# Universal Projective Synchronization of Two Different Hyperchaotic Systems with Unknown Parameters

Authors: Baojie Zhang; Hongxing Li
Journal: Journal of Applied Mathematics
 (2014)

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## 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.


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## Body

## 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 Lu¨ 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 Section2, the definition of UPS is given. In Section 3, the UPS of hyperchaotic Lu¨ 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:(1)x˙=f(t,x),(2)y˙=g(t,y)+u(t,x,y),
where x=(x1,x2,…,xn)T and y=(y1,y2,…,yn)T∈Rn are the state vectors; f,g:Rn→Rn are continuous nonlinear vector function; u(t,x,y)∈Rn 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
(3)e=(e1,e2,…,en)T≜x-A(t)y,
where A(t)=diag⁡(α1(t),α2(t),…,αn(t)); αi(t)(i=1,2,3,…,n) is continuous differentiable function with boundness, and αi(t)≠0 for all t. Obviously, the error system is
(4)e˙=f(t,x)-A˙(t)y-A(t)(g(t,y)+u(t,x,y)).
Suppose there exists a reference system
(5)z˙=h(t,z),
where z=(z1,z2,…,zn)T∈Rn, h:Rn→Rn is a continuous differentiable function. Suppose the reference system is an attractor.Definition 1.
For the scaling diagonal matrix functionA(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,
(6)lim⁡t→∞∥e-z∥=0.Now we give another dual definition of UPS. We define the state error vector(7)e*=(e1*,e2*,…,en*)T≜A(t)x-y;
then the error system is
(8)e˙*=A˙(t)x+A(t)f(t,x)-g(t,y)-u(t,x,y).Definition 2.
For the scaling diagonal matrix functionA(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,
(9)lim⁡t→∞∥e*-z∥=0.Remark 3.
It is easy to see that UPS degenerated to MFPS if the reference system denotes constant vector0; 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 Definition1.


## 3. UPS of Hyperchaotic Systems with Uncertain Parameters

We take hyperchaotic Lu¨ system [17] as the drive system, which is described by
(10)x˙=a(y-x)+w,y˙=-xz+cy,z˙=xy-bz,w˙=xz+dw,
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
(11)x˙1=a1(y1-x1)+u1,y˙1=b1y1-x1z1+w1+u2,z˙1=x1y1-3.36z1+u3,w˙1=-x1z1+w1+u4,
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^, b^, c^, d^, a^1, b^1, respectively. a~=a^-a, b~=b^-b, c~=c^-c, d~=d^-d, a~1=a^1-a1, and b~1=b^1-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 functionβ(t)=(β1(t),β2(t),β3(t),β4(t))T as the reference system. Let e~=(e~1,e~2,e~3,e~4)T≜(e1-β1,e2-β2,e3-β3,e4-β4)T.Theorem 5.
For the given scaling matrixA(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:
(12)u1=1α1(t)(β˙1a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-β˙1(t)+k1e~1),u2=1α2(t)(β˙2-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-β˙2(t)+k2e~2),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-β˙3(t)+k3e~3),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1-β˙4(t)+k4e~4),(13)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2,
where k1,k2,k3,k4 are positive control gains.Proof.
Choose the following Lyapunov function:(14)V=12(e~Te~+a~2+b~2+c~2+d~2+a~12+b~12).
Then the derivative of V is given by
(15)V˙=12(2e~Te~˙+2a~a~˙+2b~b~˙+2c~c~˙+2d~d~˙+2a~1a~˙1+2b~1b~˙1)=e~Te~˙+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1=e~1(β˙1a(y-x)+w-α˙1x1-α1a1(y1-x1)-α1u1-β˙1)+e~2(β˙1-xz+cy-α˙2y1-α2(b1y1-x1z1+w1)-α2u2-β˙2)+e~3(xy-bz-α˙3z1-α3(x1y1-c1z1)-α3u3-β˙3)+e~4(xz+dw-α˙4w1-α4(-x1z1+w1)-α4u4-β˙4)+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1.
With the controls (12) and the parameter update laws (13), we have
(16)V˙=-k1e~12-k2e~22-k3e~32-k4e~42.
Because V˙ is negative simidefinite, e~1,e~2,e~3,e~4∈L∞, a~,b~,c~,d~,a~1,b~1∈L∞. From the definition of e~, we know e~˙1,e~˙2,e~˙3,e~˙4∈L∞. From V˙=-k1e~12-k2e~22-k3e~32-k4e~42=-e~TKe~ (K=diag⁡(k1,k2,k3,k4)), we have
(17)∫0t‍λmin⁡(K)∥e~∥2dt≤∫0t‍e~TKe~dt≤∫0t‍-V˙dt=V(0)-V(t)≤V(0),
where λmin⁡(K) is the minimum eigenvalue of K, so e~1,e~2,e~3,e~4∈L∞; according to Barbalat’s Lemma we have
(18)lim⁡t→∞∥e~(t)∥=0.
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 functionβ(t)=(2sint+cos⁡2t,32sin(3t)+6,6cos⁡t,2.7sin(2t)+3cos⁡t)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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, and b^1(0)=3. Simulation results are as Figures 1, 2, 3, 4, 5, 6, 7, and 8.Figure 1
The trajectories projection onx-y(x1-y1) plane of the drive system and the response system.Figure 2
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 3
The time evolution of the errore1 and the function β1(t).Figure 4
The time evolution of the errore2 and the function β2(t).Figure 5
The time evolution of the errore3 and the function β3(t).Figure 6
The time evolution of the errore4 and the function β4(t).Figure 7
The time evolution of the estimated valuesa^, b^, c^, and d^.Figure 8
The time evolution of the estimated valuesa^1 and b^1.


### 3.2. UPS in the Sense of a Hyperchaotic System

We take hyperchaotic Wang system [18] as the reference system, which is given by
(19)x˙2=a2(-x2+y2)+w2,y˙2=c2x2-y2-x2z2,z˙2=x2y2-b2z2,w˙2=-y2z2+d2w2,
where x2, y2, z2, w2 are state variables and a2=10, b2=8/3, c2=28, and d2=-1 are parameters.Lete~=(e~1,e~2,e~3,e~4)T≜(e1-x2,e2-y2,e3-z2,e4-w2)T.Theorem 6.
For the given scaling matrixA(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:
(20)u1=1α1(t)(a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-(a2(-x2+y2)+w2)+k1e~1),u2=1α2(t)(-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-(c2x2-y2-x2z2)+k2e~2b^1),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-(x2y2-b2z2)+k3e~3b^),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1-(-y2z2+d2w2)+k4e~4d^),(21)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2.The proof is similar to Theorem5 (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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, and b^1(0)=3. Simulation results are as Figures 9, 10, 11, 12, 13, 14, 15, and 16.Figure 9
The trajectories projection onx-y(x1-y1) plane of the drive system and the response system.Figure 10
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 11
The time evolution of the errore1 and the state x2.Figure 12
The time evolution of the errore2 and the state y2.Figure 13
The time evolution of the errore3 and the state z2.Figure 14
The time evolution of the errore4 and the state w2.Figure 15
The time evolution of the estimated valuesa^, b^, c^, and d^.Figure 16
The time evolution of the estimated valuesa^1 and b^1.


### 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 parametersa2, b2, c2, d3 are uncertain. Suppose the estimated values of a2, b2, c2, d3 are a^2, b^2, c^2, d^2, respectively; the parameter errors are (a~2,b~2,c~2,d~2)=(a^2-a2,b^2-b2,c^2-c2,d^2-d2). Let e~=(e~1,e~2,e~3,e~4)T≜(e1-x2,e2-y2,e3-z2,e4-w2)T.Theorem 7.
For the given scaling matrixA(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:
(22)u1=1α1(t)(a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-a^2(-x2+y2)-w2+k1e~1),u2=1α2(t)(-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-c^2x2+y2+x2z2+k2e~2b^1),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-x2y2+b^2z2+k3e~3),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1+y2z2-d^2w2+k4e~4),(23)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2,(24)a^˙2=-(-x2+y2)e~1,b^˙2=z2e~3,c^˙2=-x2e~2,d^˙2=-w2e~4.Proof.
Choose the following Lyapunov function:(25)V=12(e~Te~+a~2+b~2+c~2+d~2+a~12+b~12+a~22+b~22+c~22+d~22).
Then the derivative of V is given by
(26)V˙=12(2e~Te~˙+2a~a~˙+2b~b~˙+2c~c~˙+2d~d~˙+2a~1a~˙1+2b~1b~˙1+2a~2a~˙2+2b~2b~˙2+2c~2c~˙2+2d~2d~˙2)=e~Te~˙+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1+a~2a~˙2+b~2b~˙2+c~2c~˙2+d~2d~˙2=e~1(a(y-x)+w-α˙1x1-α1a1(y1-x1)-α1u1-a2(-x2+y2)-w2)+e~2(-xz+cy-α˙2y1-α2(b1y1-x1z1+w1)-α2u2-c2x2+y2+x2z2)+e~3(xy-bz-α˙3z1-α3(x1y1-c1z1)-α3u3-x2y2+b2z2)+e~4(xz+dw-α˙4w1-α4(-x1z1+w1)-α4u4+y2z2-d2w2)+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1+a~2a~˙2+b~2b~˙2+c~2c~˙2+d~2d~˙2.
With the controls (22) and the parameter update laws (23) and (24), we have
(27)V˙=-k1e~12-k2e~22-k3e~32-k4e~42.
Because V˙ is negative simidefinite, e~1,e~2,e~3,e~4∈L∞, a~,b~,c~,d~,a~1,b~1,a~2,b~2,c~2,d~2∈L∞. From the definition of e~, we know e~˙1,e~˙2,e~˙3,e~˙4∈L∞. From V˙=-k1e~12-k2e~22-k3e~32-k4e~42=-e~TKe~ (K=diag⁡(k1,k2,k3,k4)), we have
(28)∫0t‍λmin⁡(K)∥e~∥2dt≤∫0t‍e~TKe~dt≤∫0t‍-V˙dt=V(0)-V(t)≤V(0),
where λmin⁡(K) is the minimum eigenvalue of K, so e~1,e~2,e~3,e~4∈L∞; according to Barbalat’s Lemma we have
(29)lim⁡t→∞∥e~(t)∥=0.
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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, b^1(0)=3, a^2(0)=4, b^2(0)=4, c^2(0)=4, and d^2(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 onx-y(x1-y1) plane of the drive system and the response system.Figure 18
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 19
The time evolution of the errore1 and the state x2.Figure 20
The time evolution of the errore2 and the state y2.Figure 21
The time evolution of the errore3 and the state z2.Figure 22
The time evolution of the errore4 and the state w2.Figure 23
The time evolution of the estimated valuesa^,b^,c^,d^.Figure 24
The time evolution of the estimated valuesa^1,b^1.Figure 25
The time evolution of the estimated valuesa^2,b^2,c^2,d^2.


## 3.1. UPS in the Sense of Vector Function

Take the vector functionβ(t)=(β1(t),β2(t),β3(t),β4(t))T as the reference system. Let e~=(e~1,e~2,e~3,e~4)T≜(e1-β1,e2-β2,e3-β3,e4-β4)T.Theorem 5.
For the given scaling matrixA(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:
(12)u1=1α1(t)(β˙1a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-β˙1(t)+k1e~1),u2=1α2(t)(β˙2-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-β˙2(t)+k2e~2),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-β˙3(t)+k3e~3),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1-β˙4(t)+k4e~4),(13)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2,
where k1,k2,k3,k4 are positive control gains.Proof.
Choose the following Lyapunov function:(14)V=12(e~Te~+a~2+b~2+c~2+d~2+a~12+b~12).
Then the derivative of V is given by
(15)V˙=12(2e~Te~˙+2a~a~˙+2b~b~˙+2c~c~˙+2d~d~˙+2a~1a~˙1+2b~1b~˙1)=e~Te~˙+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1=e~1(β˙1a(y-x)+w-α˙1x1-α1a1(y1-x1)-α1u1-β˙1)+e~2(β˙1-xz+cy-α˙2y1-α2(b1y1-x1z1+w1)-α2u2-β˙2)+e~3(xy-bz-α˙3z1-α3(x1y1-c1z1)-α3u3-β˙3)+e~4(xz+dw-α˙4w1-α4(-x1z1+w1)-α4u4-β˙4)+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1.
With the controls (12) and the parameter update laws (13), we have
(16)V˙=-k1e~12-k2e~22-k3e~32-k4e~42.
Because V˙ is negative simidefinite, e~1,e~2,e~3,e~4∈L∞, a~,b~,c~,d~,a~1,b~1∈L∞. From the definition of e~, we know e~˙1,e~˙2,e~˙3,e~˙4∈L∞. From V˙=-k1e~12-k2e~22-k3e~32-k4e~42=-e~TKe~ (K=diag⁡(k1,k2,k3,k4)), we have
(17)∫0t‍λmin⁡(K)∥e~∥2dt≤∫0t‍e~TKe~dt≤∫0t‍-V˙dt=V(0)-V(t)≤V(0),
where λmin⁡(K) is the minimum eigenvalue of K, so e~1,e~2,e~3,e~4∈L∞; according to Barbalat’s Lemma we have
(18)lim⁡t→∞∥e~(t)∥=0.
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 functionβ(t)=(2sint+cos⁡2t,32sin(3t)+6,6cos⁡t,2.7sin(2t)+3cos⁡t)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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, and b^1(0)=3. Simulation results are as Figures 1, 2, 3, 4, 5, 6, 7, and 8.Figure 1
The trajectories projection onx-y(x1-y1) plane of the drive system and the response system.Figure 2
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 3
The time evolution of the errore1 and the function β1(t).Figure 4
The time evolution of the errore2 and the function β2(t).Figure 5
The time evolution of the errore3 and the function β3(t).Figure 6
The time evolution of the errore4 and the function β4(t).Figure 7
The time evolution of the estimated valuesa^, b^, c^, and d^.Figure 8
The time evolution of the estimated valuesa^1 and b^1.


## 3.2. UPS in the Sense of a Hyperchaotic System

We take hyperchaotic Wang system [18] as the reference system, which is given by
(19)x˙2=a2(-x2+y2)+w2,y˙2=c2x2-y2-x2z2,z˙2=x2y2-b2z2,w˙2=-y2z2+d2w2,
where x2, y2, z2, w2 are state variables and a2=10, b2=8/3, c2=28, and d2=-1 are parameters.Lete~=(e~1,e~2,e~3,e~4)T≜(e1-x2,e2-y2,e3-z2,e4-w2)T.Theorem 6.
For the given scaling matrixA(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:
(20)u1=1α1(t)(a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-(a2(-x2+y2)+w2)+k1e~1),u2=1α2(t)(-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-(c2x2-y2-x2z2)+k2e~2b^1),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-(x2y2-b2z2)+k3e~3b^),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1-(-y2z2+d2w2)+k4e~4d^),(21)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2.The proof is similar to Theorem5 (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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, and b^1(0)=3. Simulation results are as Figures 9, 10, 11, 12, 13, 14, 15, and 16.Figure 9
The trajectories projection onx-y(x1-y1) plane of the drive system and the response system.Figure 10
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 11
The time evolution of the errore1 and the state x2.Figure 12
The time evolution of the errore2 and the state y2.Figure 13
The time evolution of the errore3 and the state z2.Figure 14
The time evolution of the errore4 and the state w2.Figure 15
The time evolution of the estimated valuesa^, b^, c^, and d^.Figure 16
The time evolution of the estimated valuesa^1 and b^1.


## 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 parametersa2, b2, c2, d3 are uncertain. Suppose the estimated values of a2, b2, c2, d3 are a^2, b^2, c^2, d^2, respectively; the parameter errors are (a~2,b~2,c~2,d~2)=(a^2-a2,b^2-b2,c^2-c2,d^2-d2). Let e~=(e~1,e~2,e~3,e~4)T≜(e1-x2,e2-y2,e3-z2,e4-w2)T.Theorem 7.
For the given scaling matrixA(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:
(22)u1=1α1(t)(a^(y-x)+w-α˙1(t)x1-α1(t)a^1(y1-x1)-a^2(-x2+y2)-w2+k1e~1),u2=1α2(t)(-xz+c^y-α˙2(t)y1+α2(t)x1z1-α2(t)b^1y1-α2(t)w1-c^2x2+y2+x2z2+k2e~2b^1),u3=1α3(t)(xy-b^z-α˙3(t)z1-α3(t)x1y1+3.36α3(t)z1-x2y2+b^2z2+k3e~3),u4=1α4(t)(xz+d^w-α˙4(t)w1+α4(t)x1z1-α4(t)w1+y2z2-d^2w2+k4e~4),(23)a^˙=(y-x)e~1,b^˙=-ze~3,c^˙=ye~2,d^˙=we~4,a^˙1=-α1(t)(y1-x1)e~1,b^˙1=-α2(t)y1e~2,(24)a^˙2=-(-x2+y2)e~1,b^˙2=z2e~3,c^˙2=-x2e~2,d^˙2=-w2e~4.Proof.
Choose the following Lyapunov function:(25)V=12(e~Te~+a~2+b~2+c~2+d~2+a~12+b~12+a~22+b~22+c~22+d~22).
Then the derivative of V is given by
(26)V˙=12(2e~Te~˙+2a~a~˙+2b~b~˙+2c~c~˙+2d~d~˙+2a~1a~˙1+2b~1b~˙1+2a~2a~˙2+2b~2b~˙2+2c~2c~˙2+2d~2d~˙2)=e~Te~˙+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1+a~2a~˙2+b~2b~˙2+c~2c~˙2+d~2d~˙2=e~1(a(y-x)+w-α˙1x1-α1a1(y1-x1)-α1u1-a2(-x2+y2)-w2)+e~2(-xz+cy-α˙2y1-α2(b1y1-x1z1+w1)-α2u2-c2x2+y2+x2z2)+e~3(xy-bz-α˙3z1-α3(x1y1-c1z1)-α3u3-x2y2+b2z2)+e~4(xz+dw-α˙4w1-α4(-x1z1+w1)-α4u4+y2z2-d2w2)+a~a~˙+b~b~˙+c~c~˙+d~d~˙+a~1a~˙1+b~1b~˙1+a~2a~˙2+b~2b~˙2+c~2c~˙2+d~2d~˙2.
With the controls (22) and the parameter update laws (23) and (24), we have
(27)V˙=-k1e~12-k2e~22-k3e~32-k4e~42.
Because V˙ is negative simidefinite, e~1,e~2,e~3,e~4∈L∞, a~,b~,c~,d~,a~1,b~1,a~2,b~2,c~2,d~2∈L∞. From the definition of e~, we know e~˙1,e~˙2,e~˙3,e~˙4∈L∞. From V˙=-k1e~12-k2e~22-k3e~32-k4e~42=-e~TKe~ (K=diag⁡(k1,k2,k3,k4)), we have
(28)∫0t‍λmin⁡(K)∥e~∥2dt≤∫0t‍e~TKe~dt≤∫0t‍-V˙dt=V(0)-V(t)≤V(0),
where λmin⁡(K) is the minimum eigenvalue of K, so e~1,e~2,e~3,e~4∈L∞; according to Barbalat’s Lemma we have
(29)lim⁡t→∞∥e~(t)∥=0.
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,-cos⁡3t-3,sint+cos⁡t+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, b^(0)=-3, c^(0)=6, d^(0)=6, a^1(0)=6, b^1(0)=3, a^2(0)=4, b^2(0)=4, c^2(0)=4, and d^2(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 onx-y(x1-y1) plane of the drive system and the response system.Figure 18
The trajectories projection onz-w(z1-w1) plane of the drive system and the response system.Figure 19
The time evolution of the errore1 and the state x2.Figure 20
The time evolution of the errore2 and the state y2.Figure 21
The time evolution of the errore3 and the state z2.Figure 22
The time evolution of the errore4 and the state w2.Figure 23
The time evolution of the estimated valuesa^,b^,c^,d^.Figure 24
The time evolution of the estimated valuesa^1,b^1.Figure 25
The time evolution of the estimated valuesa^2,b^2,c^2,d^2.


## 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.



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*Source: 549201-2014-03-12.xml*