Universal Projective Synchronization of Two Different Hyperchaotic Systems with Unknown Parameters

Abstract

Universal projective synchronization (UPS) of two chaotic systems is defined. Based on the Lyapunov stability theory, an adaptive control method is derived such that UPS of two different hyperchaotic systems with unknown parameters is realized, which is up to a scaling function matrix and three kinds of reference systems, respectively. Numerical simulations are used to verify the effectiveness of the scheme.

1. Introduction

Chaotic synchronization has gained extensively and intensively attention in recent decades since Pecora and Carroll’s pioneering and meaningful work [1]. It has applications in vast areas of physics, chemistry, biology, and others particularly in secure communication. Various synchronization modes are proposed, such as complete synchronization [1], phase synchronization [2], projective synchronization [3], and antisynchronization [4]. Projective synchronization (PS) [5] as a generalization of complete synchronization (CS) and antisynchronization (AS), has attracted much attention for the scaling factor in which the state vectors become proportional.

The projective synchronization is firstly applied in a class of systems with partial linearity. Yan and Li [6] applied this notion in a general class of chaotic systems including nonpartially-linear systems in 2005: this synchronization mode was called generalized projective synchronization (GPS). In [7], the modified projective synchronization (MPS) is presented to synchronize two identical systems up to a scaling constant matrix. In 2007, Chen and Li [8] gave a new synchronization mode—function projective synchronization (FPS) in which scaling function matrix substitutes the scaling constant matrix. Tang et al. [9] presented FPS in another familiar way by using a function instead of the scaling factor of GPS. Du et al. [10] applied Tang’s definition to synchronize different chaotic systems with uncertain parameters. A more general definition called modified function projective synchronization (MFPS) was given by Du et al. [11] in 1999 and was used in secure communication. In recent years, many efforts are concentrated on MFPS [12–14].

From the development, we know that the scaling factor is expanded from constant to constant matrix and from function to matrix function.

In this paper, we will introduce a new type of synchronization called universal projective synchronization (UPS) which concerns the displacement. Adaptive synchronization is an effective method for dealing with unknown parameters [10, 15]. Zhang et al. [16] proposed a new hyperchaotic system. UPS of hyperchaotic L𝑢̈ system [17] and the new one with unknown parameters will be investigated by using adaptive control method. Three kinds of reference systems are chosen for the UPS. The introduction of UPS will expand the scope of synchronization.

The rest of this paper is organized as follows. In Section 2, the definition of UPS is given. In Section 3, the UPS of hyperchaotic L𝑢̈ system and the new one is studied. Adaptive control schemes are proposed. Simulation results verify the effectiveness of the schemes. The conclusion is given in Section 4.

2. The Definition of UPS

The drive system and the response system are defined as follows: ̇(),̇()(),𝐱=𝐟𝑡,𝐱𝐲=𝐠𝑡,𝐲+𝐮𝑡,𝐱,𝐲 (1)(2) where 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 and 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛 are the state vectors; 𝐟,𝐠∶𝑅^𝑛→𝑅^𝑛 are continuous nonlinear vector function; 𝐮(𝑡,𝐱,𝐲)∈𝑅^𝑛 is the controller to be designed. If there is no controller 𝐮(𝑡,𝐱,𝐲), (1) and (2) are chaotic systems. We define the state error vector 𝐞=𝑒₁,𝑒₂,…,𝑒_𝑛^𝑇≜𝐱−𝐀(𝑡)𝐲, (3) where 𝐀(𝑡)=diag(𝛼₁(𝑡),𝛼₂(𝑡),…,𝛼_𝑛(𝑡)); 𝛼_𝑖(𝑡)(𝑖=1,2,3,…,𝑛) is continuous differentiable function with boundness, and 𝛼_𝑖(𝑡)≠0 for all 𝑡. Obviously, the error system is 𝐞̇=𝐟(𝑡,𝐱)−𝐀̇(𝑡)𝐲−𝐀(𝑡)(𝐠(𝑡,𝐲)+𝐮(𝑡,𝐱,𝐲)). (4) Suppose there exists a reference system 𝐳̇=𝐡(𝑡,𝐳), (5) where 𝐳=(𝑧₁,𝑧₂,…,𝑧_𝑛)^𝑇∈𝑅^𝑛, 𝐡∶𝑅^𝑛→𝑅^𝑛 is a continuous differentiable function. Suppose the reference system is an attractor.

Definition 1. For the scaling diagonal matrix function 𝐀(𝑡), it is said that the drive system (1) and the response system (2) are universal projective synchronization (UPS) in the sense of the system (5), if the error system (4) and the reference system (5) are completely synchronized; that is, lim_𝑡→∞‖𝐞−𝐳‖=0. (6)

Now we give another dual definition of UPS. We define the state error vector 𝐞^∗=𝑒^∗₁,𝑒^∗₂,…,𝑒^∗_𝑛^𝑇≜𝐀(𝑡)𝐱−𝐲; (7) then the error system is 𝐞̇^∗=𝐀̇(𝑡)𝐱+𝐀(𝑡)𝐟(𝑡,𝐱)−𝐠(𝑡,𝐲)−𝐮(𝑡,𝐱,𝐲). (8)

Definition 2. For the scaling diagonal matrix function 𝐀(𝑡), it is said that the drive system (1) and the response system (2) are universal projective synchronization (UPS) in the sense of the system (5), if there exists a scaling diagonal matrix function 𝐀(𝑡) such that the error system (8) and the reference system (5) are completely synchronized; that is, lim_𝑡→∞𝐞^∗−𝐳=0. (9)

Remark 3. It is easy to see that UPS degenerated to MFPS if the reference system denotes constant vector 𝟎; that is, the initial value and state change rate of system (5) are all 𝟎.

Remark 4. The reference system (5) expresses a constant vector 𝝃, if its initial value is 𝝃 and the state change rate is 𝟎. UPS is in the sense of constant vector 𝝃; that is, there exists a displacement 𝝃 between the drive system and response system. System (5) can be other attractors, such as periodic function, quasiperiod function, chaos, and hyperchaos.
Thereinafter UPS is always in the sense of Definition 1.

3. UPS of Hyperchaotic Systems with Uncertain Parameters

We take hyperchaotic L𝑢̈ system [17] as the drive system, which is described by ̇̇̇̇𝑥=𝑎𝑦−𝑥+𝑤,𝑦=−𝑥𝑧+𝑐𝑦,𝑧=𝑥𝑦−𝑏𝑧,𝑤=𝑥𝑧+𝑑𝑤, (10) where 𝑥, 𝑦, 𝑧, 𝑤 are state variables and 𝑎, 𝑏, 𝑐, 𝑑 are uncertain parameters, which need to be estimated. The new hyperchaotic system presented by Zhang et al. [16], as the response system, is given by 𝑥̇₁=𝑎₁𝑦₁−𝑥₁+𝑢₁,𝑦̇₁=𝑏₁𝑦₁−𝑥₁𝑧₁+𝑤₁+𝑢₂,𝑧̇₁=𝑥₁𝑦₁−3.36𝑧₁+𝑢₃,𝑤̇₁=−𝑥₁𝑧₁+𝑤₁+𝑢₄, (11) where 𝑥₁, 𝑦₁, 𝑧₁, 𝑤₁ are state variables, 𝑎₁, 𝑏₁ are uncertain parameters, which need to be estimated, and 𝑢₁,𝑢₂,𝑢₃,𝑢₄ are the nonlinear control laws. Suppose the scaling matrix 𝐀(𝑡)=diag(𝛼₁(𝑡),𝛼₂(𝑡),𝛼₃(𝑡),𝛼₄(𝑡)). Obviously the errors 𝑒₁=𝑥−𝛼₁(𝑡)𝑥₁, 𝑒₂=𝑦−𝛼₂(𝑡)𝑦₁, 𝑒₃=𝑧−𝛼₃(𝑡)𝑧₁, and 𝑒₄=𝑤−𝛼₄(𝑡)𝑤₁. The estimations of 𝑎, 𝑏, 𝑐, 𝑑, 𝑎₁, 𝑏₁ are 𝑎, 𝑏̂, 𝑐̂, 𝑑, 𝑎₁, 𝑏̂₁, respectively. 𝑎=𝑎−𝑎, 𝑏̃=𝑏̂−𝑏, 𝑐̃=𝑐̂−𝑐, 𝑑=𝑑−𝑑, 𝑎₁=𝑎₁−𝑎₁, and 𝑏̃₁=𝑏̂₁−𝑏₁ are the corresponding parameter estimate errors.

Now we discuss the UPS with three kinds of reference systems.

3.1. UPS in the Sense of Vector Function

Take the vector function 𝜷(𝑡)=(𝛽₁(𝑡),𝛽₂(𝑡),𝛽₃(𝑡),𝛽₄(𝑡))^𝑇 as the reference system. Let 𝐞̃=(𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃)^𝑇≜(𝑒₁−𝛽₁,𝑒₂−𝛽₂,𝑒₃−𝛽₃,𝑒₄−𝛽₄)^𝑇.

Theorem 5. For the given scaling matrix 𝐀(𝑡) and the vector function 𝜷(𝑡), the UPS between drive system (10) and response system (11) with unknown parameters will occur by the control 𝐮=(𝑢₁,𝑢₂,𝑢₃,𝑢₄) and parameter update law as follows: 𝑢₁=1𝛼₁(𝑡)𝑎𝑦−𝑥+𝑤−𝛼̇₁(𝑡)𝑥₁−𝛼₁(𝑡)𝑎₁𝑦₁−𝑥₁−𝛽̇₁(𝑡)+𝑘₁𝑒₁̃,𝑢₂=1𝛼₂(𝑡)−𝑥𝑧+𝑐𝑦̂−𝛼̇₂(𝑡)𝑦₁+𝛼₂(𝑡)𝑥₁𝑧₁−𝛼₂(𝑡)𝑏̂₁𝑦₁−𝛼₂(𝑡)𝑤₁−𝛽̇₂(𝑡)+𝑘₂𝑒₂̃,𝑢₃=1𝛼₃(𝑡)𝑥𝑦−𝑏̂𝑧−𝛼̇₃(𝑡)𝑧₁−𝛼₃(𝑡)𝑥₁𝑦₁+3.36𝛼₃(𝑡)𝑧₁−𝛽̇₃(𝑡)+𝑘₃𝑒₃̃,𝑢₄=1𝛼₄(𝑡)𝑥𝑧+𝑑𝑤−𝛼̇₄(𝑡)𝑤₁+𝛼₄(𝑡)𝑥₁𝑧₁−𝛼₄(𝑡)𝑤₁−𝛽̇₄(𝑡)+𝑘₄𝑒₄̃,𝑎̇=𝑦−𝑥𝑒₁̃,̇̂̃𝑏=−𝑧𝑒₃,𝑐̇̂=𝑦𝑒₂̃,̇̃𝑑=𝑤𝑒₄,𝑎̇₁=−𝛼₁(𝑡)𝑦₁−𝑥₁𝑒₁̃,̇̂𝑏₁=−𝛼₂(𝑡)𝑦₁𝑒₂̃, (12)(13) where 𝑘₁,𝑘₂,𝑘₃,𝑘₄ are positive control gains.

Proof. Choose the following Lyapunov function: 1𝑉=2𝐞̃^𝑇𝐞̃+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑎²₁+𝑏̃²₁. (14) Then the derivative of 𝑉 is given by 𝑉̇=122𝐞̃^𝑇̃̇𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝑑𝑎𝐞+2𝑎+2𝑏+2𝑐+2+2₁̇𝑎₁+2𝑏̃₁̃̇𝑏₁=𝐞̃^𝑇̃̇𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝐞+𝑎+𝑏+𝑐+𝑑+𝑎₁̇𝑎₁+𝑏̃₁̃̇𝑏₁=𝑒₁̃𝑎𝑦−𝑥+𝑤−𝛼̇₁𝑥₁−𝛼₁𝑎₁𝑦₁−𝑥₁−𝛼₁𝑢₁−𝛽̇₁+𝑒₂̃−𝑥𝑧+𝑐𝑦−𝛼̇₂𝑦₁−𝛼₂𝑏₁𝑦₁−𝑥₁𝑧₁+𝑤₁−𝛼₂𝑢₂−𝛽̇₂+𝑒₃̃𝑥𝑦−𝑏𝑧−𝛼̇₃𝑧₁−𝛼₃𝑥₁𝑦₁−𝑐₁𝑧₁−𝛼₃𝑢₃−𝛽̇₃+𝑒₄̃𝑥𝑧+𝑑𝑤−𝛼̇₄𝑤₁−𝛼₄−𝑥₁𝑧₁+𝑤₁−𝛼₄𝑢₄−𝛽̇₄+𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝑎+𝑏+𝑐+𝑑+𝑎₁̇𝑎₁+𝑏̃₁̃̇𝑏₁. (15) With the controls (12) and the parameter update laws (13), we have 𝑉̇=−𝑘₁𝑒²₁̃−𝑘₂𝑒²₂̃−𝑘₃𝑒²₃̃−𝑘₄𝑒²₄̃. (16) Because 𝑉̇ is negative simidefinite, 𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃∈𝐿_∞, 𝑎,𝑏̃,𝑐,̃𝑑,𝑎₁,𝑏̃₁∈𝐿_∞. From the definition of 𝐞̃, we know ̇𝑒₁̃,̇𝑒₂̃,̇𝑒₃̃,̇𝑒₄̃∈𝐿_∞. From 𝑉̇=−𝑘₁𝑒²₁̃−𝑘₂𝑒²₂̃−𝑘₃𝑒²₃̃−𝑘₄𝑒²₄̃=−𝑒^𝑇̃𝐾𝑒̃ (𝐾=diag(𝑘₁,𝑘₂,𝑘₃,𝑘₄)), we have ^𝑡₀𝜆_min(𝐾)‖𝑒‖̃²𝑑𝑡≤^𝑡₀𝑒^𝑇̃𝐾̃≤𝑒𝑑𝑡_𝑡0−𝑉̇𝑑𝑡=𝑉(0)−𝑉(𝑡)≤𝑉(0), (17) where 𝜆_min(𝐾) is the minimum eigenvalue of 𝐾, so 𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃∈𝐿_∞; according to Barbalat’s Lemma we have lim_𝑡→∞‖𝐞̃(𝑡)‖=0. (18) The UPS of system (10) and system (11) in the sense of vector function 𝜷(𝑡) is achieved.
Now we illustrate the effectiveness of the method. Suppose the vector function 𝜷(𝑡)=(2sin𝑡+cos2𝑡,32sin(3𝑡)+6,6cos𝑡,2.7sin(2𝑡)+3cos𝑡)^𝑇. The “unknown” parameters are 𝑎=36, 𝑏=3, 𝑐=20, 𝑑=1.3, 𝑎₁=26, and 𝑏₁=14. Control gain is (𝑘₁,𝑘₂,𝑘₃,𝑘₄)=(2,3,4,5). The scaling matrix is 𝐀(𝑡)=diag(sin2𝑡+2,−cos3𝑡−3,sin𝑡+cos𝑡+3,2sin𝑡−3). The initial values of the drive system and response system are (3,−4,2,2) and (5,7,9,11), respectively. The parameter estimate initial values are 𝑎(0)=6, 𝑏̂(0)=−3, 𝑐(̂0)=6, 𝑑(0)=6, 𝑎₁(0)=6, and 𝑏̂₁(0)=3. Simulation results are as Figures 1, 2, 3, 4, 5, 6, 7, and 8.

Figure 1 

The trajectories projection on 𝑥−𝑦(𝑥1−𝑦1) plane of the drive system and the response system.

Figure 2 

The trajectories projection on 𝑧−𝑤(𝑧1−𝑤1) plane of the drive system and the response system.

Figure 3 

The time evolution of the error 𝑒1 and the function 𝛽1(𝑡).

Figure 4 

The time evolution of the error 𝑒2 and the function 𝛽2(𝑡).

Figure 5 

The time evolution of the error 𝑒3 and the function 𝛽3(𝑡).

Figure 6 

The time evolution of the error 𝑒4 and the function 𝛽4(𝑡).

Figure 7 

The time evolution of the estimated values 𝑎, 𝑏̂, 𝑐̂, and 𝑑.

Figure 8 

The time evolution of the estimated values 𝑎1 and 𝑏̂1.

3.2. UPS in the Sense of a Hyperchaotic System

We take hyperchaotic Wang system [18] as the reference system, which is given by 𝑥̇₂=𝑎₂−𝑥₂+𝑦₂+𝑤₂,𝑦̇₂=𝑐₂𝑥₂−𝑦₂−𝑥₂𝑧₂,𝑧̇₂=𝑥₂𝑦₂−𝑏₂𝑧₂,𝑤̇₂=−𝑦₂𝑧₂+𝑑₂𝑤₂, (19) where 𝑥₂, 𝑦₂, 𝑧₂, 𝑤₂ are state variables and 𝑎₂=10, 𝑏₂=8/3, 𝑐₂=28, and 𝑑₂=−1 are parameters.

Let 𝐞̃=(𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃)^𝑇≜(𝑒₁−𝑥₂,𝑒₂−𝑦₂,𝑒₃−𝑧₂,𝑒₄−𝑤₂)^𝑇.

Theorem 6. For the given scaling matrix 𝐀(𝑡) and the reference system which is hyperchaotic Wang system, the UPS between drive system (10) and response system (11) with unknown parameters will occur by the control 𝐮=(𝑢₁,𝑢₂,𝑢₃,𝑢₄) and parameter update law as follows: 𝑢₁=1𝛼₁(𝑡)𝑎𝑦−𝑥+𝑤−𝛼̇₁(𝑡)𝑥₁−𝛼₁(𝑡)𝑎₁𝑦₁−𝑥₁−𝑎₂−𝑥₂+𝑦₂+𝑤₂+𝑘₁𝑒₁̃,𝑢₂=1𝛼₂(𝑡)−𝑥𝑧+𝑐𝑦̂−𝛼̇₂(𝑡)𝑦₁+𝛼₂(𝑡)𝑥₁𝑧₁−𝛼₂(𝑡)𝑏̂₁𝑦₁−𝛼₂(𝑡)𝑤₁−𝑐₂𝑥₂−𝑦₂−𝑥₂𝑧₂+𝑘₂𝑒₂̃,𝑢₃=1𝛼₃(𝑡)𝑥𝑦−𝑏̂𝑧−𝛼̇₃(𝑡)𝑧₁−𝛼₃(𝑡)𝑥₁𝑦₁+3.36𝛼₃(𝑡)𝑧₁−𝑥₂𝑦₂−𝑏₂𝑧₂+𝑘₃𝑒₃̃,𝑢₄=1𝛼₄(𝑡)𝑥𝑧+𝑑𝑤−𝛼̇₄(𝑡)𝑤₁+𝛼₄(𝑡)𝑥₁𝑧₁−𝛼₄(𝑡)𝑤₁−−𝑦₂𝑧₂+𝑑₂𝑤₂+𝑘₄𝑒₄̃,𝑎̇=𝑦−𝑥𝑒₁̃,̇̂̃𝑏=−𝑧𝑒₃,𝑐̇̂=𝑦𝑒₂̃,̇̃𝑑=𝑤𝑒₄,𝑎̇₁=−𝛼₁(𝑡)𝑦₁−𝑥₁𝑒₁̃,̇̂𝑏₁=−𝛼₂(𝑡)𝑦₁𝑒₂̃. (20)(21)

The proof is similar to Theorem 5 (omitted!).

Now we illustrate the effectiveness of this method. Suppose the initial value of hyperchaotic Wang is (1,6,6,3). The “unknown” parameters are 𝑎=36, 𝑏=3, 𝑐=20, 𝑑=1.3, 𝑎₁=26, and 𝑏₁=14. Control gain is (𝑘₁,𝑘₂,𝑘₃,𝑘₄)=(12,13,14,15). The scaling matrix is 𝐀(𝑡)=diag(sin2𝑡+2,−cos3𝑡−3,sin𝑡+cos𝑡+3,2sin𝑡−3). The initial values of the drive system and response system are (3,−4,2,2) and (5,7,9,11), respectively. The parameter estimate initial values are 𝑎(0)=6, 𝑏̂(0)=−3, 𝑐(̂0)=6, 𝑑(0)=6, 𝑎₁(0)=6, and 𝑏̂₁(0)=3. Simulation results are as Figures 9, 10, 11, 12, 13, 14, 15, and 16.

Figure 9 

The trajectories projection on 𝑥−𝑦(𝑥1−𝑦1) plane of the drive system and the response system.

Figure 10 

The trajectories projection on 𝑧−𝑤(𝑧1−𝑤1) plane of the drive system and the response system.

Figure 11 

The time evolution of the error 𝑒1 and the state 𝑥2.

Figure 12 

The time evolution of the error 𝑒2 and the state 𝑦2.

Figure 13 

The time evolution of the error 𝑒3 and the state 𝑧2.

Figure 14 

The time evolution of the error 𝑒4 and the state 𝑤2.

Figure 15 

The time evolution of the estimated values 𝑎, 𝑏̂, 𝑐̂, and 𝑑.

Figure 16 

The time evolution of the estimated values 𝑎1 and 𝑏̂1.

3.3. UPS in the Sense of a Hyperchaotic Wang System with Unknown Parameters

We still use hyperchaotic Wang system as the reference system, but the parameters 𝑎₂, 𝑏₂, 𝑐₂, 𝑑₃ are uncertain. Suppose the estimated values of 𝑎₂, 𝑏₂, 𝑐₂, 𝑑₃ are 𝑎₂, 𝑏̂₂, 𝑐₂̂, 𝑑₂, respectively; the parameter errors are (𝑎₂,𝑏̃₂,𝑐₂̃,𝑑₂)=(𝑎₂−𝑎₂,𝑏̂₂−𝑏₂,𝑐₂̂−𝑐₂,𝑑₂−𝑑₂). Let 𝐞̃=(𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃)^𝑇≜(𝑒₁−𝑥₂,𝑒₂−𝑦₂,𝑒₃−𝑧₂,𝑒₄−𝑤₂)^𝑇.

Theorem 7. For the given scaling matrix 𝐀(𝑡) and the reference system which is hyperchaotic Wang system with unknown parameters, the UPS between drive system (10) and response system (11) with unknown parameters will occur by the control 𝐮=(𝑢₁,𝑢₂,𝑢₃,𝑢₄) and parameter update law as follows: 𝑢₁=1𝛼₁(𝑡)𝑎𝑦−𝑥+𝑤−𝛼̇₁(𝑡)𝑥₁−𝛼₁(𝑡)𝑎₁𝑦₁−𝑥₁−𝑎₂−𝑥₂+𝑦₂−𝑤₂+𝑘₁𝑒₁̃,𝑢₂=1𝛼₂(𝑡)−𝑥𝑧+𝑐𝑦̂−𝛼̇₂(𝑡)𝑦₁+𝛼₂(𝑡)𝑥₁𝑧₁−𝛼₂(𝑡)𝑏̂₁𝑦₁−𝛼₂(𝑡)𝑤₁−𝑐₂̂𝑥₂+𝑦₂+𝑥₂𝑧₂+𝑘₂𝑒₂̃,𝑢₃=1𝛼₃(𝑡)𝑥𝑦−𝑏̂𝑧−𝛼̇₃(𝑡)𝑧₁−𝛼₃(𝑡)𝑥₁𝑦₁+3.36𝛼₃(𝑡)𝑧₁−𝑥₂𝑦₂+𝑏̂₂𝑧₂+𝑘₃𝑒₃̃,𝑢₄=1𝛼₄(𝑡)𝑥𝑧+𝑑𝑤−𝛼̇₄(𝑡)𝑤₁+𝛼₄(𝑡)𝑥₁𝑧₁−𝛼₄(𝑡)𝑤₁+𝑦₂𝑧₂−𝑑₂𝑤₂+𝑘₄𝑒₄̃,𝑎̇=𝑦−𝑥𝑒₁̃,̇̂̃𝑏=−𝑧𝑒₃,𝑐̇̂=𝑦𝑒₂̃,̇̃𝑑=𝑤𝑒₄,𝑎̇₁=−𝛼₁(𝑡)𝑦₁−𝑥₁𝑒₁̃,̇̂𝑏₁=−𝛼₂(𝑡)𝑦₁𝑒₂̃,𝑎̇₂=−−𝑥₂+𝑦₂𝑒₁̃,̇̂𝑏₂=𝑧₂𝑒₃̃,𝑐₂̇̂=−𝑥₂𝑒₂̃,̇𝑑₂=−𝑤₂𝑒₄̃. (22)(23)(24)

Proof. Choose the following Lyapunov function: 1𝑉=2𝐞̃^𝑇𝐞̃+𝑎²+𝑏̃²+𝑐²̃+𝑑²+𝑎²₁+𝑏̃²₁+𝑎²₂+𝑏̃²₂+𝑐²₂̃+𝑑²₂. (25) Then the derivative of 𝑉 is given by 𝑉̇=122𝐞̃^𝑇̃̇𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝐞+2𝑎+2𝑏+2𝑐+2𝑑+2𝑎₁̇𝑎₁+2𝑏̃₁̃̇𝑏₁+2𝑎₂̇𝑎₂+2𝑏̃₂̃̇𝑏₂+2𝑐₂̃̇𝑐₂̃+2𝑑₂̇𝑑₂=𝐞̃^𝑇̃̇𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝐞+𝑎+𝑏+𝑐+𝑑+𝑎₁̇𝑎₁+𝑏̃₁̃̇𝑏₁+𝑎₂̇𝑎₂+𝑏̃₂̃̇𝑏₂+𝑐₂̃̇𝑐₂̃+𝑑₂̇𝑑₂=𝑒₁̃𝑎𝑦−𝑥+𝑤−𝛼̇₁𝑥₁−𝛼₁𝑎₁𝑦₁−𝑥₁−𝛼₁𝑢₁−𝑎₂−𝑥₂+𝑦₂−𝑤₂+𝑒₂̃−𝑥𝑧+𝑐𝑦−𝛼̇₂𝑦₁−𝛼₂𝑏₁𝑦₁−𝑥₁𝑧₁+𝑤₁−𝛼₂𝑢₂−𝑐₂𝑥₂+𝑦₂+𝑥₂𝑧₂+𝑒₃̃𝑥𝑦−𝑏𝑧−𝛼̇₃𝑧₁−𝛼₃𝑥₁𝑦₁−𝑐₁𝑧₁−𝛼₃𝑢₃−𝑥₂𝑦₂+𝑏₂𝑧₂+𝑒₄̃𝑥𝑧+𝑑𝑤−𝛼̇₄𝑤₁−𝛼₄−𝑥₁𝑧₁+𝑤₁−𝛼₄𝑢₄+𝑦₂𝑧₂−𝑑₂𝑤₂+𝑎̇̃𝑏̃̇𝑐̃̇̃𝑑̇𝑎+𝑏+𝑐+𝑑+𝑎₁̇𝑎₁+𝑏̃₁̃̇𝑏₁+𝑎₂̇𝑎₂+𝑏̃₂̃̇𝑏₂+𝑐₂̃̇𝑐₂̃+𝑑₂̇𝑑₂. (26) With the controls (22) and the parameter update laws (23) and (24), we have 𝑉̇=−𝑘₁𝑒²₁̃−𝑘₂𝑒²₂̃−𝑘₃𝑒²₃̃−𝑘₄𝑒²₄̃. (27) Because 𝑉̇ is negative simidefinite, 𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃∈𝐿_∞, 𝑎,𝑏̃,𝑐,̃𝑑,𝑎₁,𝑏̃₁,𝑎₂,𝑏̃₂,𝑐₂̃,𝑑₂∈𝐿_∞. From the definition of 𝐞̃, we know ̇𝑒₁̃,̇𝑒₂̃,̇𝑒₃̃,̇𝑒₄̃∈𝐿_∞. From 𝑉̇=−𝑘₁𝑒²₁̃−𝑘₂𝑒²₂̃−𝑘₃𝑒²₃̃−𝑘₄𝑒²₄̃=−𝑒^𝑇̃𝐾𝑒̃ (𝐾=diag(𝑘₁,𝑘₂,𝑘₃,𝑘₄)), we have ^𝑡₀𝜆_min(𝐾)‖𝑒‖̃²𝑑𝑡≤^𝑡₀𝑒^𝑇̃𝐾̃≤𝑒𝑑𝑡_𝑡0−𝑉̇𝑑𝑡=𝑉(0)−𝑉(𝑡)≤𝑉(0), (28) where 𝜆_min(𝐾) is the minimum eigenvalue of 𝐾, so 𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃∈𝐿_∞; according to Barbalat’s Lemma we have lim_𝑡→∞‖𝐞̃(𝑡)‖=0. (29)
The UPS of system (10) and system (11) in the sense of hyperchaotic Wang system (19) with uncertain parameters is achieved.
Now we illustrate the effectiveness of this method. Suppose the initial value of hyperchaotic Wang system is (1,6,6,3). The “unknown” parameters are 𝑎=36, 𝑏=3, 𝑐=20, 𝑑=1.3, 𝑎₁=26, 𝑏₁=14, 𝑎₂=10, 𝑏₂=8/3, 𝑐₂=28, and 𝑑₂=−1. Control gain is (𝑘₁,𝑘₂,𝑘₃,𝑘₄)=(12,13,14,15). The scaling matrix is 𝐀(𝑡)=diag(sin2𝑡+2,−cos3𝑡−3,sin𝑡+cos𝑡+3,2sin𝑡−3). The initial values of the drive system and the response system are (3,−4,2,2) and (5,7,9,11), respectively. The parameter estimate initial values are 𝑎(0)=6, 𝑏̂(0)=−3, 𝑐(̂0)=6, 𝑑(0)=6, 𝑎₁(0)=6, 𝑏̂₁(0)=3, 𝑎₂(0)=4, 𝑏̂₂(0)=4, 𝑐₂̂(0)=4, and 𝑑₂(0)=4. Simulation results are as Figures 17, 18, 19, 20, 21, 22, 23, 24, and 25.
From the above analysis, we find that, in the form, the adaptive laws equations (13), (21), and (23) corresponding three kinds of reference systems are the same.

Figure 17 

The trajectories projection on 𝑥−𝑦(𝑥1−𝑦1) plane of the drive system and the response system.

Figure 18 

The trajectories projection on 𝑧−𝑤(𝑧1−𝑤1) plane of the drive system and the response system.

Figure 19 

The time evolution of the error 𝑒1 and the state 𝑥2.

Figure 20 

The time evolution of the error 𝑒2 and the state 𝑦2.

Figure 21 

The time evolution of the error 𝑒3 and the state 𝑧2.

Figure 22 

The time evolution of the error 𝑒4 and the state 𝑤2.

Figure 23 

The time evolution of the estimated values 𝑎,𝑏̂,𝑐,̂𝑑.

Figure 24 

The time evolution of the estimated values 𝑎1,𝑏̂1.

Figure 25 

The time evolution of the estimated values 𝑎2,𝑏̂2,𝑐2̂,𝑑2.

4. Conclusion

In this paper, we have defined universal projective synchronization. For different hyperchaotic systems with uncertain parameters, we have given adaptive control method such that UPS has occured and the parameters have been determined. Three kinds of reference systems have been concerned and we find that, in the form, the adaptive laws are the same. Numerical simulations have verified the effectiveness of the scheme.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported partly by the National Natural Science Foundation of China (Grant no. 61374118), the program of Yunnan Provincial Science and Technology Department (Grant no. 2013FD046), and Youth Foundation of Qujing Normal University (Grant no. 2008QN034).