# A New 4D Hyperchaotic System and Its Generalized Function Projective Synchronization

Authors: Yuan Gao; Chenghua Liang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A new four-dimensional hyperchaotic system is investigated. Numerical and analytical studies are carried out on its basic dynamical properties, such as equilibrium point, Lyapunov exponents, Poincaré maps, and chaotic dynamical behaviors. We verify the realizability of the new system via an electronic circuit by using Multisim software. Furthermore, a generalized function projective synchronization scheme of two different hyperchaotic systems with uncertain parameters is proposed, which includes some existing projective synchronization schemes, such as generalized projection synchronization and function projective synchronization. Based on the Lyapunov stability theory, a controller with parameters update laws is designed to realize synchronization. Using this controller, we realize the synchronization between Chen hyperchaotic system and the new system to verify the validity and feasibility of our method.


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## Body

## 1. Introduction

Chaos is a very fascinating nonlinear phenomenon, which exhibits extreme sensitivity to initial conditions and has noise-like behaviors. Since Lorenz discovered the first chaotic attractor in a three-dimensional (3D) autonomous system in 1963 [1], chaos has been extensively investigated in mathematics, physics, and engineering field. In recent decades, there has been increasing interest in creating the chaos, particularly, since Chen chaotic system is founded by an feedback control (or chaotification) [2]. By now, some typical chaotic systems, such as modified Lorenz system [3], Lü system [4], unified system [5], have been constructed by the consideration of chaotification.In general, hyperchaotic system is defined as a chaotic system with more than one positive Lyapunov exponent, this implies that its chaotic dynamics extend in several different directions simultaneously. Therefore, comparing with the traditional chaotic system, hyperchaotic system has more complex dynamical behaviors which can be used to enhance the security of chaotic communication system. Consequently, the topic of theoretical design and circuitry realization of various hyperchaotic systems has recently become hotspot in the nonlinear research field. In recent years, hyperchaos has been found numerically and experimentally by adding a simple state feedback controller to Chua’s circuit [6], Chen system [7], or Lorenz equation [8].Inspired by the above works, in this paper, we firstly introduce a new 4D hyperchaotic system via modifying 3D Liu system [9]. The basic nonlinear dynamical properties of the new system, such as equilibrium point, Lyapunov exponents, Poincaré maps, and chaotic behaviors, are studied numerically and analytically. Numerical results demonstrate that the new system can exhibit complex hyperchaos over a wide range of system parameters. This hyperchaotic system can be realized by a simple electronic circuit. The Multisim circuit experiment results show good agreement with the numerical simulation. Therefore, the main merit of the proposed hyperchaotic system is suitable as a generator to enhance the security of chaos communication.On the other hand, in the past decade, chaos synchronization has attracted much more attention for the wide-scope potential applications in physics, biology, secure communication, and so forth [10–13]. Since the pioneering work of Pecora and Carroll [14], different types of synchronization have been investigated, such as complete synchronization [15], phase synchronization [16, 17], lag synchronization [18], generalized synchronization [19, 20], and projective synchronization [21–23]. Amongst all the types of chaos synchronization, projective synchronization with proportional feature between the drive system and the response system, first reported by Mainieri and Rehacek in 1999 [21], has aroused more interest because this feature can be used to extend binary digital to M-nary digital communication for achieving faster secure communication [24, 25].In recent years, inspired by the projective synchronization, increasing attention has been devoted to the problem of the synchronization with more complex feature between the drive system and the response system. Wu and Lu [26] studied generalized projective synchronization (GPS) of the fractional-order Chen hyperchaotic system. In [27], a modified projective synchronization (MPS) scheme was investigated, in which the dynamical states of the drive system and response system synchronized up to a constant scaling matrix. Then, Chen et al. [28] proposed a new projective synchronization called function projective synchronization (FPS), in which the drive and response vectors can be synchronized up to a desired scaling function, but not a constant. Therefore, the FPS is very useful to obtain more secure communication, due to the unpredictability of the scaling function which can additionally enhance the security in the communication. In [29], Du et al. introduced a modified function projective synchronization (MFPS) of chaotic system, where the dynamical states of the drive and response system can be synchronized up to a desired scaling function matrix. By now, there have been some research works about the MFPS [29–35]. Particularly, in [36], Yu et al. further proposed a generalized function projective synchronization (GFPS) of two different chaotic systems with fully unknown parameters. The GFPS scheme is an extension of many existing projective synchronization schemes, such as MPS and FPS in which the responses of synchronized dynamical states can be also synchronized up to a scaling function matrix.From the existing results, one can see that the scaling function matrix in the MFPS is separately considered as a function matrix of drive system variable [30–32, 36] or a time function matrix [29, 33–35]. Obviously, if the combination of two kinds of matrixes is introduced into the synchronization, the scaling function factors become more unpredictable, and better security of chaos communication can be obtained. However, to the best of our knowledge, so far, no research results have been presented about the MFPS where the scaling matrix is composed of time function matrix and system variable function matrix. On the other hand, in the most of practical situations, chaos synchronization can be discussed for two strictly different systems, such as in social science and biological science. Furthermore, the parameters of many systems cannot be known entirely; the synchronization problem of two hyperchaotic systems is more significant for improving the security of communication. Therefore, the above discussions motivate our study.In this paper, we dedicate to further investigate a new type of GFPS scheme between two different uncertain hyperchaotic systems. In our GFPS scheme, a significant advantage is that the scaling function matrix is constructed by explicit time function matrix and function matrix of drive system variables. Thus, synchronization relationship is more complex. Moreover, complete synchronization, anti-phase synchronization, GPS, and FPS are the special cases of the proposed GFPS. In order to realize the GFPS of two different uncertain hyperchaotic systems, we make an attempt to design a controller with parameters update laws based on Lyapunov stability theory. The GFPS of Chen hyperchaotic system and the proposed hyperchaotic system is considered as an example; the simulation results of this example verify the validity of our method.This paper is organized as follows. In Section2, a new 4D hyperchaotic system is introduced, and its nonlinear dynamical properties and circuit implementation are studied. Furthermore, a GFPS of different hyperchaotic systems with uncertain parameters is investigated in Section 3. The controller for GFPS based on Lyapunov stability theory is put forward, and simulation results of an example are provided to realize the synchronization between Chen hyperchaotic system and the new system in this section. Finally conclusion in Section 4 closes the paper.


## 2. The Proposed 4D Dynamic System

By modifying the 3D Liu chaotic system [9], we introduce the following 4D quadratic smooth autonomous system:
(1)x˙1=-a1x1+a2x2,x˙2=a3x1-x1x3-x2+x4,x˙3=x12-a4(x1+x3),x˙4=-a5x1,
where (x1,x2,x3,x4)T∈R4 is the state vector and ai(i=1,2,…,5) are positive constant parameters of the system. The values of the parameters in system (1) are a1=25, a2=60, a3=40, a4=4, and a5=5, so the new system is hyperchaotic in this case. For the purpose to reveal the hyperchaotic dynamics of this new system, in the following paragraphs, some basic properties of dynamic system (1) are analyzed.


### 2.1. Dissipativity and the Existence of Attractor

The divergence of the system (1) is written by
(2)∇V=∂x˙1∂x1+∂x˙2∂x2+∂x˙3∂x3+∂x˙4∂x4=-a1-1-a4<0.
Therefore, system (1) is dissipative system with an exponential contraction rate V˙=-(a1+a4+1)V. It means that a volume element V0 including the system orbit is shrunken to zero as t→∞ at an exponential rate ∇V. That is, to say, all the system orbits will finally be restricted to a special subset of zero volume and the asymptotic motion settles onto an attractor of the system (1).


### 2.2. Equilibrium Point and Stability

The only equilibrium pointO(0,0,0,0) of system (1) can be found, while x˙1=x˙2=x˙3=x˙4=0. By linearizing the system (1) at O, one obtains Jacobian matrix
(3)J=[-a1a200a3-101-a40-a40-a5000].
Thus, the characteristic equation of the linearized system is expressed as
(4)F(λ)=λ4+(1+a1+a4)λ3+(a1-a2a3+a1a4+a4)λ2+(a1a4+a2a5-a2a3a4)λ+a2a4a5=0.
Setting the parameters a1=25, a2=60, a3=40, and a4=4 and using Routh-Hurwitz criterion to solve (4), one can find two eigenvalues with positive real part at the range of a5>0. For example, when a5=5, the eigenvalues λ1=-4, λ2=-63.48, λ3=37.35, and λ4=0.126, respectively. Characteristic equation has two positive roots and two negative roots, which implies that the fixed point O(0,0,0,0) is an unstable saddle point. While the parameters a1=25, a2=60, a4=4, a5=5, and a3∈(0,∞), characteristic equation has also two roots with positive real part. For example, if a3=10, the eigenvalues λ1=-4, λ2=-40.41, λ3=13.87, λ4=0.534, respectively. Therefore, O is still an unstable saddle point.


### 2.3. Lyapunov Spectra and Poincaré Map

The numerical simulations are carried out, by means of Rouge-Kutta integration algorithm, to solve the differential equations. Lyapunov exponents are calculated out by Wolf algorithm [37]. The initial values of system are set to (x1(0),x2(0),x3(0),x4(0))T=(2.0,6.0,9.0,2.0)T.When the parametersa1=25, a2=60, a3=40, and a4=4 are fixed and the parameter a5 is varied from zero to 20, the Lyapunov exponent spectrum of system (1) with respect to a5 is shown in Figure 1. Figure 2 indicates the Lyapunov spectrum corresponding the system parameters a1=25, a2=60, a4=4, a5=5, and a3∈[10,50]. From Figures 1 and 2, one can see clearly that the 4D system has two positive Lyapunov exponents over a wide range of parameters a3 and a5; thus, the presented system is hyperchaotic.Lyapunov exponents versus parametera5(a1=25, a2=60, a3=40, and a4=4).
(a)
(b)Lyapunov exponents versus parametera3(a1=25, a2=60, a4=4, and a5=5).
(a)
(b)For the parameter values(a1,a2,a3,a4,a5)=(25,60,40,4,5), four Lyapunov exponents of system (1) are calculated as L1=3.0057, L2=0.0304, L3=-0.1631, and L4=-46.1578, respectively. Therefore, the Lyapunov dimension of system (1) is
(5)D=j+1|Lj+1|∑i=1jLi=3+L1+L2+L3|L4|=3.0622.
The Lyapunov dimension is fractional, which implies that system (1) is really a dissipative system. In Figures 3(a) and 3(b), we show phase portraits of the orbits on x1-x2 and x1-x3 planes, it is evident that the system has a double-scroll hyperchaotic attractor. Taking
(6)∑1={[x1,x2,x3,x4]T∈R4∣x3=50},∑2={[x1,x2,x3,x4]T∈R4∣x2=0},
as cross sections, Poincaré maps on x1-x2 and x1-x3 planes are illustrated in Figures 4(a) and 4(b), respectively, which show the existence of the chaotic properties in the dynamical system. In the discussion below, the parameters of the new hyperchaotic system a1=25, a2=60, a3=40, a4=4, and a5=5 are used.Phase portraits of the hyperchaotic attractor(a1=25, a2=60, a4=4, a3=40, and a5=5).
(a)
projection on thex1-x2 plane
(b)
projection on thex1-x3 planePoincaré maps of the hyperchaotic attractor on different cross sections(a1=25,a2=60, a4=4,a3=40, and a5=5).
(a)
x
3
=
50
(b)
x
2
=
0


### 2.4. Verification via Circuit Simulation

In order to verify the realizability of the proposed hyperchaotic system in practical applications, we design an electronic circuit based on the mathematical model (1) with parameters (a1,a2,a3,a4,a5)=(25,60,40,4,5). By proper choice of the resistors and capacitors, this circuit demonstrates chaotic behavior. In our implementation, output signals of the system states are all compressed to 10% of theoretical values. The operational amplifiers (TL741 chip) and associated circuitry perform the basic operations of addition, subtraction, and integration. The nonlinear terms in the equation are implemented with the analog multipliers (AD633 chip). Figure 5 gives the circuit schematic of the hyperchaotic system. Based on Kirchhoff’s current law, the circuit equations can be written as follows:
(2.4)u˙C1=-1R2C1uC1+R10R9R1C1uC2,u˙C2=R4R3R5C2uC1-R1510R14R6C2uC1uC3-1R8C2uC2+R18R17R7C2uC4,u˙C3=-1R12C3uC12-1R11C3uC1-1R13C3uC3,u˙C4=-1R16C4uC1,
where uCi(i=1,2,3,4) are the voltage on the capacitor terminals, which correspond to the system states xi(i=1,2,3,4), respectively. The parameters in the circuit are designed as R1=16.6667 kΩ, R2=40 kΩ, R5=25 kΩ, R8=1000 kΩ, R16=200 kΩ, R11=R13=250 kΩ, R6=R7=R12=100 kΩ, R3=R4=R9=R10=R14=R15=R17=R18=10 kΩ, and C1=C2=C3=C4=1 μF.Figure 5
The circuit schematic of the new hyperchaotic system (1).Multisim software is used to simulate the realized circuit of the hyperchaotic system. The chaotic attractors measured by Multisim experiment are shown in Figure6, which are almost same as the numerical results shown in Figure 3. The comparison between the numerical results and the circuit experimental simulation demonstrates a very good qualitative agreement to each other. Therefore, the functionality of the proposed system is confirmed.Phase portraits of the hyperchaotic attractor by Multisim experiment.
(a)
(b)


## 2.1. Dissipativity and the Existence of Attractor

The divergence of the system (1) is written by
(2)∇V=∂x˙1∂x1+∂x˙2∂x2+∂x˙3∂x3+∂x˙4∂x4=-a1-1-a4<0.
Therefore, system (1) is dissipative system with an exponential contraction rate V˙=-(a1+a4+1)V. It means that a volume element V0 including the system orbit is shrunken to zero as t→∞ at an exponential rate ∇V. That is, to say, all the system orbits will finally be restricted to a special subset of zero volume and the asymptotic motion settles onto an attractor of the system (1).


## 2.2. Equilibrium Point and Stability

The only equilibrium pointO(0,0,0,0) of system (1) can be found, while x˙1=x˙2=x˙3=x˙4=0. By linearizing the system (1) at O, one obtains Jacobian matrix
(3)J=[-a1a200a3-101-a40-a40-a5000].
Thus, the characteristic equation of the linearized system is expressed as
(4)F(λ)=λ4+(1+a1+a4)λ3+(a1-a2a3+a1a4+a4)λ2+(a1a4+a2a5-a2a3a4)λ+a2a4a5=0.
Setting the parameters a1=25, a2=60, a3=40, and a4=4 and using Routh-Hurwitz criterion to solve (4), one can find two eigenvalues with positive real part at the range of a5>0. For example, when a5=5, the eigenvalues λ1=-4, λ2=-63.48, λ3=37.35, and λ4=0.126, respectively. Characteristic equation has two positive roots and two negative roots, which implies that the fixed point O(0,0,0,0) is an unstable saddle point. While the parameters a1=25, a2=60, a4=4, a5=5, and a3∈(0,∞), characteristic equation has also two roots with positive real part. For example, if a3=10, the eigenvalues λ1=-4, λ2=-40.41, λ3=13.87, λ4=0.534, respectively. Therefore, O is still an unstable saddle point.


## 2.3. Lyapunov Spectra and Poincaré Map

The numerical simulations are carried out, by means of Rouge-Kutta integration algorithm, to solve the differential equations. Lyapunov exponents are calculated out by Wolf algorithm [37]. The initial values of system are set to (x1(0),x2(0),x3(0),x4(0))T=(2.0,6.0,9.0,2.0)T.When the parametersa1=25, a2=60, a3=40, and a4=4 are fixed and the parameter a5 is varied from zero to 20, the Lyapunov exponent spectrum of system (1) with respect to a5 is shown in Figure 1. Figure 2 indicates the Lyapunov spectrum corresponding the system parameters a1=25, a2=60, a4=4, a5=5, and a3∈[10,50]. From Figures 1 and 2, one can see clearly that the 4D system has two positive Lyapunov exponents over a wide range of parameters a3 and a5; thus, the presented system is hyperchaotic.Lyapunov exponents versus parametera5(a1=25, a2=60, a3=40, and a4=4).
(a)
(b)Lyapunov exponents versus parametera3(a1=25, a2=60, a4=4, and a5=5).
(a)
(b)For the parameter values(a1,a2,a3,a4,a5)=(25,60,40,4,5), four Lyapunov exponents of system (1) are calculated as L1=3.0057, L2=0.0304, L3=-0.1631, and L4=-46.1578, respectively. Therefore, the Lyapunov dimension of system (1) is
(5)D=j+1|Lj+1|∑i=1jLi=3+L1+L2+L3|L4|=3.0622.
The Lyapunov dimension is fractional, which implies that system (1) is really a dissipative system. In Figures 3(a) and 3(b), we show phase portraits of the orbits on x1-x2 and x1-x3 planes, it is evident that the system has a double-scroll hyperchaotic attractor. Taking
(6)∑1={[x1,x2,x3,x4]T∈R4∣x3=50},∑2={[x1,x2,x3,x4]T∈R4∣x2=0},
as cross sections, Poincaré maps on x1-x2 and x1-x3 planes are illustrated in Figures 4(a) and 4(b), respectively, which show the existence of the chaotic properties in the dynamical system. In the discussion below, the parameters of the new hyperchaotic system a1=25, a2=60, a3=40, a4=4, and a5=5 are used.Phase portraits of the hyperchaotic attractor(a1=25, a2=60, a4=4, a3=40, and a5=5).
(a)
projection on thex1-x2 plane
(b)
projection on thex1-x3 planePoincaré maps of the hyperchaotic attractor on different cross sections(a1=25,a2=60, a4=4,a3=40, and a5=5).
(a)
x
3
=
50
(b)
x
2
=
0


## 2.4. Verification via Circuit Simulation

In order to verify the realizability of the proposed hyperchaotic system in practical applications, we design an electronic circuit based on the mathematical model (1) with parameters (a1,a2,a3,a4,a5)=(25,60,40,4,5). By proper choice of the resistors and capacitors, this circuit demonstrates chaotic behavior. In our implementation, output signals of the system states are all compressed to 10% of theoretical values. The operational amplifiers (TL741 chip) and associated circuitry perform the basic operations of addition, subtraction, and integration. The nonlinear terms in the equation are implemented with the analog multipliers (AD633 chip). Figure 5 gives the circuit schematic of the hyperchaotic system. Based on Kirchhoff’s current law, the circuit equations can be written as follows:
(2.4)u˙C1=-1R2C1uC1+R10R9R1C1uC2,u˙C2=R4R3R5C2uC1-R1510R14R6C2uC1uC3-1R8C2uC2+R18R17R7C2uC4,u˙C3=-1R12C3uC12-1R11C3uC1-1R13C3uC3,u˙C4=-1R16C4uC1,
where uCi(i=1,2,3,4) are the voltage on the capacitor terminals, which correspond to the system states xi(i=1,2,3,4), respectively. The parameters in the circuit are designed as R1=16.6667 kΩ, R2=40 kΩ, R5=25 kΩ, R8=1000 kΩ, R16=200 kΩ, R11=R13=250 kΩ, R6=R7=R12=100 kΩ, R3=R4=R9=R10=R14=R15=R17=R18=10 kΩ, and C1=C2=C3=C4=1 μF.Figure 5
The circuit schematic of the new hyperchaotic system (1).Multisim software is used to simulate the realized circuit of the hyperchaotic system. The chaotic attractors measured by Multisim experiment are shown in Figure6, which are almost same as the numerical results shown in Figure 3. The comparison between the numerical results and the circuit experimental simulation demonstrates a very good qualitative agreement to each other. Therefore, the functionality of the proposed system is confirmed.Phase portraits of the hyperchaotic attractor by Multisim experiment.
(a)
(b)


## 3. GFPS of Hyperchaotic Systems

### 3.1. The Definition of GFPS

From the viewpoint of control, the synchronization task is to design a proper controller which obtains signals from the drive system to tune the behavior of the response system. Here, we consider the hyperchaotic (drive and response) systems as the following form:(8)x˙=f(x),(9)y˙=g(y)+u(x,y),
where x=(x1,x2,…,xn)T∈Rn, y=(y1,y2,…,yn)T∈Rn are the state vectors of the systems (8) and (9), respectively, f,g:Rn→Rn are two continuous vector functions, and u=(u1,u2,…,un)T∈Rn is a controller which need to be designed.Definition 1.
For the drive system (8) and the unidirectionally coupled response system (9), they are said to be (GFPS) if there exists vector function u(x,y) such that
(10)limt→∞∥y-Φ(t)H(x)x∥⟶0,
where H(x)=diag(h1(x),h2(x),…,hn(x)), hi(x):Rn→R are continuous functions,Φ(t)=diag(φ1(t),φ2(t),…,φn(t)), φi(t)≠0(i=1,2,…,n) are continuously differentiable functions with bounded, and ∥·∥ represents the matrix norm.Remark 2.
The function matrixΦ(t)H(x) is called the scaling matrix, and φ1(t)h1(x), φ2(t)h2(x),…,φn(t)hn(x) are called scaling function factors. In the existing results [29–36], the scaling matrix Φ(t) or H(x) is always treated separately. The function matrix in our work is a complex integrated form. Consequently, the presented GFPS can improve antidecryption and unpredictability of the secure communication. Obviously, the GFPS scheme becomes function projective synchronization (FPS) when φ1(t)h1(x)=φ2(t)h2(x)=⋯=φn(t)hn(x). Furthermore, while φi(t)hi(x)(i=1,2,…,n) are constants, GFPS will be simplified into generalized projective synchronization (GPS). In particular, the problem is further reduced to the complete synchronization and anti-phase synchronization if Φ(t)H(x)=I and Φ(t)H(x)=-I, where I is an n×n identity matrix.Remark 3.
The error vector for GFPS between the drive system (8) and the response system (9) is defined by
(11)e(t)=y-Φ(t)H(x)x,
where e(t)=(e1(t),e2(t),…,en(t))T∈Rn, and ei=yi-φi(t)hi(x)xi, (i=1,2,…,n).Remark 4.
IfΦ(t)H(x)=0, the synchronization problem turns into a chaos control problem.


### 3.2. The Scheme of GFPS with Unknown System Parameters

Considering GFPS of two different hyperchaotic systems with fully unknown parameters, we can rewrite the drive and response systems (8), (9) as
(12)x˙=f1(x)+f2(x)α,(13)y˙=g1(y)+g2(y)β+u,
respectively, where f1,g1:Rn→Rn are continuous vector functions, f2:Rn→Rn×m, g2:Rn→Rn×l are continuous matrix functions, and α∈Rm, β∈Rl are unknown parameter vectors of systems (12) and (13). We introduce α~ and β~ as the parameter estimation vectors of α and β, respectively, and define the Jacobian matrix of vector H(x)x as follows:
(14)Γ=d(H(x)x)dx.Theorem 5.
For a given continuous scaling matrix functionΦ(t)H(x) and all initial conditions x(0), y(0), the GFPS between the drive system (12) and the response system (13) can be achieved and the unknown parameters α and β can be identified by the following controller and the adaptive parameter update laws:
(15)u(t)=-g1(y)-g2(y)β~+Φ˙(t)H(x)x+Φ(t)Γ[f1(x)+f2(x)α~]-Ke(t),(16)α~˙=-f2T(x)ΓTΦ(t)e(t)-Peα,β~˙=g2T(y)e(t)-Qeβ,
respectively, where K=diag(k1,k2,…,kn), P=diag(kn+1,kn+2,…,kn+m), Q=diag(kn+m+1,kn+m+2,…,kn+m+l), ki>0(i=1,2,…,n+m+l), and eα=α~-α and eβ=β~-β are error vectors of the identified parameters.Proof.
The time derivative of the error vector (11) is
(17)e˙(t)=y˙-Φ˙(t)H(x)x-Φ˙(t)Γx˙.
Substituting (15) into (17), we have
(18)e˙(t)=-g2(y)(β~-β)+Φ(t)Γf2(x)(α~-α)-Ke(t)=-g2(y)eβ+Φ(t)Γf2(x)eα-Ke(t).
Here, we define the Lyapunov function as
(19)V(t)=12[eT(t)e(t)+eαT(t)eα(t)+eβT(t)eβ(t)].
From (16) and (18), the first derivative of (19) with respect to time is
(20)V˙(t)=eTe˙+eαTe˙α+eβTe˙β=eT[-g2(y)eβ+Φ(t)Γf2(x)eα-Ke(t)]+eαT[-f2T(x)ΓTΦ(t)e(t)Peα]+eβT[g2T(y)e(t)-Qeβ]=-eT(t)Ke(t)-eαT(t)Peα(t)-eβT(t)Qeβ(t)<0.
According to the Lyapunov stability theorem, the trivial solution of the error system (17) is asymptotically stable, which implies that the GFPS of system (12) and (13) is obtained, and error vectors of parameters eα, eβ approach zero as t→∞. The proof is complete.


### 3.3. Applications

In order to verify the effectiveness and feasibility of the proposed GFPS scheme, we apply the controller to realize the synchronization between Chen hyperchaotic system and the proposed system. Chen hyperchaotic system is described by the following state equations [38]:
(21)y˙1=b1(y2-y1)+y4,y˙2=b2y1-y1y3+b3y2,y˙3=y1y2-b4y3,y˙4=y2y3+b5y4,
where y=(y1,y2,y3,y4)T is the state vector and β=(b1,b2,b3,b4,b5)T is parameter vector. As the parameters (b1,b2,b3,b4,b5)=(35,7,12,3,0.5), Chen system shows hyperchaotic properties.We take the proposed system and Chen hyperchaotic system as the drive system and the response system, respectively. Thus, the parameter vector of the drive system (1) is defined as α=(a1,a2,a3,a4,a5)T. The response system can be written as
(22)y˙1=b1(y2-y1)+y4+u1,y˙2=b2y1+y1y3+b3y2+u2,y˙3=y1y2-b4y3+u3,y˙4=y2y3-b5y4+u4,
where u1, u2, u3, and u4 are the control functions to be designed. Without loss of generality, we consider the continuous function matrixes
(23)Φ(t)=diag{φ1(t),φ2(t),φ3(t),φ4(t)}=diag{c11sinω1t+c12,c21sinω2t+c22,c31sinω3t+c32,c41sinω4t+c42},(24)H(x)=diag{h1(x),h2(x),h3(x),h4(x)}=diag{d11x1+d12,d21x2+d22,d31x3+d32,d41x4+d42},
where cij, dij(i=1,2,3,4;j=1,2) are constants. According to the GPFS scheme presented above, the synchronization errors are defined as
(25)ei=yi-φi(t)hi(x)xi=yi-(ci1sinωit+ci2)×(di1xi+di2)xi,i=1,2,3,4.
Substituting (1) and (22) into (25), one can obtain the time derivative of error
(26)e˙1=y4+b1(y2-y1)-c11ω1cosω1t×(d11x1+d12)x1-(c11sinω1t+c12)×(2d11x1+d12)(-a1x1+a2x2)+u1,e˙2=b2y1+b3y2-y1y3-c21ω2cosω2t×(d21x2+d22)x2-(c21sinω2t+c22)×(2d21x2+d22)(a3x1-x1x3-x2+x4)+u2,e˙3=y1y2-b4y3-c31ω3cosω3t×(d31x3+d32)x3-(c31sinω3t+c32)×(2d31x3+d32)[x12-a4(x1+x3)]+u3,e˙4=y2y3+b5y4-c41ω4cosω4t(d41x4+d42)x4+a5x1(c41sinω4t+c42)(2d41x4+d42)+u4.
From (15) and (16), we get the controller
(27)u1=-y4-b~1(y2-y1)-c11ω1cosω1t×(d11x1+d12)x1+(c11sinω1t+c12)×(2d11x1+d12)(-a~1x1+a~2x2)-k1e1,u2=y1y3-b~2y1-b~3y2+c21ω2cosω2t×(d21x2+d22)x2+(c21sinω2t+c22)×(2d21x1+d22)(a~3x1-x1x3-x2+x4)-k2e2,u2=b~4y3-y1y2+c31ω3cosω3t×(d31x3+d32)x3+(c31sinω3t+c32)×(2d31x1+d32)[x1x2-a~4(x1+x3)]-k3e3,u4=-y2y3-b~5y4+c41ω4cosω4t(d41x4+d42)x4-a~5x1(c41sinω4t+c42)(2d41x4+d42)-k4e4,
and the adaptive parameter update laws
(28)a~˙1=x1(c11sinω1t+c12)(2d11x1+d12)e1-k5ea1,a~˙2=-x2(c11sinω1t+c12)(2d11x1+d12)e1-k6ea2,a~˙3=-x1(c21sinω2t+c22)(2d21x2+d22)e2-k7ea3,a~˙4=(x1+x5)(c31sinω3t+c32)×(2d31x3+d32)e3-k8ea4,a~˙5=x1(c41sinω4t+c42)(2d41x4+d42)e4-k9ea5,b~˙1=(y2-y1)e1-k10eb1,b~˙2=y1e2-k11eb2,b~˙3=y2e2-k12eb3,b~˙4=-y3e3-k12eb4,b~˙5=y4e4-k14eb5,
where the control gains ki>0(i=1,2,…,14), eai=a~i-ai(i=1,2,3,4,5) and ebi=b~i-bi(i=1,2,3,4,5), are the corresponding errors of the identified parameters.


### 3.4. Simulation Results

In the simulation, the initial values of the drive and response system are set to be(x1(0),x2(0),x3(0),x4(0))=(2.0,6.0,9.0,2.0) and (y1(0),y2(0),y3(0),y4(0))=(3.0,-4.0,2.0,2.0), respectively. For the simulation purpose, we select all the gains ki=70(i=1,2,…,14). The true values of “unknown” parameters of new system and Chen system are arbitrarily taken as a~1=a~2=a~3=a~4=a~5=0 and b~1=b~2=b~3=b~4=b~5=0, respectively. In order to indicate the effectiveness of the controller (27), the control input is applied to the response system after t=4 s.We choose the coefficients of the scaling function matrixΦ(t)Η(x) according to (24) and (25) as cij=0.1, di1=0.02, di2=-10(i=1,2,3,4;j=1,2), and (ω1,ω2,ω3,ω4)=(1,2,3,4). Figure 7 provides the trajectories of GFPS between Chen system (22) and new system (1). Synchronization error curves are displayed in Figure 8. From Figures 7 and 8, one can see that the following mathematic relationships,
(29)yi=φi(t)hi(x)xi=(ci1sinωit+ci2)(di1xi+di2)xi,i=1,2,3,4
are satisfied rapidly after control; it means that the GFPS is achieved. The estimated values of unknown parameters for the drive and response system are shown by Figures 9 and 10, respectively. From Figures 9 and 10, it is obvious that the unknown parameters are converged to the true values quickly by applying the parameters update laws.The state trajectories of the drive and response systems(φi(t)h(xi)xi=(ci1sinωt+ci2)(di1xi+di2)xi, cij=0.1, di1=0.02, di2=-10,(i=1,2,3,4;j=1,2), and (ω1,ω2,ω3,ω4)=(1,2,3,4)).
(a)
(b)
(c)
(d)Error curves of GFPS(ei=yi-φi(t)h(xi)xi,i=1,2,3,4).
(a)
(b)
(c)
(d)The estimated values of the unknown parametersai(i=1,2,3,4,5) for drive system with true parameters (a1,a2,a3,a4,a5)=(25,60,40,4,5).
(a)
(b)The estimated values of the unknown parametersbi(i=1,2,3,4,5) for response system with true parameters (b1,b2,b3,b4,b5)=(35,7,12,3,0.5).
(a)
(b)Whenci1=di1=0, ci2=-di2=1(i=1,2,3,4), the scaling functions are simplified as φi(t)hi(x)=-1(i=1,2,3,4). Under the same conditions of the system parameters, the initial statuses, and the control gains, the simulation is operated. Figure 11 displays the time evolutions of anti-phase synchronization errors ei=yi+xi(i=1,2,3,4), and it is obvious that ei(i=1,2,3,4) are nonzero and irregular before control and converge to zero rapidly after control. Figure 12 exhibits the synchronization relationship yi=-xi(i=1,2,3,4), which means that anti-phase synchronization between the drive system (1) and the response system (22) is gained.The evolution of anti-phase synchronization error(ei=yi+xi,i=1,2,3,4).
(a)
(b)
(c)
(d)Anti-phase synchronization withyi=-xi(i=1,2,3,4).
(a)
(b)
(c)
(d)From the above simulation results of the GPFS, we can find that the synchronization error can be quickly converged to zero, and the unknown parameters of the drive system and the response system all are accurately estimated to true values. This means that our GPFS algorithm based on Lyapunov stability theory is convergent and effective.


## 3.1. The Definition of GFPS

From the viewpoint of control, the synchronization task is to design a proper controller which obtains signals from the drive system to tune the behavior of the response system. Here, we consider the hyperchaotic (drive and response) systems as the following form:(8)x˙=f(x),(9)y˙=g(y)+u(x,y),
where x=(x1,x2,…,xn)T∈Rn, y=(y1,y2,…,yn)T∈Rn are the state vectors of the systems (8) and (9), respectively, f,g:Rn→Rn are two continuous vector functions, and u=(u1,u2,…,un)T∈Rn is a controller which need to be designed.Definition 1.
For the drive system (8) and the unidirectionally coupled response system (9), they are said to be (GFPS) if there exists vector function u(x,y) such that
(10)limt→∞∥y-Φ(t)H(x)x∥⟶0,
where H(x)=diag(h1(x),h2(x),…,hn(x)), hi(x):Rn→R are continuous functions,Φ(t)=diag(φ1(t),φ2(t),…,φn(t)), φi(t)≠0(i=1,2,…,n) are continuously differentiable functions with bounded, and ∥·∥ represents the matrix norm.Remark 2.
The function matrixΦ(t)H(x) is called the scaling matrix, and φ1(t)h1(x), φ2(t)h2(x),…,φn(t)hn(x) are called scaling function factors. In the existing results [29–36], the scaling matrix Φ(t) or H(x) is always treated separately. The function matrix in our work is a complex integrated form. Consequently, the presented GFPS can improve antidecryption and unpredictability of the secure communication. Obviously, the GFPS scheme becomes function projective synchronization (FPS) when φ1(t)h1(x)=φ2(t)h2(x)=⋯=φn(t)hn(x). Furthermore, while φi(t)hi(x)(i=1,2,…,n) are constants, GFPS will be simplified into generalized projective synchronization (GPS). In particular, the problem is further reduced to the complete synchronization and anti-phase synchronization if Φ(t)H(x)=I and Φ(t)H(x)=-I, where I is an n×n identity matrix.Remark 3.
The error vector for GFPS between the drive system (8) and the response system (9) is defined by
(11)e(t)=y-Φ(t)H(x)x,
where e(t)=(e1(t),e2(t),…,en(t))T∈Rn, and ei=yi-φi(t)hi(x)xi, (i=1,2,…,n).Remark 4.
IfΦ(t)H(x)=0, the synchronization problem turns into a chaos control problem.


## 3.2. The Scheme of GFPS with Unknown System Parameters

Considering GFPS of two different hyperchaotic systems with fully unknown parameters, we can rewrite the drive and response systems (8), (9) as
(12)x˙=f1(x)+f2(x)α,(13)y˙=g1(y)+g2(y)β+u,
respectively, where f1,g1:Rn→Rn are continuous vector functions, f2:Rn→Rn×m, g2:Rn→Rn×l are continuous matrix functions, and α∈Rm, β∈Rl are unknown parameter vectors of systems (12) and (13). We introduce α~ and β~ as the parameter estimation vectors of α and β, respectively, and define the Jacobian matrix of vector H(x)x as follows:
(14)Γ=d(H(x)x)dx.Theorem 5.
For a given continuous scaling matrix functionΦ(t)H(x) and all initial conditions x(0), y(0), the GFPS between the drive system (12) and the response system (13) can be achieved and the unknown parameters α and β can be identified by the following controller and the adaptive parameter update laws:
(15)u(t)=-g1(y)-g2(y)β~+Φ˙(t)H(x)x+Φ(t)Γ[f1(x)+f2(x)α~]-Ke(t),(16)α~˙=-f2T(x)ΓTΦ(t)e(t)-Peα,β~˙=g2T(y)e(t)-Qeβ,
respectively, where K=diag(k1,k2,…,kn), P=diag(kn+1,kn+2,…,kn+m), Q=diag(kn+m+1,kn+m+2,…,kn+m+l), ki>0(i=1,2,…,n+m+l), and eα=α~-α and eβ=β~-β are error vectors of the identified parameters.Proof.
The time derivative of the error vector (11) is
(17)e˙(t)=y˙-Φ˙(t)H(x)x-Φ˙(t)Γx˙.
Substituting (15) into (17), we have
(18)e˙(t)=-g2(y)(β~-β)+Φ(t)Γf2(x)(α~-α)-Ke(t)=-g2(y)eβ+Φ(t)Γf2(x)eα-Ke(t).
Here, we define the Lyapunov function as
(19)V(t)=12[eT(t)e(t)+eαT(t)eα(t)+eβT(t)eβ(t)].
From (16) and (18), the first derivative of (19) with respect to time is
(20)V˙(t)=eTe˙+eαTe˙α+eβTe˙β=eT[-g2(y)eβ+Φ(t)Γf2(x)eα-Ke(t)]+eαT[-f2T(x)ΓTΦ(t)e(t)Peα]+eβT[g2T(y)e(t)-Qeβ]=-eT(t)Ke(t)-eαT(t)Peα(t)-eβT(t)Qeβ(t)<0.
According to the Lyapunov stability theorem, the trivial solution of the error system (17) is asymptotically stable, which implies that the GFPS of system (12) and (13) is obtained, and error vectors of parameters eα, eβ approach zero as t→∞. The proof is complete.


## 3.3. Applications

In order to verify the effectiveness and feasibility of the proposed GFPS scheme, we apply the controller to realize the synchronization between Chen hyperchaotic system and the proposed system. Chen hyperchaotic system is described by the following state equations [38]:
(21)y˙1=b1(y2-y1)+y4,y˙2=b2y1-y1y3+b3y2,y˙3=y1y2-b4y3,y˙4=y2y3+b5y4,
where y=(y1,y2,y3,y4)T is the state vector and β=(b1,b2,b3,b4,b5)T is parameter vector. As the parameters (b1,b2,b3,b4,b5)=(35,7,12,3,0.5), Chen system shows hyperchaotic properties.We take the proposed system and Chen hyperchaotic system as the drive system and the response system, respectively. Thus, the parameter vector of the drive system (1) is defined as α=(a1,a2,a3,a4,a5)T. The response system can be written as
(22)y˙1=b1(y2-y1)+y4+u1,y˙2=b2y1+y1y3+b3y2+u2,y˙3=y1y2-b4y3+u3,y˙4=y2y3-b5y4+u4,
where u1, u2, u3, and u4 are the control functions to be designed. Without loss of generality, we consider the continuous function matrixes
(23)Φ(t)=diag{φ1(t),φ2(t),φ3(t),φ4(t)}=diag{c11sinω1t+c12,c21sinω2t+c22,c31sinω3t+c32,c41sinω4t+c42},(24)H(x)=diag{h1(x),h2(x),h3(x),h4(x)}=diag{d11x1+d12,d21x2+d22,d31x3+d32,d41x4+d42},
where cij, dij(i=1,2,3,4;j=1,2) are constants. According to the GPFS scheme presented above, the synchronization errors are defined as
(25)ei=yi-φi(t)hi(x)xi=yi-(ci1sinωit+ci2)×(di1xi+di2)xi,i=1,2,3,4.
Substituting (1) and (22) into (25), one can obtain the time derivative of error
(26)e˙1=y4+b1(y2-y1)-c11ω1cosω1t×(d11x1+d12)x1-(c11sinω1t+c12)×(2d11x1+d12)(-a1x1+a2x2)+u1,e˙2=b2y1+b3y2-y1y3-c21ω2cosω2t×(d21x2+d22)x2-(c21sinω2t+c22)×(2d21x2+d22)(a3x1-x1x3-x2+x4)+u2,e˙3=y1y2-b4y3-c31ω3cosω3t×(d31x3+d32)x3-(c31sinω3t+c32)×(2d31x3+d32)[x12-a4(x1+x3)]+u3,e˙4=y2y3+b5y4-c41ω4cosω4t(d41x4+d42)x4+a5x1(c41sinω4t+c42)(2d41x4+d42)+u4.
From (15) and (16), we get the controller
(27)u1=-y4-b~1(y2-y1)-c11ω1cosω1t×(d11x1+d12)x1+(c11sinω1t+c12)×(2d11x1+d12)(-a~1x1+a~2x2)-k1e1,u2=y1y3-b~2y1-b~3y2+c21ω2cosω2t×(d21x2+d22)x2+(c21sinω2t+c22)×(2d21x1+d22)(a~3x1-x1x3-x2+x4)-k2e2,u2=b~4y3-y1y2+c31ω3cosω3t×(d31x3+d32)x3+(c31sinω3t+c32)×(2d31x1+d32)[x1x2-a~4(x1+x3)]-k3e3,u4=-y2y3-b~5y4+c41ω4cosω4t(d41x4+d42)x4-a~5x1(c41sinω4t+c42)(2d41x4+d42)-k4e4,
and the adaptive parameter update laws
(28)a~˙1=x1(c11sinω1t+c12)(2d11x1+d12)e1-k5ea1,a~˙2=-x2(c11sinω1t+c12)(2d11x1+d12)e1-k6ea2,a~˙3=-x1(c21sinω2t+c22)(2d21x2+d22)e2-k7ea3,a~˙4=(x1+x5)(c31sinω3t+c32)×(2d31x3+d32)e3-k8ea4,a~˙5=x1(c41sinω4t+c42)(2d41x4+d42)e4-k9ea5,b~˙1=(y2-y1)e1-k10eb1,b~˙2=y1e2-k11eb2,b~˙3=y2e2-k12eb3,b~˙4=-y3e3-k12eb4,b~˙5=y4e4-k14eb5,
where the control gains ki>0(i=1,2,…,14), eai=a~i-ai(i=1,2,3,4,5) and ebi=b~i-bi(i=1,2,3,4,5), are the corresponding errors of the identified parameters.


## 3.4. Simulation Results

In the simulation, the initial values of the drive and response system are set to be(x1(0),x2(0),x3(0),x4(0))=(2.0,6.0,9.0,2.0) and (y1(0),y2(0),y3(0),y4(0))=(3.0,-4.0,2.0,2.0), respectively. For the simulation purpose, we select all the gains ki=70(i=1,2,…,14). The true values of “unknown” parameters of new system and Chen system are arbitrarily taken as a~1=a~2=a~3=a~4=a~5=0 and b~1=b~2=b~3=b~4=b~5=0, respectively. In order to indicate the effectiveness of the controller (27), the control input is applied to the response system after t=4 s.We choose the coefficients of the scaling function matrixΦ(t)Η(x) according to (24) and (25) as cij=0.1, di1=0.02, di2=-10(i=1,2,3,4;j=1,2), and (ω1,ω2,ω3,ω4)=(1,2,3,4). Figure 7 provides the trajectories of GFPS between Chen system (22) and new system (1). Synchronization error curves are displayed in Figure 8. From Figures 7 and 8, one can see that the following mathematic relationships,
(29)yi=φi(t)hi(x)xi=(ci1sinωit+ci2)(di1xi+di2)xi,i=1,2,3,4
are satisfied rapidly after control; it means that the GFPS is achieved. The estimated values of unknown parameters for the drive and response system are shown by Figures 9 and 10, respectively. From Figures 9 and 10, it is obvious that the unknown parameters are converged to the true values quickly by applying the parameters update laws.The state trajectories of the drive and response systems(φi(t)h(xi)xi=(ci1sinωt+ci2)(di1xi+di2)xi, cij=0.1, di1=0.02, di2=-10,(i=1,2,3,4;j=1,2), and (ω1,ω2,ω3,ω4)=(1,2,3,4)).
(a)
(b)
(c)
(d)Error curves of GFPS(ei=yi-φi(t)h(xi)xi,i=1,2,3,4).
(a)
(b)
(c)
(d)The estimated values of the unknown parametersai(i=1,2,3,4,5) for drive system with true parameters (a1,a2,a3,a4,a5)=(25,60,40,4,5).
(a)
(b)The estimated values of the unknown parametersbi(i=1,2,3,4,5) for response system with true parameters (b1,b2,b3,b4,b5)=(35,7,12,3,0.5).
(a)
(b)Whenci1=di1=0, ci2=-di2=1(i=1,2,3,4), the scaling functions are simplified as φi(t)hi(x)=-1(i=1,2,3,4). Under the same conditions of the system parameters, the initial statuses, and the control gains, the simulation is operated. Figure 11 displays the time evolutions of anti-phase synchronization errors ei=yi+xi(i=1,2,3,4), and it is obvious that ei(i=1,2,3,4) are nonzero and irregular before control and converge to zero rapidly after control. Figure 12 exhibits the synchronization relationship yi=-xi(i=1,2,3,4), which means that anti-phase synchronization between the drive system (1) and the response system (22) is gained.The evolution of anti-phase synchronization error(ei=yi+xi,i=1,2,3,4).
(a)
(b)
(c)
(d)Anti-phase synchronization withyi=-xi(i=1,2,3,4).
(a)
(b)
(c)
(d)From the above simulation results of the GPFS, we can find that the synchronization error can be quickly converged to zero, and the unknown parameters of the drive system and the response system all are accurately estimated to true values. This means that our GPFS algorithm based on Lyapunov stability theory is convergent and effective.


## 4. Conclusion

In this paper, a new 4D hyperchaotic system is introduced at first, and some basic nonlinear dynamical properties of the system are studied. The new system shows complex hyperchaotical behaviors over a wide range of system parameters and can be realized by a simple electronic circuit. At this point, this new hyperchaotic system is suitable as a chaotic sequences generator in the secure communication field. Secondly, a new type of GFPS scheme of two different uncertain hyperchaotic systems is investigated. In our case, the scaling matrixesΦ(t) and H(x) are no more treated separately but as an integrated form; thus, synchronization relationship is more complex. It should be noted that complete synchronization, anti-phase synchronization, general projective synchronization, and function projective synchronization are the special cases of the presented GFPS. A controller with parameters update laws is designed based on Lyapunov stability theory. The GFPS of Chen hyperchaotic system and the proposed hyperchaotic system is considered as an example; the simulation results of the example verify the validity of our method.



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*Source: 701756-2013-12-23.xml*