Adaptive Synchronization and Antisynchronization of a Hyperchaotic Complex Chen System with Unknown Parameters Based on Passive Control

Abstract

This paper investigates the synchronization and antisynchronization problems of a hyperchaotic complex Chen system with unknown parameters based on the properties of a passive system. The essential conditions are derived under which the synchronization or antisynchronization error dynamical system could be equivalent to a passive system and be globally asymptotically stabilized at a zero equilibrium point via smooth state feedback. Corresponding parameter estimation update laws are obtained to estimate the unknown parameters as well. Numerical simulations verify the effectiveness of the theoretical analysis.

1. Introduction

Hyperchaos [1] is generally characterized as a chaotic attractor with more than one positive Lyapunov exponent and has richer dynamical behaviors than chaos. Over the past three decades, hyperchaotic systems with real variables have been investigated extensively [2–5]. Since Fowler et al. [6] introduced a complex Lorenz model to generalize the real Lorenz model in 1982, chaotic and hyperchaotic complex systems have attracted increasing attention to the systems with complex variables which can be used to describe the physics of a detuned laser, rotating fluids, disk dynamos, electronic circuits, and particle beam dynamics in high energy accelerators [7]. When applying the complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information.

In recent years, chaos synchronization has attracted increasing attention among scientists due to its potential applications in the fields of secure communications; optical, chemical, physical, and biological systems; neural networks; and so forth [8, 9]. Several types of synchronization have been investigated on complex chaotic and hyperchaotic systems including complete synchronization [7], antisynchronization [10, 11], phase and antiphase synchronization [12], lag and antilag synchronization [13, 14], hybrid projective synchronization [15], and modified function projective synchronization [16]. Among the abovementioned synchronization phenomena, the most widely investigated one is complete synchronization (synchronization for short hereafter), which implies that the differences of state variables of synchronized systems starting from different initial values converge to zero eventually. On the other hand, antisynchronization is an another interesting phenomenon, which is characterized by the vanishing of the sum of the relevant state variables of synchronized systems. When applying antisynchronization to communication systems, the security and secrecy of communication can be strengthened while transmitting digital signals by the transform between synchronization and antisynchronization continuously.

Recently, many researchers have begun to give their attention to the concept of passivity of nonlinear systems. The passivity theory is considered to be an alternative tool for analyzing the stability of nonlinear systems. The main idea of passivity theory is that the passive properties of a system can keep the system internally stable. In order to make a system stable, one can design a controller which renders the closed-loop system passive with the help of passivity theory. For the past decade, the passivity theory has played an important role in designing an asymptotically stabilizing controller for control and synchronization of chaotic and hyperchaotic systems with real variables [17–20]. For complex nonlinear systems, only Mahmoud et al. [21] applied the passive control to investigate the control of 𝑛-dimensional chaotic complex nonlinear systems. In this paper, we apply the passive control to investigate the synchronization and antisynchronization problems of the newly reported hyperchaotic complex Chen system [22].

This paper is organized as follows. Section 2 introduces the passive control theory and a new hyperchaotic complex Chen system. Sections 3 and 4 investigate the synchronization and antisynchronization problems of two identical hyperchaotic complex Chen systems with unknown parameters, respectively. In Section 5, two numerical examples are provided to illustrate the analytical results. Finally, conclusions are given in Section 6.

2. The Passive Control Theory and a New Hyperchaotic Complex Chen System

Consider the following nonlinear affine system: ̇(𝑥)(𝑥)(𝑥),𝑥=𝑓+𝑔𝑢,𝑤=ℎ (1) where 𝑥∈𝑅^𝑛 is the state variable and 𝑢∈𝑅^𝑚 and 𝑤∈𝑅^𝑚 are input and output values, respectively. 𝑓(𝑥) and 𝑔(𝑥) are smooth vector fields, 𝑓(0)=0, and ℎ(𝑥) is a smooth mapping.

Definition 1 (see [23]). System (1) is a minimum phase system if 𝐿_𝑔ℎ(0) is nonsingular and 𝑥=0 is one of the asymptotically stabilized equilibrium points of 𝑓(𝑥).

Definition 2 (see [24]). System (1) is passive if there exists a real constant 𝛽 such that for all 𝑡≥0, the following inequality holds: ^𝑡₀𝑢^𝑇(𝜏)𝑤(𝜏)𝑑𝜏≥𝛽, (2) or there exists a 𝜌≥0 and a real constant 𝛽 such that ^𝑡₀𝑢^𝑇(𝜏)𝑤(𝜏)𝑑𝜏+𝛽≥^𝑡₀𝜌𝑤^𝑇(𝜏)𝑤(𝜏)𝑑𝜏. (3)

If system (1) has relative degree [1,…,1] at 𝑥=0 (i.e., 𝐿_𝑔ℎ(0) is nonsingular) and the distribution spanned by the vector field 𝑔₁(𝑥),…,𝑔_𝑚(𝑥) is innovative, then it can be represented as the following normal form: 𝑧̇=𝑓₀(𝑧)()̇()()+𝑝𝑧,𝑤𝑤,𝑤=𝑏𝑧,𝑤+𝑎𝑧,𝑤𝑢, (4) where 𝑎(𝑧,𝑤) is nonsingular for any (𝑧,𝑤).

Theorem 3 (see [24]). Suppose the system (4) is passive with a storage function 𝑉, which is positive-definite, and the system (4) is locally zero-state detectable. Let 𝜙 be a smooth function such that 𝜙(0)=0 and 𝑤^𝑇𝜙(𝑤)>0 for each nonzero 𝑤. Then the control law 𝑢=−𝜙(𝑤) asymptotically stabilizes the equilibrium of system (4).

Recently, the authors [22] introduced six different versions of the hyperchaotic complex Chen system by adding a state feedback controller to the chaotic complex Chen system [25], and the dynamics of the six hyperchaotic complex Chen systems were studied in detail in [22]. In this paper, we apply the passive control to investigate the synchronization and antisynchronization problems of the following form of hyperchaotic complex Chen system: ̇̇̇1𝑥=𝛼𝑦−𝑥+𝑤,𝑦=𝛾−𝛼𝑥−𝑥𝑧+𝛾𝑦,𝑧=2𝑥𝑦+𝑥𝑦̇1−𝛽𝑧+𝑤,𝑤=2𝑥𝑦+𝑥𝑦−𝑑𝑤, (5) where 𝛼, 𝛽, 𝛾, and 𝑑 are positive real parameters, 𝑥=𝑢₁+𝑖𝑢₂ and 𝑦=𝑢₃+𝑖𝑢₄ are complex functions, 𝑖=√−1,  𝑧=𝑢₅, and 𝑤=𝑢₆. 𝑢_𝑗(𝑗=1,2,3,4,5,6) are real functions. The overbar represents complex conjugate function.

3. Synchronization of the Hyperchaotic Complex Chen System with Unknown Parameters

In this section, we study the synchronization problem of the hyperchaotic complex Chen system (5) with unknown parameters using the technique of passive control. We consider system (5) as the drive system, and the response system is described by 𝑥̇₁=𝛼𝑦₁−𝑥₁+𝑤₁,𝑦̇₁=𝛾−𝛼𝑥₁−𝑥₁𝑧₁+𝛾𝑦₁+𝜇₁+𝑖𝜇₂,𝑧̇₁=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝛽𝑧₁+𝑤₁+𝜇₃,𝑤̇₁=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝑑𝑤₁+𝜇₄, (6) where 𝑥₁=𝑣₁+𝑖𝑣₂, 𝑦₁=𝑣₃+𝑖𝑣₄, and 𝑧₁=𝑣₅, 𝑤₁=𝑣₆. 𝑣_𝑗(𝑗=1,2,3,4,5,6) are real functions, and 𝜇₁, 𝜇₂, 𝜇₃, and 𝜇₄ are real control functions to be determined to achieve synchronization.

By subtracting the drive system (5) from the response system (6), we obtain the following error dynamical system 𝑒₁̇+𝑖𝑒₂̇=𝛼𝑦₁−𝑥₁+𝑤₁−𝛼,𝑒̇𝑦−𝑥+𝑤₃+𝑖𝑒₄̇=𝛾−𝛼𝑥₁−𝑥₁𝑧₁+𝛾𝑦₁−𝛾−𝛼𝑥−𝑥𝑧+𝛾𝑦+𝜇₁+𝑖𝜇₂,𝑒₅̇=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝛽𝑧₁+𝑤₁−12𝑥𝑦+𝑥𝑦−𝛽𝑧+𝑤+𝜇₃,𝑒₆̇=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝑑𝑤₁−12𝑥𝑦+𝑥𝑦−𝑑𝑤+𝜇₄, (7) where 𝑒_𝑗=𝑣_𝑗−𝑢_𝑗(𝑗=1,2,3,4,5,6) are error states and 𝛼, 𝛽, 𝛾, and 𝑑 are unknown parameters.

Separating the real and imaginary parts of error dynamical system (7), we obtain the following real system: 𝑒₁̇=𝛼𝑒₃−𝑒₁+𝑒₆,𝑒₂̇=𝛼𝑒₄−𝑒₂,𝑒₃̇=𝛾−𝛼𝑒₁+𝛾𝑒₃−𝑒₁𝑒₅−𝑒₁𝑢₅−𝑢₁𝑒₅+𝜇₁,𝑒₄̇=𝛾−𝛼𝑒₂+𝛾𝑒₄−𝑒₂𝑒₅−𝑒₂𝑢₅−𝑢₂𝑒₅+𝜇₂,𝑒₅̇=𝑒₁𝑒₃+𝑒₁𝑢₃+𝑢₁𝑒₃+𝑒₂𝑒₄+𝑒₂𝑢₄+𝑢₂𝑒₄−𝛽𝑒₅+𝑒₆+𝜇₃,𝑒₆̇=𝑒₁𝑒₃+𝑒₁𝑢₃+𝑢₁𝑒₃+𝑒₂𝑒₄+𝑒₂𝑢₄+𝑢₂𝑒₄−𝑑𝑒₆+𝜇₄. (8)

Let 𝑧₁=𝑒₁,  𝑧₂=𝑒₂, 𝑦₁=𝑒₃, 𝑦₂=𝑒₄, 𝑦₃=𝑒₅, and 𝑦₄=𝑒₆; then the error dynamical system (8) can be rewritten as 𝑧̇₁=𝛼𝑦₁−𝑧₁+𝑦₄,𝑧̇₂=𝛼𝑦₂−𝑧₂,𝑦̇₁=𝛾−𝛼𝑧₁+𝛾𝑦₁−𝑧₁𝑦₃−𝑧₁𝑢₅−𝑢₁𝑦₃+𝜇₁,𝑦̇₂=𝛾−𝛼𝑧₂+𝛾𝑦₂−𝑧₂𝑦₃−𝑧₂𝑢₅−𝑢₂𝑦₃+𝜇₂,𝑦̇₃=𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁+𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−𝛽𝑦₃+𝑦₄+𝜇₃,𝑦̇₄=𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁+𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−𝑑𝑦₄+𝜇₄, (9)

which is the following normal form: 𝑧̇=𝑓₀(𝑧)̇+𝑝𝑧,𝑦𝑦,𝑦=𝑏𝑧,𝑦+𝑎𝑧,𝑦𝜇, (10) where 𝑧=[𝑧₁,𝑧₂]^𝑇, 𝑦=[𝑦₁,𝑦₂,𝑦₃,𝑦₄]^𝑇, 𝜇=[𝜇₁,𝜇₂,𝜇₃,𝜇₄]^𝑇 and 𝑓₀(𝑧)=−𝛼𝑧₁,−𝛼𝑧₂^𝑇=,𝑎=,𝑏=𝑧,𝑝𝑧,𝑦𝛼0010𝛼00𝑧,𝑦1000010000100001𝑧,𝑦𝛾−𝛼₁+𝛾𝑦₁−𝑧₁𝑦₃−𝑧₁𝑢₅−𝑢₁𝑦₃𝛾−𝛼𝑧₂+𝛾𝑦₂−𝑧₂𝑦₃−𝑧₂𝑢₅−𝑢₂𝑦₃𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁+𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−𝛽𝑦₃+𝑦₄𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁+𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−𝑑𝑦₄. (11)

Then we can arrive at the following result.

Theorem 4. If the passive controllers are designed as 𝜇₁=−𝛾𝑧₁−𝛾𝑦₁+𝑧₁𝑦₃+𝑧₁𝑢₅+𝑢₁𝑦₃−𝑘₁𝑦₁+𝜈₁,𝜇₂=−𝛾𝑧₂−𝛾𝑦₂+𝑧₂𝑦₃+𝑧₂𝑢₅+𝑢₂𝑦₃−𝑘₂𝑦₂+𝜈₂,𝜇₃=−𝑧₁𝑦₁−𝑧₁𝑢₃−𝑢₁𝑦₁−𝑧₂𝑦₂−𝑧₂𝑢₄−𝑢₂𝑦₂+𝛽𝑦₃−𝑦₄−𝑘₃𝑦₃+𝜈₃,𝜇₄=−𝑧₁𝑦₁−𝑧₁𝑢₃−𝑢₁𝑦₁−𝑧₂𝑦₂−𝑧₂𝑢₄−𝑢₂𝑦₂+𝑑𝑦₄−𝑧₁−𝑘₄𝑦₄+𝜈₄ (12) and the parameter estimation update laws as 𝛽=−𝑦²₃,𝛾=𝑧₁𝑦₁+𝑦²₁+𝑧₂𝑦₂+𝑦²₂,𝑑=−𝑦²₄, (13) where 𝑘=[𝑘₁,𝑘₂,𝑘₃,𝑘₄]^𝑇 is a positive constant vector, 𝜈=[𝜈₁,𝜈₂,𝜈₃,𝜈₄]^𝑇 is an external signal vector which is connected with the reference input, and 𝛽, 𝛾, and 𝑑 are estimated values of the unknown parameters 𝛼, 𝛽, 𝛾, and 𝑑, respectively, then the error dynamical system (8) will be rendered passive and will be asymptotically stable at its equilibrium (0,0,0,0) and the two systems (5) and (6) starting from different initial values will be synchronized.

Proof. Construct the following storage function: 𝑉(𝑧)+1𝑧,𝑦=𝑊2𝑦^𝑇1𝑦+2(𝛼−𝛼)²+12𝛽−𝛽²+12𝛾−𝛾²+12𝑑−𝑑², (14) where 𝑊(𝑧)=(1/2)(𝑧²₁+𝑧²₂) is a Lyapunov function of 𝑓₀(𝑧).
The zero dynamics of system (10) describe the internal dynamics, which are consistent with the external constraint 𝑦=0; that is, 𝑧̇=𝑓₀(𝑧).
Differentiating 𝑊(𝑧) with respect to 𝑡, we have 𝑑𝑊(𝑧)=(𝑧)𝑑𝑡𝜕𝑊𝑓𝜕𝑧⁰(𝑧)=−𝛼𝑧²₁−𝛼𝑧²₂≤0. (15)
Then 𝑓₀(𝑧) is globally asymptotically stable; that is, the zero dynamics of system (8) is Lyapunov stable. In the light of Definition 1, system (8) is a minimum phase system.
Furthermore, taking the time derivative of 𝑉(𝑧,𝑦) along the trajectory of the error dynamical system (8) yields 𝑑𝑉=(𝑧)𝑑𝑡𝑧,𝑦𝜕𝑊̇𝜕𝑧𝑧+𝑦^𝑇̇()̇𝛽̇+̇𝑑̇=(𝑧)𝑦+𝛼−𝛼𝛼+𝛽−𝛽𝛾−𝛾𝛾+𝑑−𝑑𝜕𝑊𝑓𝜕𝑧⁰(𝑧)+𝜕𝑊(𝑧)𝑝𝑦𝜕𝑧𝑧,𝑦+𝑦^𝑇𝑏𝑧,𝑦+𝑦^𝑇𝑎()𝛼̇+̇̇̇𝑧,𝑦𝜇+𝛼−𝛼𝛽−𝛽𝛽+𝛾−𝛾𝛾+𝑑−𝑑𝑑. (16)
Since the error dynamical system (8) is a minimum phase system; that is, ((𝜕𝑊(𝑧))/𝜕𝑧)𝑓₀(𝑧)≤0, then (15) becomes 𝑑𝑉≤(𝑧)𝑑𝑡𝑧,𝑦𝜕𝑊𝑝𝜕𝑧𝑧,𝑦𝑦+𝑦^𝑇𝑏𝑧,𝑦+𝑦^𝑇𝑎𝜇+()̇̇𝛾̇+̇𝑧,𝑦𝛼−𝛼𝛼+𝛽−𝛽𝛽+𝛾−𝛾𝑑−𝑑𝑑. (17)
Substituting (12) and (13) into (17) yields 𝑑𝑉𝑑𝑡𝑧,𝑦≤−𝑘𝑦^𝑇𝑦+𝜈^𝑇𝑦. (18)
Then, taking integration on both sides of (18), we get 𝑉𝑧,𝑦−𝑉𝑧₀,𝑦₀≤−^𝑡₀𝑘𝑦^𝑇(𝜏)𝑦(𝜏)+𝑑𝜏_𝑡0𝜈^𝑇(𝜏)𝑦(𝜏)𝑑𝜏. (19)
For 𝑉(𝑧,𝑦)≥0, let 𝑉(𝑧₀,𝑦₀)=𝑢; then the above inequality can be rewritten as ^𝑡₀𝑣^𝑇(𝜏)𝑦(𝜏)𝑑𝜏+𝑢≥^𝑡₀𝑘𝑦^𝑇(𝜏)𝑦(𝜏)≥𝑑𝜏+𝑉𝑧,𝑦_𝑡0𝑘𝑦^𝑇(𝜏)𝑦(𝜏)𝑑𝜏. (20)
Letting 𝜈=[𝜈₁,𝜈₂,𝜈₃,𝜈₄]^𝑇=[0,0,0,0]^𝑇, in light of Definition 2 and Theorem 3, system (8) will be stabilized at its equilibrium (0,0,0,0) with the controllers (12) and the parameter estimation update laws (13); that is, the drive system (5) and response system (6) with different initial conditions will be synchronized with each other asymptotically.
This completes the proof.

Remark 5. The controllers (12) are only related to the parameters 𝛽, 𝛾, and 𝑑, and so we do not need to estimate the parameter 𝛼.

4. Antisynchronization of the Hyperchaotic Complex Chen System with Unknown Parameters

To investigate the antisynchronization of the hyperchaotic complex Chen system, we need to add system (6) to system (5) and obtain the following antisynchronization error dynamical system: 𝑒₁̇+𝑖𝑒₂̇=𝛼𝑦₁−𝑥₁+𝑤₁+𝛼,𝑒̇𝑦−𝑥+𝑤₃+𝑖𝑒₄̇=𝛾−𝛼𝑥₁−𝑥₁𝑧₁+𝛾𝑦₁+𝛾−𝛼𝑥−𝑥𝑧+𝛾𝑦+𝜇₁+𝑖𝜇₂,𝑒₅̇=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝛽𝑧₁+𝑤₁+12𝑥𝑦+𝑥𝑦−𝛽𝑧+𝑤+𝜇₃,𝑒₆̇=12𝑥₁𝑦₁+𝑥₁𝑦₁−𝑑𝑤₁+12𝑥𝑦+𝑥𝑦−𝑑𝑤+𝜇₄, (21) where 𝑒_𝑗=𝑣_𝑗+𝑢_𝑗(𝑗=1,2,3,4,5,6) are antisynchronization error states and 𝛼, 𝛽, 𝛾, and 𝑑 are unknown parameters. By separating the real and imaginary parts of the antisynchronization error dynamical system (21), we get the following real system: 𝑒₁̇=𝛼𝑒₃−𝑒₁+𝑒₆,𝑒₂̇=𝛼𝑒₄−𝑒₂,𝑒₃̇=𝛾−𝛼𝑒₁+𝛾𝑒₃−𝑒₁𝑒₅+𝑒₁𝑢₅+𝑢₁𝑒₅−2𝑢₁𝑢₅+𝜇₁,𝑒₄̇=𝛾−𝛼𝑒₂+𝛾𝑒₄−𝑒₂𝑒₅+𝑒₂𝑢₅+𝑢₂𝑒₅−2𝑢₂𝑢₅+𝜇₂,𝑒₅̇=𝑒₁𝑒₃−𝑒₁𝑢₃−𝑢₁𝑒₃+2𝑢₁𝑢₃+𝑒₂𝑒₄−𝑒₂𝑢₄−𝑢₂𝑒₄+2𝑢₂𝑢₄−𝛽𝑒₅+𝑒₆+𝜇₃,𝑒₆̇=𝑒₁𝑒₃−𝑒₁𝑢₃−𝑢₁𝑒₃+2𝑢₁𝑢₃+𝑒₂𝑒₄−𝑒₂𝑢₄−𝑢₂𝑒₄+2𝑢₂𝑢₄−𝑑𝑒₆+𝜇₄. (22)

Let 𝑧₁=𝑒₁, 𝑧₂=𝑒₂, 𝑦₁=𝑒₃, 𝑦₂=𝑒₄, 𝑦₃=𝑒₅, and 𝑦₄=𝑒₆; then the error dynamical system (22) can be rewritten as 𝑧̇₁=𝛼𝑦₁−𝑧₁+𝑦₄,𝑧̇₂=𝛼𝑦₂−𝑧₂,𝑦̇₁=𝛾−𝛼𝑧₁+𝛾𝑦₁−𝑧₁𝑦₃+𝑧₁𝑢₅+𝑢₁𝑦₃−2𝑢₁𝑢₅+𝜇₁,𝑦̇₂=𝛾−𝛼𝑧₂+𝛾𝑦₂−𝑧₂𝑦₃+𝑧₂𝑢₅+𝑢₂𝑦₃−2𝑢₂𝑢₅+𝜇₂,𝑦̇₃=𝑧₁𝑦₁−𝑧₁𝑢₃−𝑢₁𝑦₁+2𝑢₁𝑢₃+𝑧₂𝑦₂−𝑧₂𝑢₄−𝑢₂𝑦₂+2𝑢₂𝑢₄−𝛽𝑦₃+𝑦₄+𝜇₃,𝑦̇₄=𝑧₁𝑦₁−𝑧₁𝑢₃−𝑢₁𝑦₁+2𝑢₁𝑢₃+𝑧₂𝑦₂−𝑧₂𝑢₄−𝑢₂𝑦₂+2𝑢₂𝑢₄−𝑑𝑦₄+𝜇₄. (23) Thus, we can establish the following theorem.

Theorem 6. If the passive controllers are designed as 𝜇₁=−𝛾𝑧₁−𝛾𝑦₁+𝑧₁𝑦₃−𝑧₁𝑢₅−𝑢₁𝑦₃+2𝑢₁𝑢₅−𝑘₁𝑦₁+𝜈₁,𝜇₂=−𝛾𝑧₂−𝛾𝑦₂+𝑧₂𝑦₃−𝑧₂𝑢₅−𝑢₂𝑦₃+2𝑢₂𝑢₅−𝑘₂𝑦₂+𝜈₂,𝜇₃=−𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁−2𝑢₁𝑢₃−𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−2𝑢₂𝑢₄+𝛽𝑦₃−𝑦₄−𝑘₃𝑦₃+𝜈₃,𝜇₄=−𝑧₁𝑦₁+𝑧₁𝑢₃+𝑢₁𝑦₁−2𝑢₁𝑢₃−𝑧₂𝑦₂+𝑧₂𝑢₄+𝑢₂𝑦₂−2𝑢₂𝑢₄+𝑑𝑦₄−𝑧₁−𝑘₄𝑦₄+𝜈₄ (24) and the parameter estimation update laws as 𝛽=−𝑦²₃,𝛾=𝑧₁𝑦₁+𝑦²₁+𝑧₂𝑦₂+𝑦²₂,𝑑=−𝑦²₄, (25) where 𝑘=[𝑘₁,𝑘₂,𝑘₃,𝑘₄]^𝑇 is a positive constant vector, 𝜈=[𝜈₁,𝜈₂,𝜈₃,𝜈₄]^𝑇 is an external signal vector which is connected with the reference input, and 𝛽, 𝛾, and 𝑑 are estimated values of the unknown parameters 𝛽, 𝛾, and 𝑑, respectively, then the antisynchronization error dynamical system (22) will be rendered passive and will be asymptotically stable at its equilibrium (0,0,0,0) and the two systems (5) and (6) starting from different initial values will be antisynchronized.

Proof. The proof of Theorem 6 is similar to that of Theorem 4, so it is omitted here.

5. Numerical Simulations

In this section, we perform two numerical simulations to demonstrate the effectiveness of the above synchronization and antisynchronization schemes. In the following numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. The system parameters are selected as 𝛼=32, 𝛽=5, 𝛾=25, and 𝑑=6 so that the hyperchaotic complex Chen system (5) exhibits hyperchaos.

Example 7. For the synchronization of the hyperchaotic complex Chen system, we consider the drive system (5) and the response system (6) with the controllers (12) and update laws (13). The initial values for the drive system (5) and response system (6) are given as (𝑥(0),𝑦(0),𝑧(0),𝑤(0))=(1+2𝑖,3−𝑖,−2,−3) and (𝑥₁(0),𝑦₁(0),𝑧₁(0),𝑤₁(0))=(1.5+3𝑖,4.5+2𝑖,1,5); thus (𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑢₅,𝑢₆)=(1,2,3,−1,−2,−3) and (𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅,𝑣₆)=(1.5,3,4.5,2,1,5), respectively. The initial errors are (𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅,𝑒₆)=(0.5,1,1.5,3,3,8). And the initial values of the parameter estimation update laws are 𝛽(0)=𝛾(0)=𝑑(0)=0.1. We choose 𝑘₁=𝑘₂=𝑘₃=𝑘₄=1 and 𝜈₁=𝜈₂=𝜈₃=𝜈₄=0. Figure 1 shows the time response of states determined by the drive system (5) and the response system (6) with the controllers (12) and the parameter estimation update laws (13). Figure 2 shows the time response of error states for the error dynamical system (8). From Figures 1 and 2, we can see that the two systems (5) and (6) starting from different initial conditions synchronize with each other immediately and the trajectories of the error dynamical system (8) are asymptotically stabilized at the equilibrium point 𝑂(0,0,0,0).

Figure 1 

The time response of states for the drive system (5) and the response system (6) with controllers (12) and parameter estimation update laws (13).

Figure 2 

The time response of synchronization error states 𝑒𝑗(𝑗=1,2,3,4,5,6) for the error dynamical system (8).

Example 8. For the antisynchronization of the hyperchaotic complex Chen system, we also consider the drive system (5) and the response system (6) but with the controllers (24) and update laws (25). The initial values for the drive system (5) and response system (6) are also given as (𝑥(0),𝑦(0),𝑧(0),𝑤(0))=(1+2𝑖,3−𝑖,−2,−3) and (𝑥₁(0),𝑦₁(0),𝑧₁(0),𝑤₁(0))=(1.5+3𝑖,4.5+2𝑖,1,5); thus (𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑢₅,𝑢₆)=(1,2,3,−1,−2,−3) and (𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅,𝑣₆)=(1.5,3,4.5,2,1,5), respectively. But the initial errors are (𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅,𝑒₆)=(2.5,5,7.5,1,−1,2). And the initial values of the parameter estimation update laws are 𝛽(0)=𝛾(0)=𝑑(0)=0.1. We choose 𝑘₁=𝑘₂=𝑘₃=𝑘₄=1 and 𝜈₁=𝜈₂=𝜈₃=𝜈₄=0. Figure 3 shows the time response of states determined by the drive system (5) and the response system (6) with the controllers (24) and the parameter estimation update laws (25). The time response of error states for the antisynchronization error dynamical system (22) is shown in Figure 4. From Figures 3 and 4, we can see that the two systems starting from different initial conditions antisynchronize with each other immediately and the trajectories of the antisynchronization error dynamical system (22) are asymptotically stabilized at the zero equilibrium.

Figure 3 

The time response of states for the drive system (5) and the response system (6) with controllers (24) and parameter estimation update laws (25).

Figure 4 

The time response of antisynchronization error states 𝑒𝑗(𝑗=1,2,3,4,5,6) for the error dynamical system (22).

6. Conclusions

Hyperchaotic systems with real variables have been investigated extensively over the past three decades. But hyperchaotic complex systems have attracted increasing attention due to the fact that they have much wider applications. So we investigate the synchronization and antisynchronization problems of a hyperchaotic complex Chen system by applying the passive control technique. Based on the fact that once a system is passive, there exists a control law that makes the passive system stable, then the passivity-based controller can be proposed to asymptotically stabilize the error dynamical system. Then corresponding passive controllers and update laws of the parameters are proposed to achieve synchronization and antisynchronization between two hyperchaotic complex Chen systems with different initial conditions, respectively. Furthermore, this work can be extended to achieve synchronization and antisynchronization of other versions of the hyperchaotic complex Chen system, even other types of hyperchaotic complex systems, such as Lorenz system [26] and Lü system [27].

Acknowledgments

This work was supported by the Youth Foundation of Yunnan University of Nationalities under Grant no. 11QN07, the Natural Science Foundation of Yunnan Province under Grants no. 2009CD019 and no. 2011FZ172, and the Natural Science Foundation of China under Grants no. 61065008, no. 61005087, and no. 61263042.