄,𝑦̇₂=𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃,𝑦̇₃=−𝑏̂𝑦₃+𝑦₁𝑦₂,𝑦̇₄=−𝑦₄−𝑐𝑦̂₁, (12) where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑟 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 𝑎=11,  𝑏=2.9,   𝑐=5,  𝑑=23, and 𝑟=1.9.

The controllers, 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄, are added to the four equations of (12), respectively: 𝑦̇₁=𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄+𝑢₁,𝑦̇₂=𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑢₂,𝑦̇₃=−𝑏̂𝑦₃+𝑦₁𝑦₂+𝑢₃,𝑦̇₄=−𝑦₄−𝑐𝑦̂₁+𝑢₄. (13)

The initial values of the states of the master system and of the slave system are taken as 𝑥₁(0)=10, 𝑥₂(0)=−13, 𝑥₃(0)=12, 𝑥₄(0)=−5, 𝑦₁(0)=−3, 𝑦₂(0)=−4, 𝑦₃(0)=3, and 𝑦₄(0)=−5.

Case 1. A time delay symplectic synchronization.
We take 𝐹₁(𝑡)=𝑥₁(𝑡−𝑇), 𝐹₂(𝑡)=𝑥₂(𝑡−𝑇), 𝐹₃(𝑡)=𝑥₃(𝑡−𝑇), and 𝐹₄(𝑡)=𝑥₄(𝑡−𝑇). They are chaotic functions of time, where time delay 𝑇=0.5 sec. The 𝐻_𝑖(𝑥,𝑦,𝑡)=(𝑥²_𝑖+𝑦_𝑖)(sin𝑤𝑡+3)(𝑖=1,2,3,4) are given. By (5) we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞𝑥²_𝑖+𝑦_𝑖(sin𝜔𝑡+3)−𝑦_𝑖+𝑥_𝑖(𝑡−𝑇)=0,𝑖=1,2,3,4. (14) From (14) we have 𝑒_𝑖̇=2𝑥_𝑖𝑥̇_𝑖+𝑦̇_𝑖(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²_𝑖+𝑦_𝑖−𝑦̇_𝑖+𝑥̇_𝑖(𝑡−𝑇),𝑖=1,2,3,4, (15) where 𝜔=19.

Equation (15) can be expressed as 𝑒₁̇=2𝑥₁𝑎𝑥₂−𝑥₁+𝑟𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝑎𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)+𝑟𝑥₄(),𝑒̇𝑡−𝑇₂=2𝑥₂𝑑𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝑢₂+𝑑𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(),𝑒̇𝑡−𝑇₃=2𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝑏𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(),𝑒̇𝑡−𝑇₄=2𝑥₄−𝑥₄−𝑐𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄(𝑡−𝑇)−𝑐𝑥₁(𝑡−𝑇), (16) where 𝑒₁=𝑥²₁+𝑦₁(sin𝜔𝑡+3)−𝑦₁+𝑥₁(),𝑒𝑡−𝑇₂=𝑥²₂+𝑦₂(sin𝜔𝑡+3)−𝑦₂+𝑥₂(),𝑒𝑡−𝑇₃=𝑥²₃+𝑦₃(sin𝜔𝑡+3)−𝑦₃+𝑥₃(),𝑒𝑡−𝑇₄=𝑥²₄+𝑦₄(sin𝜔𝑡+3)−𝑦₄+𝑥₄(𝑡−𝑇). (17)

Choose a positive definite Lyapunov function: 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄,̃̃𝑟̃=1𝑎,𝑏,𝑐,𝑑,2𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃, (18) where 𝑎=𝑎−𝑎,  𝑏̃=𝑏−𝑏̂,  𝑐̃=𝑐−𝑐̂,  𝑑=𝑑−𝑑,  𝑟̃=𝑟−𝑟̂, and 𝑎, 𝑏̂, 𝑐̂, 𝑑, and 𝑟̂ are estimates of uncertain parameters a, b, c, d, and r, respectively.

Its time derivative along any solution of (16) is 𝑉̇=𝑒₁2𝑥₁𝑎𝑥₂−𝑥₁+𝑟𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝑎𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)+𝑟𝑥₄(𝑡−𝑇)+𝑒₂2𝑥₂𝑑𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃−𝑢₂+𝑑𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(𝑡−𝑇)+𝑒₃2𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝑏𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(𝑡−𝑇)+𝑒₄2𝑥₄−𝑥₄−𝑐𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄(𝑡−𝑇)−𝑐𝑥₁(+𝑎̇𝑏̃𝑏̇̃+𝑐̃̇̃𝑑̇𝑟̃̇̃𝑡−𝑇)𝑎+𝑐+𝑑+𝑟. (19)

The adaptive controllers are chosen as 𝑢₁=2𝑥₁𝑎𝑥₂−𝑥₁+𝑟𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦₄+𝑎𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)+𝑟𝑥₄(𝑡−𝑇)+𝑒₁,𝑢₂=2𝑥₂𝑑𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝑑𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(𝑡−𝑇)+𝑒₂,𝑢₃=2𝑥₃−𝑏𝑥₃+𝑥₁𝑥₂−𝑏𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏𝑦₃−𝑦₁𝑦₂−𝑏𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(𝑡−𝑇)+𝑒₃,𝑢₄=2𝑥₄−𝑥₄−𝑐𝑥₁−𝑦₄−𝑐𝑦₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦₁−𝑥₄(𝑡−𝑇)−𝑐𝑥₁(𝑡−𝑇)+𝑒₄, (20) and the update laws are chosen as 𝑎̇=−𝑎̇=2𝑥₁𝑦₂−𝑦₁(sin𝜔𝑡+3)𝑒₁+𝑦₂−𝑦₁𝑒₁−̇̃̇̂𝑎,𝑏=−𝑏=−2𝑥₃𝑦₃𝑒₃(sin𝜔𝑡+3)−𝑦₃𝑒₃−̃̇̃̇̂𝑏,𝑐=−𝑐=−2𝑥₄𝑦₁𝑒₄(sin𝜔𝑡+3)−𝑦₁𝑒₄−̃̇̇𝑐,𝑑=−𝑑=2𝑥₂𝑦₁𝑒₂(sin𝜔𝑡+3)+𝑦₁𝑒₂−̇̃̇̂𝑑,𝑟=−𝑟=2𝑥₁𝑦₄𝑒₁(sin𝜔𝑡+3)+𝑦₄𝑒₁−𝑟.̃ (21) The initial values of estimate for uncertain parameters are 𝑎(0)=𝑏̂(0)=𝑐(̂0)=𝑑(0)=𝑟(̂0)=0.

Equation (19) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃<0, (22) 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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(b)

(c)

(d)

Figure 1 

Projections of phase portrait for master system (11).

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Figure 2 

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

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Figure 3 

Time histories of states, state errors, parameter differences, 𝐹1, 𝐹2, 𝐹3, 𝐹4, 𝐻1, 𝐻2, 𝐻3, and 𝐻4 for Case 1.

Case 2. A time delay symplectic synchronization with uncertain chaotic parameters.
The Lorenz-Stenflo master system with uncertain chaotic parameters is 𝑥̇₁=𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄,𝑥̇₂=𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂,𝑥̇₄=−𝑥₄−𝐶(𝑡)𝑥₁, (23) where 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡) are uncertain chaotic parameters. In simulation, we take 𝐴(𝑡)=𝑎1+𝑘₁𝑧₁,𝐵(𝑡)=𝑏1+𝑘₂𝑧₃,𝐶(𝑡)=𝑐1+𝑘₃𝑧₄,𝐷(𝑡)=𝑑1+𝑘₄𝑧₂,𝑅(𝑡)=𝑟1+𝑘₅𝑧₁, (24) where 𝑘₁, 𝑘₂, 𝑘₃, 𝑘₄, and 𝑘₅ are positive constants. Chosen are 𝑘₁=0.07, 𝑘₂=0.08, 𝑘₃=0.06, 𝑘₄=0.07, and 𝑘₅=0.08. The system (23) is chaotic dynamic motion, shown in Figure 4.

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(d)

Figure 4 

Projections of the phase portrait for chaotic system (23).

The 𝐹(𝑡) is chaotic system, and the chaotic signal of goal system can be described as 𝑧̇₁=𝑎𝑧₂−𝑧₁+𝑟𝑧₄,𝑧̇₂=𝑑𝑧₁−𝑧₂−𝑧₁𝑧₃,𝑧̇₃=−𝑏𝑧₃+𝑧₁𝑧₂,𝑧̇₄=−𝑧₄−𝑐𝑧₁, (25) where initial conditions of the chaotic signal of system are 𝑧₁(0) = 2, 𝑧₂(0) = 5, 𝑧₃(0) = −4, and 𝑧₄(0) = −6. The 𝐻_𝑖(𝑥,𝑦,𝑡)=(𝑥²_𝑖+𝑦_𝑖)(sin𝑤𝑡+3)(𝑖=1,2,3,4) are given. By (5) we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞𝑥²_𝑖+𝑦_𝑖(sin𝜔𝑡+3)−𝑦_𝑖+𝑥_𝑖(𝑡−𝑇)=0,𝑖=1,2,3,4. (26) From (26) we have 𝑒_𝑖̇=2𝑥_𝑖𝑥̇_𝑖+𝑦̇_𝑖(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²_𝑖+𝑦_𝑖−𝑦̇_𝑖+𝑥̇_𝑖(𝑡−𝑇),𝑖=1,2,3,4. (27) Equation (27) can be expressed as 𝑒₁̇=2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝐴(𝑡)𝑥₂(𝑡−𝑇)−𝑥₁((𝑡)𝑥𝑡−𝑇)+𝑅₄(),𝑒̇𝑡−𝑇₂=2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝑢₂+𝐷(𝑡)𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(),𝑒̇𝑡−𝑇₃=2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝐵(𝑡)𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(),𝑒̇𝑡−𝑇₄=2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄(𝑡−𝑇)−𝐶(𝑡)𝑥₁(𝑡−𝑇), (28) where 𝑒₁=𝑥²₁+𝑦₁(sin𝜔𝑡+3)−𝑦₁+𝑥₁(),𝑒𝑡−𝑇₂=𝑥²₂+𝑦₂(sin𝜔𝑡+3)−𝑦₂+𝑥₂(),𝑒𝑡−𝑇₃=𝑥²₃+𝑦₃(sin𝜔𝑡+3)−𝑦₃+𝑥₃(),𝑒𝑡−𝑇₄=𝑥²₄+𝑦₄(sin𝜔𝑡+3)−𝑦₄+𝑥₄(𝑡−𝑇). (29)

Choose a positive definite Lyapunov function: 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄,̃̃𝑟̃=1𝑎,𝑏,𝑐,𝑑,2𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃, (30) where 𝑎=𝐴(𝑡)−𝑎,  𝑏̃=𝐵(𝑡)−𝑏̂,  𝑐̃=𝐶(𝑡)−𝑐̂,  𝑑=𝐷(𝑡)−𝑑,   𝑟̃=𝑅(𝑡)−𝑟̂, and 𝑎, 𝑏̂, 𝑐̂, 𝑑, and 𝑟̂ are estimates of uncertain parameters 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡), respectively.

Its time derivative along any solution of (28) is 𝑉̇=𝑒₁2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝐴(𝑡)𝑥₂(𝑡−𝑇)−𝑥₁()(𝑡)𝑥𝑡−𝑇+𝑅₄(𝑡−𝑇)+𝑒₂2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃−𝑢₂+𝐷(𝑡)𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(𝑡−𝑇)+𝑒₃2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝐵(𝑡)𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(𝑡−𝑇)+𝑒₄2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄()(𝑡)𝑥𝑡−𝑇−𝐶₁(𝑡−𝑇)+𝑎𝑎̇+𝑏̃𝑏̇̃+𝑐𝑐̃̇̃+𝑑𝑑̇+𝑟𝑟̃.̇̃ (31)

The adaptive controllers are chosen as 𝑢₁=2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦₄+𝐴(𝑡)𝑥₂(𝑡−𝑇)−𝑥₁((𝑡)𝑥𝑡−𝑇)+𝑅₄(𝑡−𝑇)+𝑒₁,𝑢₂=2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝐷(𝑡)𝑥₁(𝑡−𝑇)−𝑥₂(𝑡−𝑇)−𝑥₁(𝑡−𝑇)𝑥₃(𝑡−𝑇)+𝑒₂,𝑢₃=2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏𝑦₃−𝑦₁𝑦₂−𝐵(𝑡)𝑥₃(𝑡−𝑇)+𝑥₁(𝑡−𝑇)𝑥₂(𝑡−𝑇)+𝑒₃,𝑢₄=2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦₁−𝑥₄()(𝑡)𝑥𝑡−𝑇−𝐶₁(𝑡−𝑇)+𝑒₄, (32) and the update laws are chosen as 𝑎̇=2𝑥₁𝑦₂−𝑦₁(sin𝜔𝑡+3)𝑒₁+𝑦₂−𝑦₁𝑒₁−̇̃𝑎,𝑏=−2𝑥₃𝑦₃𝑒₃(sin𝜔𝑡+3)−𝑦₃𝑒₃−̃̇̃𝑏,𝑐=−2𝑥₄𝑦₁𝑒₄(sin𝜔𝑡+3)−𝑦₁𝑒₄−̃̇𝑐,𝑑=2𝑥₂𝑦₁𝑒₂(sin𝜔𝑡+3)+𝑦₁𝑒₂−̇̃𝑑,𝑟=2𝑥₁𝑦₄𝑒₁(sin𝜔𝑡+3)+𝑦₄𝑒₁−𝑟.̃ (33) The initial values of estimate for uncertain parameters are 𝑎(0)=𝑏̂(0)=𝑐(̂0)=𝑑(0)=𝑟(̂0)=0.

Equation (31) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃<0, (34) 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.

(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.

(a)

(b)

Figure 7 

Time histories of 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡) and parameter differences for Case 2.

Case 3. A multitime delay symplectic synchronization with uncertain chaotic parameters.
We take 𝐹₁(𝑡)=𝑥₁(𝑡−𝑇₁), 𝐹₂(𝑡)=𝑥₂(𝑡−𝑇₂), 𝐹₃(𝑡)=𝑥₃(𝑡−𝑇₃), and 𝐹₄(𝑡)=𝑥₄(𝑡−𝑇₄). They are chaotic functions of time, where multi time delay 𝑇₁, 𝑇₂, 𝑇₃, and 𝑇₄ are positive constants, 𝑇₁ = 0.5 sec, 𝑇₂=0.7 sec, 𝑇₃=0.8 sec, and 𝑇₄=0.9 sec. The 𝐻_𝑖(𝑥,𝑦,𝑡)=(𝑥²_𝑖+𝑦_𝑖)(sin𝑤𝑡+3)(𝑖=1,2,3,4) are given. By (5) we have lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞𝑥²_𝑖+𝑦_𝑖(sin𝜔𝑡+3)−𝑦_𝑖+𝑥_𝑖𝑡−𝑇_𝑖=0,𝑖=1,2,3,4. (35) From (35) we have 𝑒_𝑖̇=2𝑥_𝑖𝑥̇_𝑖+𝑦̇_𝑖(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²_𝑖+𝑦_𝑖−𝑦̇_𝑖+𝑥̇_𝑖𝑡−𝑇_𝑖,𝑖=1,2,3,4. (36) Equation (36) can be expressed as 𝑒₁̇=2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝐴(𝑡)𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁+𝑅(𝑡)𝑥₄𝑡−𝑇₄,𝑒₂̇=2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝑢₂+𝐷(𝑡)𝑥₁𝑡−𝑇₁−𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁𝑥₃𝑡−𝑇₃,𝑒₃̇=2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝐵(𝑡)𝑥₃𝑡−𝑇₃+𝑥₁𝑡−𝑇₁𝑥₂𝑡−𝑇₂,𝑒₄̇=2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄𝑡−𝑇₄−𝐶(𝑡)𝑥₁𝑡−𝑇₁, (37) where 𝑒₁=𝑥²₁+𝑦₁(sin𝜔𝑡+3)−𝑦₁+𝑥₁𝑡−𝑇₁,𝑒₂=𝑥²₂+𝑦₂(sin𝜔𝑡+3)−𝑦₂+𝑥₂𝑡−𝑇₂,𝑒₃=𝑥²₃+𝑦₃(sin𝜔𝑡+3)−𝑦₃+𝑥₃𝑡−𝑇₃,𝑒₄=𝑥²₄+𝑦₄(sin𝜔𝑡+3)−𝑦₄+𝑥₄𝑡−𝑇₄. (38)
Choose a positive definite Lyapunov function: 𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄,̃̃𝑟̃=1𝑎,𝑏,𝑐,𝑑,2𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃, (39) where 𝑎=𝐴(𝑡)−𝑎,  𝑏̃=𝐵(𝑡)−𝑏̂,  𝑐̃=𝐶(𝑡)−𝑐̂,  𝑑=𝐷(𝑡)−𝑑,  𝑟̃=𝑅(𝑡)−𝑟̂, and 𝑎,  𝑏̂,  𝑐̂,  𝑑,  and 𝑟̂ are estimates of uncertain parameters 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡), respectively.

Its time derivative along any solution of (37) is 𝑉̇=𝑒₁2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦̂₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦̂₄−𝑢₁+𝐴(𝑡)𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁+𝑅(𝑡)𝑥₄𝑡−𝑇₄+𝑒₂2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃−𝑢₂+𝐷(𝑡)𝑥₁𝑡−𝑇₁−𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁𝑥₃𝑡−𝑇₃+𝑒₃2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏̂𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏̂𝑦₃−𝑦₁𝑦₂−𝑢₃−𝐵(𝑡)𝑥₃𝑡−𝑇₃+𝑥₁𝑡−𝑇₁𝑥₂𝑡−𝑇₂+𝑒₄2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦̂₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦̂₁−𝑢₄−𝑥₄𝑡−𝑇₄−𝐶(𝑡)𝑥₁𝑡−𝑇₁+𝑎𝑎̇+𝑏̃𝑏̇̃+𝑐𝑐̃̇̃+𝑑𝑑̇+𝑟𝑟̃̇.̃ (40)

The adaptive controllers are chosen as 𝑢₁=2𝑥₁𝐴(𝑡)𝑥₂−𝑥₁+𝑅(𝑡)𝑥₄+𝑎𝑦₂−𝑦₁+𝑟𝑦₄×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₁+𝑦₁−𝑎𝑦₂−𝑦₁−𝑟𝑦₄+𝐴(𝑡)𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁+𝑅(𝑡)𝑥₄𝑡−𝑇₄+𝑒₁,𝑢₂=2𝑥₂𝐷(𝑡)𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑑𝑦₁−𝑦₂−𝑦₁𝑦₃×(sin𝜔𝑡+3)+𝜔cos𝜔𝑡𝑥²₂+𝑦₂−𝑑𝑦₁+𝑦₂+𝑦₁𝑦₃+𝐷(𝑡)𝑥₁𝑡−𝑇₁−𝑥₂𝑡−𝑇₂−𝑥₁𝑡−𝑇₁𝑥₃𝑡−𝑇₃+𝑒₂,𝑢₃=2𝑥₃−𝐵(𝑡)𝑥₃+𝑥₁𝑥₂−𝑏𝑦₃+𝑦₁𝑦₂)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₃+𝑦₃+𝑏𝑦₃−𝑦₁𝑦₂−𝐵(𝑡)𝑥₃𝑡−𝑇₃+𝑥₁𝑡−𝑇₁𝑥₂𝑡−𝑇₂+𝑒₃,𝑢₄=2𝑥₄−𝑥₄−𝐶(𝑡)𝑥₁−𝑦₄−𝑐𝑦₁)𝑥(sin𝜔𝑡+3+𝜔cos𝜔𝑡²₄+𝑦₄+𝑦₄+𝑐𝑦₁−𝑥₄𝑡−𝑇₄−𝐶(𝑡)𝑥₁𝑡−𝑇₁+𝑒₄, (41) and the update laws are chosen as 𝑎̇=2𝑥₁𝑦₂−𝑦₁(sin𝜔𝑡+3)𝑒₁+𝑦₂−𝑦₁𝑒₁−̇̃𝑎,𝑏=−2𝑥₃𝑦₃𝑒₃(sin𝜔𝑡+3)−𝑦₃𝑒₃−̃̇̃𝑏,𝑐=−2𝑥₄𝑦₁𝑒₄(sin𝜔𝑡+3)−𝑦₁𝑒₄−̃̇𝑐,𝑑=2𝑥₂𝑦₁𝑒₂(sin𝜔𝑡+3)+𝑦₁𝑒₂−̇̃𝑑,𝑟=2𝑥₁𝑦₄𝑒₁(sin𝜔𝑡+3)+𝑦₄𝑒₁−𝑟.̃ (42) The initial values of estimate for uncertain parameters are 𝑎(0)=𝑏̂(0)=𝑐(̂0)=𝑑(0)=𝑟(̂0)=0.

Equation (40) becomes 𝑉̇=−𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑟²̃<0, (43) 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.

(a)

(b)

(c)

(d)

Figure 8 

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

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9 

Time histories of states, state errors, parameter differences, 𝐹1, 𝐹2, 𝐹3, 𝐹4, 𝐻1, 𝐻2, 𝐻3, and 𝐻4 for Case 3.

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.