Generalized Projective Synchronization for Different Hyperchaotic Dynamical Systems

Abstract

Projective synchronization and generalized projective synchronization have recently been observed in the coupled hyperchaotic systems. In this paper a generalized projective synchronization technique is applied in the hyperchaotic Lorenz system and the hyperchaotic Lü. The sufficient conditions for achieving projective synchronization of two different hyperchaotic systems are derived. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.

1. Introduction

Chaos is an interesting phenomenon in nonlinear dynamical systems research area. In the last three decades, chaos has been extensively studied within the scientific, engineering, and mathematical communities [1–6].

A chaotic system is a nonlinear deterministic system that displays complex, noisy-like and unpredictable behavior. These motions are trajectories in which infinite unstable periodic orbits (UPOs) are embedded. Chaos is generally undesirable in many fields. This irregular and complex phenomenon can lead systems to harmful or even catastrophic situations. In these troublesome cases chaos should be suppressed as much as possible or totally eliminated. Therefore controlling chaos has become one of the most considerable research area in the nonlinear problems ranging from biology, physics and chemistry to economics.

Since Pecora and Carroll [7, 8] showed that it is possible to synchronize two identical chaotic systems, chaos synchronization has been intensively and extensively studied due to its potential applications in secure communication, ecological systems, system identification, and so forth.

Among all kinds of chaos synchronizations, projective synchronization is one of the most noticeable ones. This kind of synchronization was first observed in continuous systems [9] where a part of state variables possessed some partial linearity [10]. Its typical feature is that the state variables of the two-coupled system may synchronize up to a scaling factor but the Lyapunov exponents and fractal dimensions remain unchanged. Such synchronization has been relatively understood well [11–14].

In 1999, Mainieri and Rehacek [10] first reported the projective synchronization phenomenon and explained the mechanism of the formation of projective synchronization in three-dimensional systems and further attempted to predict the scaling factor by introducing a vector field. However, they only provided a guideline of predicting the scaling factor rather than a concrete theoretical solution.

Generalized synchronization [15–23] is another interesting chaos synchronization technique. It means that there exists a transformation which is able to map asymptotically the trajectories of the master attractor into those of the slave one. To understand such kind of synchronization needs much mathematics. Till now, there are relatively few publications for generalized synchronization.

A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems. Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant. Recently, Huang proposed a simple adaptive feedback control method, which dose not need to estimate or find feedback constant, to effectively synchronize two almost arbitrary identical hyperchaotic systems [24–26]. This technique has been adopted by some authors to realize the identical synchronization of almost all kinds of coupled identical neural networks with time—varying delay [27] and the complete synchronization in uncertain complex networks [28].

In this paper, we introduce a new synchronization technique, which is different from projective synchronization, but share the same typical feature of projective synchronization, that is, the Lyapunov exponents and fractal dimensions are also invariant during the synchronization process. To some extent, the synchronization presented here is very similar to the generalized synchronization.

The rest of the paper is organized as follows. In Section 2, a mathematical description of generalized projective synchronization is presented. In Section 3 system description is introduced. In Section 4, the projective synchronization problem of a hyperchaotic Lorenz system is investigated and numerical simulation results are demonstrated in Subsection 4.1. In Section 5, the generalized projective synchronization problem of hyperchaotic Lü system is presented, and numerical simulation results are given in Subsection 5.1. In Section 6, the generalized projective synchronization problem between hyperchaotic Lorenz system and hyperchaotic Lü system is presented, and numerical simulation results are given in Subsection 6.1. Finally, in Section 7 the conclusion of the paper is given.

2. Generalized Projective Synchronization of Chaotic Systems

First, both projective and generalized synchronization are introduced.

A partial linear system is often expressed as𝑢̇(𝐭)(𝐳)𝑧̇(𝐭)(),=𝐌𝐮,=𝐟𝐮,𝐳 (2.1) in which the state vector 𝑢 is linearly related to 𝑢̇ with respect to 𝑡, while the matrix 𝐌(𝐳) only depends upon the variable 𝑧 which is nonlinearly related to the variable 𝑢. Projective synchronization often occurs when two identical system are coupled through the variable 𝑧 in the form as𝑢̇_𝑑=𝐌(𝐳)𝐮_𝑑,𝑧̇(𝐭)=𝐟𝐮_𝑑,𝑢̇,𝐳_𝑟=𝐌(𝐳)𝐮_𝑟. (2.2) The subscripts 𝑑 and 𝑟 stand for the driver (or master) and response (or slave) systems, respectively.

If there exists a constant 𝛼∈𝑅 (𝛼≠0) such that lim_𝑡→∞‖𝑢_𝑟−𝛼𝑢_𝑑‖=0, then the projective synchronization between the drive system and response system is achieved, and we call 𝛼 as “scaling factor.”

Consider the following coupled system:𝑥̇_𝑑=𝑓𝑥_𝑑,𝑦̇_𝑟=𝑔𝑦_𝑟,ℎ_𝜇𝑥_𝑑, (2.3) where 𝑥_𝑑∈𝑅^𝑛,𝑦_𝑟∈𝑅^𝑘,𝑓∶𝑅^𝑛→𝑅^𝑛,ℎ∶𝑅^𝑛→𝑅^𝑘, and𝑔∶𝑅^2𝑘→𝑅^𝑘. When 𝜇=0, 𝑦_𝑟 evolves independently and has no relation to 𝑥_𝑑, and we assume that both systems are chaotic. When 𝜇≠0, the chaotic trajectories of the two systems are said to be generalized synchronization if there exists a transformation 𝜑∶𝑥_𝑑→𝑦_𝑟 which is able to map asymptotically the trajectories of the master attractor into those of the slave attractor 𝑦_𝑟(𝑡)=𝜑(𝑥_𝑑(𝑡)), regardless of the initial conditions in the basin of the synchronization manifold 𝑀={(𝑥_𝑑,𝑦_𝑟∶𝑦_𝑟(𝑡)=𝜑(𝑥_𝑑(𝑡))} [22, 23]. In general. 𝜑 is difficult to be determined.

In what follows, a new definition is introduced. Consider the following chaotic equations:𝑥̇_𝑑=𝑓𝑥_𝑑,𝑥̇_𝑟=𝑔𝑥_𝑟,𝑢𝑥_𝑑,𝑥_𝑟, (2.4) where 𝑥_𝑑,𝑥_𝑟∈𝑅^𝑛,𝑢∶𝑅^2𝑛→𝑅^𝑛,𝑔∶𝑅^2𝑛→𝑅^𝑛, and𝑢(0,0)=0,𝑔(𝑥,𝑢(0,0))=𝑓(𝑥)∶𝑅^2𝑘→𝑅^𝑘. If there exists a constant 𝛼∈𝑅 (𝛼≠0) such that lim_𝑡→∞‖𝑥_𝑟−𝛼𝑥_𝑑‖=0, then we call them “generalized projective synchronization.”

Remark 2.1. (i) This definition is very similar to that of generalized synchronization, see (2.3) and (2.4). (ii) The master attractor synchronizes to the slave one up to a scaling factor 𝑎. Obviously, the Lyapnove exponents and fractal dimension remain invariant. (iii) From the last equation of (2.4), 𝑢 can be regarded as a feedback controller (or “synchronizer”), that is, similar to [7, 10], and if and only if such feedback controller 𝑢 is applied to the slave system, generalized projective synchronization may occur.

Remark 2.2. From the definition, one has lim_𝑡→∞‖𝑥_𝑟−𝛼𝑥_𝑑‖, the limit of 𝛼(𝑡) as 𝑡→∞ is still written as 𝛼. So one gets lim_𝑡→∞log|𝛼(𝑡)|=lim_𝑡→∞log|𝑥_𝑟/𝑥_𝑑|.

3. System Description

Very recently, based on Lorenz system [29, 30] and Lü system [31, 32], two hyperchaotic systems, we are constructed by introducing state feedback controller function, which were named as hyperchaotic Lorenz system and hyperchaotic Lü system, respectively.

The hyperchaotic Lorenz system is described bẏ̇̇̇𝑥=𝑎𝑦−𝑥+𝑤,𝑦=−𝑥𝑧+𝑟𝑥−𝑦,𝑧=−𝑏𝑧+𝑥𝑦,𝑤=𝑑𝑤−𝑥𝑧. (3.1)

When parameters 𝑎=10,𝑟=28,𝑏=8/3, and 0.85<𝑑≤1.3, the system (3.1) shows hyperchaotic behavior, see Figure 1

Figure 1 

It shows the attractor of hyperchaotic Lorenz dynamical system at 𝑎=10,𝑟=28,𝑏=8/3, and 𝑑=1.3 in 𝑥,𝑦,𝑧 subspace.

The hyperchaotic Lü system is described by𝑥̇=𝑎₁̇𝑦−𝑥+𝑤𝑦=−𝑥𝑧+𝑐₁𝑦𝑧̇=−𝑏₁̇𝑧+𝑥𝑦𝑤=𝑑₁𝑤+𝑥𝑧. (3.2)

When parameters 𝑎₁=36,𝑏₁=3,𝑐₁=20, and −0.35<𝑑₁≤1.3, the system (3.2) has hyperchaotic attractor, see Figure 2

Figure 2 

It shows the attractor of hyperchaotic Lü dynamical system at 𝑎1=36,𝑏1=3,𝑐1=20, and𝑑1=1.3 in 𝑥,𝑦,𝑧 subspace.

4. Generalized Projective Synchronization for Hyperchaotic Lorenz System

In order to observe generalized projective synchronization between two identical hyperchaotic Lorenz systems, we assume that the drive system with four state variables denoted by the subscript 1 and the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The drive and response systems are defined below, respectively, 𝑥̇₁=𝑎𝑦₁−𝑥₁+𝑤₁,𝑦̇₁=−𝑥₁𝑧₁+𝑟𝑥₁−𝑦₁,𝑧̇₁=−𝑏𝑧₁+𝑥₁𝑦₁,𝑤̇₁=𝑑𝑤₁−𝑥₁𝑧₁,𝑥̇₂=𝑎𝑦₂−𝑥₂+𝑤₂+𝑢₁,𝑦̇₂=−𝑥₂𝑧₂+𝑟𝑥₂−𝑦₂+𝑢₂,𝑧̇₂=−𝑏𝑧₂+𝑥₂𝑦₂+𝑢₃,𝑤̇₂=𝑑𝑤₂−𝑥₂𝑧₂+𝑢₄, (4.1)(4.2) where 𝑈=[𝑢₁𝑢₂𝑢₃𝑢₄]^𝑇 is the controller functions. The controller 𝑈 is to be determined for the purpose of projective synchronizing the two identical hyperchaotic Lorenz systems.

In order to get generalized projective synchronization, we define the error system as the difference between the system (4.2) and (4.1). Set 𝑒_𝑥=𝑥₂−𝛼𝑥₁,𝑒_𝑦=𝑦₂−𝛼𝑦₁,𝑒_𝑧=𝑧₂−𝛼𝑧₁,𝑒_𝑤=𝑤₂−𝛼𝑤₁, (4.3) then one obtains the error dynamical system between (4.2) and(4.1)𝑒_𝑥̇=𝑎𝑒_𝑦−𝑒_𝑥+𝑒_𝑤+𝑢₁,𝑒_𝑦̇=𝑟𝑒_𝑥−𝑒_𝑦−𝑥₂𝑧₂+𝛼𝑥₁𝑧₁+𝑢₂,𝑒_𝑧̇=−𝑏𝑒_𝑧+𝑥₂𝑦₂−𝛼𝑥₁𝑦₁+𝑢₃,𝑒_𝑤̇=𝑑𝑒_𝑤−𝑥₂𝑧₂+𝛼𝑥₁𝑧₁+𝑢₄. (4.4)

Let𝑉₁=𝑢₁,𝑉₂=𝛼𝑥₁𝑧₁−𝑥₂𝑧₂+𝑢₂,𝑉₃=𝑥₂𝑦₂−𝛼𝑥₁𝑦₁+𝑢₃,𝑉₄=𝛼𝑥₁𝑧₁−𝑥₂𝑧₂+𝑢₄, (4.5) then the error dynamical system can be rewritten as𝑒_𝑥̇=𝑎𝑒_𝑦−𝑒_𝑥+𝑒_𝑤+𝑉₁,𝑒_𝑦̇=𝑟𝑒_𝑥−𝑒_𝑦+𝑉₂,𝑒_𝑧̇=−𝑏𝑒_𝑧+𝑉₃,𝑒_𝑤̇=𝑑𝑒_𝑤+𝑉₄. (4.6)

To get the projective synchronization to occur, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as 𝑡→∞. As we know, if all the eigenvalues of the Jacobian matrix of closed-loop system have negative real parts, the system is stable. Based on this theory, we desired the (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 to guarantee that all the eigenvalues of closed-loop system (4.6) have negative real part. There are of course some other choices of (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇, but here the choice is very easy and convenient. For simplicity, choose (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 as follows:𝑉₁𝑉₂𝑉₃𝑉₄𝑒=𝑀^𝑥𝑒_𝑦𝑒_𝑧𝑒_𝑤,where𝑀=0−𝑎0−1−𝑟0000000000−2𝑑. (4.7)

System (4.6) has four negative eigenvalues −10,−1,−8/3, and −1.3 when setting 𝑎=10,𝑟=28,𝑏=8/3, and 𝑑=1.3. That is to say, the error states 𝑒_𝑥, 𝑒_𝑥, 𝑒_𝑧, and 𝑒_𝑤 converge to zero as 𝑡→∞. So the generalized projective synchronization is achieved.

4.1. Numerical Results

By using MAPLE 12, the systems of differential equations (4.1) and (4.2) are solved numerically. The parameters are chosen as 𝑎=10,𝑟=28,𝑏=8/3, and 𝑑=1.3 in all simulations so that the hyperchaotic Lorenz system exhibits a chaotic behavior if no control is applied (see Figure 1). The initial states of the drive system are 𝑥₁(0)=0.1,𝑦₁(0)=0.1,𝑧₁(0)=0.1, and 𝑤₁(0)=0.1,and initial states of the response system are 𝑥₂(0)=1,𝑦₂(0)=−1,𝑧₂(0)=1, and 𝑤₂(0)=1.

Choosing 𝛼=−2 then the error system (4.4) has the initial values 𝑒_𝑥(0)=1.2,𝑒_𝑦(0)=−0.8,𝑒_𝑧(0)=1.2, and 𝑒_𝑤(0)=1.2. Figure 3 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡), and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 4 shows the evaluation of the ratios log|𝑥_𝑟/𝑥_𝑑|=log|𝑥₂/𝑥₁|,log|𝑦_𝑟/𝑦_𝑑|=log|𝑦₂/𝑦₁|,log|𝑧_𝑟/𝑧_𝑑|=log|𝑧₂/𝑧₁|, and log|𝑤_𝑟/𝑤_𝑑|=log|𝑤₂/𝑤₁| whose limits are equal to log2=0.693.

Figure 3 

It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lorenz system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−2.

Figure 4 

It shows the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to log2=0.693.

Choosing 𝛼=5 then the error system (4.4) has the initial values 𝑒_𝑥(0)=0.5,𝑒_𝑦(0)=−1.5,𝑒_𝑧(0)=0.5, and 𝑒_𝑤(0)=0.5. Figure 5 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡) and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 6 shows the evaluation of the ratios log|𝑥₂/𝑥₁|,log|𝑦₂/𝑦₁|,log|𝑧₂/𝑧₁|, and log|𝑤₂/𝑤₁| whose limits are equal to log5=1.609.

Figure 5 

It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lorenz system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=5.

Figure 6 

It shows the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to to log5=1.609.

5. Generalized Projective Synchronization for Hyperchaotic Lü System

In order to observe generalized projective synchronization between two identical hyperchaotic Lü systems, we assume that the drive system with four state variables denoted subscript 1 and the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The drive and response systems are defined below, respectively,𝑥̇₁=𝑎₁𝑦₁−𝑥₁+𝑤₁,𝑦̇₁=−𝑥₁𝑧₁+𝑐₁𝑦₁,𝑧̇₁=−𝑏₁𝑧₁+𝑥₁𝑦₁,𝑤̇₁=𝑑₁𝑤₁+𝑥₁𝑧₁,𝑥̇₂=𝑎₁𝑦₂−𝑥₂+𝑤₂+𝑢₁,𝑦̇₂=−𝑥₂𝑧₂+𝑐₁𝑦₂+𝑢₂,𝑧̇₂=−𝑏₁𝑧₂+𝑥₂𝑦₂+𝑢₃,𝑤̇₂=𝑑₁𝑤₂+𝑥₂𝑧₂+𝑢₄, (5.1)(5.2) where 𝑈=[𝑢₁𝑢₂𝑢₃𝑢₄]^𝑇 is the controller functions. The controller 𝑈 is to be determined for the purpose of projective synchronizing the two identical hyperchaotic Lü systems.

In order to get generalized projective synchronization, we define the error system as the difference between the systems (5.2) and (5.1). Set 𝑒_𝑥=𝑥₂−𝛼𝑥₁,𝑒_𝑦=𝑦₂−𝛼𝑦₁,𝑒_𝑧=𝑧₂−𝛼𝑧₁,𝑒_𝑤=𝑤₂−𝛼𝑤₁, (5.3) then one obtains the error dynamical system between (5.2) and(5.1)𝑒_𝑥̇=𝑎₁𝑒_𝑦−𝑒_𝑥+𝑒_𝑤+𝑢₁,𝑒_𝑦̇=𝑐₁𝑒_𝑦−𝑥₂𝑧₂+𝛼𝑥₁𝑧₁+𝑢₂,𝑒_𝑧̇=−𝑏₁𝑒_𝑧+𝑥₂𝑦₂−𝛼𝑥₁𝑦₁+𝑢₃,𝑒_𝑤̇=𝑑₁𝑒_𝑤+𝑥₂𝑧₂−𝛼𝑥₁𝑧₁+𝑢₄. (5.4)

Let𝑉₁=𝑢₁,𝑉₂=𝛼𝑥₁𝑧₁−𝑥₂𝑧₂+𝑢₂,𝑉₃=𝑥₂𝑦₂−𝛼𝑥₁𝑦₁+𝑢₃,𝑉₄=𝑥₂𝑧₂−𝛼𝑥₁𝑧₁+𝑢₄, (5.5) then the error dynamical system can be rewritten as𝑒_𝑥̇=𝑎₁𝑒_𝑦−𝑒_𝑥+𝑒_𝑤+𝑉₁,𝑒_𝑦̇=𝑐₁𝑒_𝑦+𝑉₂,𝑒_𝑧̇=−𝑏₁𝑒_𝑧+𝑉₃,𝑒_𝑤̇=𝑑₁𝑒_𝑤+𝑉₄. (5.6)

To get the projective synchronization, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as 𝑡→∞. As we know, if all the eigenvalues of the Jacobian matrix of closed-loop system have negative real parts, the system is stable. Based on this theory, we desired the (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 to guarantee that all the eigenvalues of closed-loop system (5.6) have negative real part. There are of course some other choices of (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇, but here the choice is very easy and convenient. For simplicity, choose (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 as follows:𝑉₁𝑉₂𝑉₃𝑉₄𝑒=𝑀^𝑥𝑒_𝑦𝑒_𝑧𝑒_𝑤,where𝑀=0−𝑎₁0−10−2𝑐₁000000000−2𝑑. (5.7)

System (5.6) has four negative eigenvalues −36,−20,−3, and −1.3 when setting 𝑎₁=36,𝑏₁=3,𝑐₁=20, and 𝑑₁=1.3. That is to say, the error states 𝑒_𝑥, 𝑒_𝑥, 𝑒_𝑧, and 𝑒_𝑤 converge to zero as 𝑡→∞. So the generalized projective synchronization is achieved.

5.1. Numerical Results

By using MAPLE 12, the systems of differential equations (5.1) and (5.2) are solved numerically. The parameters are chosen as 𝑎₁=36,𝑏₁=3,𝑐₁=20, and 𝑑₁=1.3 in all simulations so that the hyperchaotic Lü system exhibits a chaotic behavior if no control is applied (see Figure 2). The initial states of the drive system are 𝑥₁(0)=−7,𝑦₁(0)=−12,𝑧₁(0)=7, and 𝑤₁(0)=11 and initial states of the response system are 𝑥₂(0)=−4,𝑦₂(0)=−6,𝑧₂(0)=1, and 𝑤₂(0)=1.

Choosing 𝛼=−2 then the error system (5.4) has the initial values 𝑒_𝑥(0)=−18,𝑒_𝑦(0)=−30,𝑒_𝑧(0)=−13, and 𝑒_𝑤(0)=21. Figure 7 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡), and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 8 shows the evaluation of the ratios log|𝑥₂/𝑥₁|,log|𝑦₂/𝑦₁|,log|𝑧₂/𝑧₁|, and log|𝑤₂/𝑤₁| whose limits are equal to log2=0.693.

Figure 7 

It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lü system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−2.

Figure 8 

It shows the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to log2=0.693.

Choosing 𝛼=5 then the error system (5.4) has the initial values 𝑒_𝑥(0)=31,𝑒_𝑦(0)=54,𝑒_𝑧(0)=−34, and 𝑒_𝑤(0)=−54. Figure 9 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡), and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 10 shows the evaluation of the ratios log|𝑥₂/𝑥₁|,log|𝑦₂/𝑦₁|,log|𝑧₂/𝑧₁|, and log|𝑤₂/𝑤₁| whose limits are equal to log5=1.609.

Figure 9 

It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lü system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=5.

Figure 10 

It shows the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to to log5=1.609.

6. Generalized Projective Synchronization between Hyperchaotic Lorenz System and Hyperchaotic Lü System

In order to observe generalized projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lü system, we assume that hyperchaotic Lorenz system is the drive system and hyperchaotic Lü system is the response system. The drive system with four state variables denoted by the subscript 1 and the response system with four state variables denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The drive and response systems are defined below, respectively,𝑥̇₁=𝑎𝑦₁−𝑥₁+𝑤₁,𝑦̇₁=−𝑥₁𝑧₁+𝑟𝑥₁−𝑦₁,𝑧̇₁=−𝑏𝑧₁+𝑥₁𝑦₁,𝑤̇₁=𝑑𝑤₁−𝑥₁𝑧₁,𝑥̇₂=𝑎₁𝑦₂−𝑥₂+𝑤₂+𝑢₁,𝑦̇₂=−𝑥₂𝑧₂+𝑐₁𝑦₂+𝑢₂,𝑧̇₂=−𝑏₁𝑧₂+𝑥₂𝑦₂+𝑢₃,𝑤̇₂=𝑑₁𝑤₂+𝑥₂𝑧₂+𝑢₄, (6.1)(6.2) where 𝑈=[𝑢₁𝑢₂𝑢₃𝑢₄]^𝑇 is the controller functions. The controller 𝑈 is to be determined for the purpose of projective synchronizing between hyperchaotic Lorenz system and hyperchaotic Lü system.

In order to get generalized projective synchronization, we define the error system as the difference between the systems (6.2) and (6.1). Set 𝑒_𝑥=𝑥₂−𝛼𝑥₁,𝑒_𝑦=𝑦₂−𝛼𝑦₁,𝑒_𝑧=𝑧₂−𝛼𝑧₁,𝑒_𝑤=𝑤₂−𝛼𝑤₁, (6.3) then one obtains the error dynamical system between (6.2) and(6.1)𝑒_𝑥̇=𝑎₁𝑦₂−𝑥₂+𝑤₂−𝑎𝛼𝑦₁−𝑥₁−𝛼𝑤₁+𝑢₁,𝑒_𝑦̇=−𝑥₂𝑧₂+𝑐₁𝑦₂+𝛼𝑥₁𝑧₁−𝛼𝑟𝑥₁+𝛼𝑦₁+𝑢₂,𝑒_𝑧̇=−𝑏₁𝑧₂+𝑥₂𝑦₂+𝛼𝑏𝑧₁−𝛼𝑥₁𝑦₁+𝑢₃,𝑒_𝑤̇=𝑑₁𝑤₂+𝑥₂𝑧₂−𝛼𝑑𝑤₁+𝛼𝑥₁𝑧₁+𝑢₄. (6.4)

Let𝑉₁=𝑎−𝑎₁𝑥₂−𝑦₂+𝑢₁,𝑉₂=𝛼𝑥₁𝑧₁−𝑥₂𝑧₂+𝑐₁+1𝑦₂−𝑟𝑥₂+𝑢₂,𝑉₃=𝑥₂𝑦₂−𝛼𝑥₁𝑦₁−𝑏₁−𝑏𝑧₂+𝑢₃,𝑉₄=𝑑₁+𝑑𝑤₂+𝑥₂𝑧₂+𝑢₄+𝛼𝑥₁𝑧₁, (6.5) then the error dynamical system can be rewritten as 𝑒_𝑥̇=𝑎𝑒_𝑦−𝑒_𝑥+𝑒_𝑤+𝑉₁𝑒_𝑦̇=𝑟𝑒_𝑥−𝑒_𝑦+𝑉₂𝑒_𝑧̇=−𝑏𝑒_𝑧+𝑉₃𝑒_𝑤̇=−𝑑𝑒_𝑤+𝑉₄. (6.6)

To get the projective synchronization, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as 𝑡→∞. As we know, if all the eigenvalues of the Jacobian matrix of closed loop system have negative real parts, the system is stable. Based on this theory, we desired the (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 to guarantee that all the eigenvalues of closed-loop system (6.7) have negative real part. There are of course some other choices of (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇, but here the choice is very easy and convenient. For simplicity, choose (𝑉₁,𝑉₂,𝑉₃,𝑉₄)^𝑇 as follows:𝑉₁𝑉₂𝑉₃𝑉₄𝑒=𝑀^𝑥𝑒_𝑦𝑒_𝑧𝑒_𝑤,where𝑀=0−𝑎0−1−𝑟00000000000. (6.7)

System (6.7) has four negative eigenvalues −10,−1,−8/3, and −1.3 when setting 𝑎=10,𝑟=28,𝑏=8/3, and 𝑑=1.3. That is to say, the error states 𝑒_𝑥, 𝑒_𝑥, 𝑒_𝑧, and 𝑒_𝑤 converge to zero as 𝑡→∞. So the generalized projective synchronization is achieved.

6.1. Numerical Results

By using MAPLE 12, the systems of differential equations (6.1) and (6.2) are solved numerically. The initial states of the drive system are 𝑥₁(0)=0.1,𝑦₁(0)=0.1,𝑧₁(0)=0.1, and 𝑤₁(0)=0.1,and initial states of the response system are 𝑥₂(0)=−7,𝑦₂(0)=−12,𝑧₂(0)=7, and 𝑤₂(0)=11.

Choosing 𝛼=−2, then the error system has the initial values 𝑒_𝑥(0)=−6.8,𝑒_𝑦(0)=−11.8,𝑒_𝑧(0)=7.2, and 𝑒_𝑤(0)=11.2. Figure 11 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡), and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 12 shows the evaluation of the ratios log|𝑥₂/𝑥₁|,log|𝑦₂/𝑦₁|,log|𝑧₂/𝑧₁|, and log|𝑤₂/𝑤₁| whose limits are equal to log2=0.693.

Figure 11 

It shows the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the error system between hyperchaotic Lorenz system and hyperchaotic Lü system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−2.

Figure 12 

It shows that the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to log2=0.693.

Choosing 𝛼=5, then the error system has the initial values 𝑒_𝑥(0)=−7.5,𝑒_𝑦(0)=−12.5,𝑒_𝑧(0)=6.5, and 𝑒_𝑤(0)=10.5. Figure 13 shows that the trajectories of 𝑒_𝑥(𝑡),𝑒_𝑦(𝑡),𝑒_𝑧(𝑡), and 𝑒_𝑤(𝑡) tended to zero after 𝑡≥5. Figure 14 shows the evaluation of the ratios log|𝑥₂/𝑥₁|,log|𝑦₂/𝑦₁|,log|𝑧₂/𝑧₁|, and log|𝑤₂/𝑤₁| whose limits are equal to log5=1.609.

Figure 13 

It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the error system between hyperchaotic Lorenz system and hyperchaotic Lü system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=5.

Figure 14 

It shows the evaluation of the ratios log|𝑥2/𝑥1|,log|𝑦2/𝑦1|,log|𝑧2/𝑧1|, and log|𝑤2/𝑤1| whose limits are equal to log5=1.609.

7. Conclusion

This paper shows that the generalized projective synchronizations for the hyperchaotic Lorenz system and the hyperchaotic Lü system can be easily achieved by using the fundamental feedback techniques, where the scaling factors can be arbitrarily manipulated (amplified or reduced). We note that the driver and response systems achieve complete synchronization when 𝛼 is equal to 1 (see Figure 15). Further, if 𝛼 is equal to −1, then the two systems are said to be in anti-synchronization (see Figure 16). Numerical simulations are used to verify the effectiveness of the proposed generalized projective synchronization techniques.

(a)

(b)

(c)

Figure 15 

(a) It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lorenz system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=1. (b) It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lü system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=1. (c) It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the error system between hyperchaotic Lorenz system and hyperchaotic Lü system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=1.

(a)

(b)

(c)

Figure 16 

(a) Its shows the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lorenz system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−1. (b) It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the hyperchaotic Lü system error system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−1. (c) It shows that the behaviour of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and𝑒𝑤of the error system between hyperchaotic Lorenz system and hyperchaotic Lü system tends to zero as 𝑡 tends to 5 when the scaling factor 𝛼=−1.