# Parameter Estimation and Hybrid Lag Synchronization in Hyperchaotic Lü Systems

Authors: Qing Wei; Zuolei Wang
Journal: Journal of Nonlinear Dynamics
 (2014)

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## Abstract

The antiphase and complete lag synchronization of hyperchaotic Lü systems with unknown parameters is investigated. Based on the Lyapunov stability theory, the sufficient conditions for achieving hybrid lag synchronization are derived. The optimized parameter observers are approached analytically via adaptive control approach. Numerical simulation results are presented to verify the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Chaos has been thoroughly studied over the past two decades for its “random” behavior and sensitive dependence on the initial conditions. Despite the complexity and unpredictability of chaotic behavior [1], it can be controlled and two chaotic systems can be synchronized [2]. Since Pecora and Carroll introduced a method to synchronize two identical chaotic dynamical systems [3], the synchronization of chaotic dynamical systems attracted much attention due to its theoretical challenge and potential application in secure communications, chemical reactions, biomedical science, social science, and many other fields [4–6]. Various synchronizations have been presented, such as complete synchronization (CS), phase synchronization (PS), lag synchronization (LS) or anticipated synchronization (AS), and generalized synchronization (GS) [7–16]. Subsequently, many effective synchronization methods have been proposed, such as linear or nonlinear feedback synchronization, adaptive synchronization, lag synchronization, Q-S synchronization, and anticipated synchronization [17–26].It is an interesting problem that part of the states of the interactive chaotic system are synchronized in one type of synchronization while other states synchronized in another type of synchronization. This phenomenon is taken as mixed synchronization. Due to the potential applications of it, some types of mixed synchronization are introduced recently. In [27], some variables may converge into synchronization while other variables are in antisynchronization state in Chen-Lee chaotic systems. In [28], using a scalar coupling, some of the state variables may be in complete synchronization while others may be in antisynchronization state in two unidirectionally coupled chaotic oscillators. It is found that lag synchronization has important technological implications. Generally, lag synchronization can be trivially accomplished by coupling the response system to a past state of the drive system or by mismatching of the system parameters. Therefore, inspired by [29, 30], it is invited to investigate the coexistence lag synchronization of chaotic systems via linear input both for theoretical research and practical applications. Mixed lag synchronization in the hyperchaotic systems could also be a useful topic, particularly for its potential application in secure communication due to its safety against attack and unmasking, the secure keys and carrier wave could be generated in more complex but safe way.In this paper, the coexistence lag synchronization of two identical hyperchaotic Lü systems will be achieved only by linear controllers. Based on the Lyapunov stability theory, the sufficient conditions for achieving hybrid lag synchronization are derived. Numerical simulation results are presented to demonstrate the effectiveness of the proposed scheme.


## 2. System Description and Problem Formulation

### 2.1. Hyperchaotic Lü System

In this paper, we consider the following hyperchaotic Lü system:(1)x˙=a(y-x)y˙=cy+w-xzz˙=xy-bzw˙=-r1x-r2y.
The dynamical behavior and its control have been studied in [31]. When a=36,  b=3,c=20,   r1=r2=2,   system (1) has two positive Lyapunov exponents, that is, λ1=1.4106, λ2=0.1232, and the hyperchaotic attractors of system (1) are shown in Figure 1 (3D overview).The hyperchaotic attractors of system (1): (a) (x,y,z); (b) (x,y,w); (c) (x,z,w).
(a)
(b)
(c)


### 2.2. Hybrid Lag Synchronization Formulation

In this section, the hybrid lag synchronization of hyperchaotic Lü system will be investigated via linear feedback control method. For convenience, the drive system is denoted as the following:(2)x˙m=a(ym-xm)y˙m=cym+wm-xmzmz˙m=xmym-bzmw˙m=-r1xm-r2ym.
The controlled response system is written as
(3)x˙s=a^(ys-xs)+u1y˙s=c^ys+ws-xszs+u2z˙s=xsys-b^zs+u3w˙s=-r^1xs-r^2ys+u4,
where u1,u2,u3,u4 are controllers to be constructed.The target of this paper is to determine the controllers, such that the state variablesxs,    ys, and ws in response system antiphase synchronize to the xm,ym, and wm in drive system with a time lag, respectively, while the third state variable zs complete-synchronizes to zm with a time lag. For this purpose, the errors of corresponding variables of hybrid synchronization with a time lag and parameters are often defined as
(4)e1=xs+xm(t-τ),e2=ys+ym(t-τ),e3=zs-zm(t-τ),e4=ws+wm(t-τ),ea=a-a^,eb=b-b^,ec=c-c^,er1=r1-r^1,er2=r2-r^2.
Then we can get the following theorem.Theorem 1.
The third corresponding pair variables of systems (2) and (3) will reach complete lag synchronization, while other three pairs of corresponding variables of systems (2) and (3) will reach lag antisynchronization for any original values, when
(5)ui=-kiei,(i=1,2,3,4),
if the feedback coefficients ki(i=1,2,3,4) are large enough and the condition
(6)e˙a=e1(ys-xs)e˙b=-e3zse˙c=e2yse˙r1=-e4xse˙r2=-e4ys
is satisfied.Proof.
According to (4), the error system can be derived as
(7)e˙1=a(e2-e1)-ea(ys-xs)+u1e˙2=ce2+e4-xm(t-τ)zm(t-τ)-xszs-ecys+u2e˙3=xsys-xm(t-τ)ym(t-τ)-be3+ebzs+u3e˙4=-r1e1-r2e2+er1xs+er2ys+u4.
Let Lyapunov stability function be
(8)V=[e12+e22+e32+e42+ea2+eb2+ec2+er12+er22]2;
then
(9)V˙=e1e˙1+e2e˙2+e3e˙3+e4e˙4+eae˙a+ebe˙b+ece˙c+er1e˙r1+er2e˙r2=e1[a(e2-e1)]+ea[e˙a-e1(ys-xs)]-k1e12+e2[ce2+e4-xm(t-τ)e3-zse1]-k2e22+ec[e˙c-e2ys]+e3[e1ys-xm(t-τ)e2-be3]+eb[e˙b+e3zs]-k3e32+e4(-r1e1-r2e2)+er1[e˙r1+e4xs]+er2[e˙r2+e4ys]-k4e42.
Let M be the bound positive value of hyperchaotic Lü system, that is, |xm|,   |ym|,   |zm|,   |wm|,|xs|,   |ys|,   |zs|, and |ws|≤M, which implies that
(10)V˙≤-(k1-a)e12-(k2-c)e22-(k3-b)e32-k4e42+(a+M)|e1||e2|+M|e1||e3|+r1|e1||e4|+2M|e2||e3|+(r2+1)|e2||e4|=-(|e1|,|e2|,|e3|,|e4|)P(|e1|,|e2|,|e3|,|e4|)T,
where
(11)P=(k1-a-(a+M)2-M2-r12-(a+M)2k2-c-M-(r2+1)2-M2-Mk3-b0-r12-(r2+1)20k4).
It is obvious that, for suitable values of ki(i=1,2,3,4), the matrix P is positive definite and V˙ is negative semidefinite. So one obtains lim⁡t→∞⁡∥ei∥=0 (i=1,2,3,4). It means that the states of response system (3) and the states of drive system (2) are ultimately hybrid lag synchronized asymptotically.


## 2.1. Hyperchaotic Lü System

In this paper, we consider the following hyperchaotic Lü system:(1)x˙=a(y-x)y˙=cy+w-xzz˙=xy-bzw˙=-r1x-r2y.
The dynamical behavior and its control have been studied in [31]. When a=36,  b=3,c=20,   r1=r2=2,   system (1) has two positive Lyapunov exponents, that is, λ1=1.4106, λ2=0.1232, and the hyperchaotic attractors of system (1) are shown in Figure 1 (3D overview).The hyperchaotic attractors of system (1): (a) (x,y,z); (b) (x,y,w); (c) (x,z,w).
(a)
(b)
(c)


## 2.2. Hybrid Lag Synchronization Formulation

In this section, the hybrid lag synchronization of hyperchaotic Lü system will be investigated via linear feedback control method. For convenience, the drive system is denoted as the following:(2)x˙m=a(ym-xm)y˙m=cym+wm-xmzmz˙m=xmym-bzmw˙m=-r1xm-r2ym.
The controlled response system is written as
(3)x˙s=a^(ys-xs)+u1y˙s=c^ys+ws-xszs+u2z˙s=xsys-b^zs+u3w˙s=-r^1xs-r^2ys+u4,
where u1,u2,u3,u4 are controllers to be constructed.The target of this paper is to determine the controllers, such that the state variablesxs,    ys, and ws in response system antiphase synchronize to the xm,ym, and wm in drive system with a time lag, respectively, while the third state variable zs complete-synchronizes to zm with a time lag. For this purpose, the errors of corresponding variables of hybrid synchronization with a time lag and parameters are often defined as
(4)e1=xs+xm(t-τ),e2=ys+ym(t-τ),e3=zs-zm(t-τ),e4=ws+wm(t-τ),ea=a-a^,eb=b-b^,ec=c-c^,er1=r1-r^1,er2=r2-r^2.
Then we can get the following theorem.Theorem 1.
The third corresponding pair variables of systems (2) and (3) will reach complete lag synchronization, while other three pairs of corresponding variables of systems (2) and (3) will reach lag antisynchronization for any original values, when
(5)ui=-kiei,(i=1,2,3,4),
if the feedback coefficients ki(i=1,2,3,4) are large enough and the condition
(6)e˙a=e1(ys-xs)e˙b=-e3zse˙c=e2yse˙r1=-e4xse˙r2=-e4ys
is satisfied.Proof.
According to (4), the error system can be derived as
(7)e˙1=a(e2-e1)-ea(ys-xs)+u1e˙2=ce2+e4-xm(t-τ)zm(t-τ)-xszs-ecys+u2e˙3=xsys-xm(t-τ)ym(t-τ)-be3+ebzs+u3e˙4=-r1e1-r2e2+er1xs+er2ys+u4.
Let Lyapunov stability function be
(8)V=[e12+e22+e32+e42+ea2+eb2+ec2+er12+er22]2;
then
(9)V˙=e1e˙1+e2e˙2+e3e˙3+e4e˙4+eae˙a+ebe˙b+ece˙c+er1e˙r1+er2e˙r2=e1[a(e2-e1)]+ea[e˙a-e1(ys-xs)]-k1e12+e2[ce2+e4-xm(t-τ)e3-zse1]-k2e22+ec[e˙c-e2ys]+e3[e1ys-xm(t-τ)e2-be3]+eb[e˙b+e3zs]-k3e32+e4(-r1e1-r2e2)+er1[e˙r1+e4xs]+er2[e˙r2+e4ys]-k4e42.
Let M be the bound positive value of hyperchaotic Lü system, that is, |xm|,   |ym|,   |zm|,   |wm|,|xs|,   |ys|,   |zs|, and |ws|≤M, which implies that
(10)V˙≤-(k1-a)e12-(k2-c)e22-(k3-b)e32-k4e42+(a+M)|e1||e2|+M|e1||e3|+r1|e1||e4|+2M|e2||e3|+(r2+1)|e2||e4|=-(|e1|,|e2|,|e3|,|e4|)P(|e1|,|e2|,|e3|,|e4|)T,
where
(11)P=(k1-a-(a+M)2-M2-r12-(a+M)2k2-c-M-(r2+1)2-M2-Mk3-b0-r12-(r2+1)20k4).
It is obvious that, for suitable values of ki(i=1,2,3,4), the matrix P is positive definite and V˙ is negative semidefinite. So one obtains lim⁡t→∞⁡∥ei∥=0 (i=1,2,3,4). It means that the states of response system (3) and the states of drive system (2) are ultimately hybrid lag synchronized asymptotically.


## 3. Numerical Simulations

In this section, an illustrative example is presented to demonstrate the effectiveness of the proposed scheme. In the simulations, the system parameters are chosen asa = 36, b=3, c=20, r1=r2=2. The time delay is τ=1. According to Theorem 1, the initial conditions can be given as any values. For simplicity, the initial conditions of the master system and slave system are set to be (xm(0),ym(0),zm(0),wm(0))=(-2.0,3.0,0.4,0.2) and (xs(0),ys(0),zs(0),ws(0))=(1.2,0.3,0.5,0.6), respectively, and the feedback gain coefficients are supposed as ki=k(i=1,2,3,4). From the theoretical analysis, it is known that conditions obtained analytically for the synchronization are only the sufficient conditions. Stronger feedback gain coefficient could induce synchronization easily, while smaller gain coefficient could not ensure the two chaotic systems reach hybrid lag synchronization Figure 2 shows the evolutions of the variables xm,ym,zm,wm in drive system (2) and xs,ys,zs,ws in response system (3) at k=2.3. The evolutions of the state variables xm(t-τ),ym(t-τ),zm(t-τ),wm(t-τ) and xs,ys,zs,ws are plotted in Figure 3. Figure 4 depicts the synchronization errors of the state variables between the master system and the slave system. Figure 5 draws the identification of unknown parameters a, b, c, r1, and r2, from which it is obvious to see that the five unknown parameters converge to their real values, respectively. From the simulation results, we know that the proposed scheme works well and the system error states are regulated to zero asymptotically. Therefore, the hybrid synchronization of systems (2) and (3) can be achieved; that is, the third corresponding pair variables of systems (2) and (3) reach complete lag synchronization, while other three pair corresponding variables of systems (2) and (3) arrive lag antisynchronization.Figure 2
The evolutions of the state variablesxs, ys, zs, ws in response system and xm,ym,zm,wm in drive system at k=2.3.Figure 3
The evolutions of the state variablesxm(t-τ),   ym(t-τ),   zm(t-τ),   wm(t-τ) and xs,ys,zs,ws.Figure 4
The synchronization errors of the state variables between the master system and the slave system.Figure 5
The identified results of the unknown parametersa,b,c,r1, and r2.


## 4. Conclusion

In this paper, mixed lag synchronization in the hyperchaotic systems is studied. Considering the unknown system parameters, an adaptive control scheme has been proposed to ensure the hybrid lag synchronization between the drive and the controlled response systems. Numerical simulations have verified the effectiveness of the proposed technique.



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*Source: 842790-2014-03-30.xml*