# Finite-Time Combination-Combination Synchronization for Hyperchaotic Systems

Authors: Huini Lin; Jianping Cai; Jincheng Wang
Journal: Journal of Chaos
 (2013)

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## Abstract

A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finite-time stability theory, step-by-step control and nonlinear control method, a suitable controller is designed to achieve finite-time combination-combination synchronization among four hyperchaotic systems. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique.


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## Body

## 1. Introduction

As a new subject in 1980s, chaos almost covers all the fields of science. It is known that chaos is an interesting nonlinear phenomenon which may lead to irregularity and unpredictability in the dynamic system, and it has been intensively studied in the last three decades. Since Pecora and Carroll proposed the PC method to synchronize two chaotic systems in 1990 [1, 2], the study of synchronization of chaotic systems has been widely investigated due to their potential applications in various fields, for instance, in chemical reactions, biological systems, and secure communication. Over the past decades, a variety of control approaches such as adaptive control [3], linear feedback control [4], active control [5], and backstepping control [6] have been proposed for various types of synchronization, which include complete synchronization [7], projective synchronization [8, 9], general synchronization [10], lag synchronization [11], and novel compound synchronization [12].Most of the aforementioned works are based on the synchronization scheme which consists of one drive system and one response system and can be seen as one-to-one system. However, we found it not secure and flexible enough in many real world applications, for instance, in secure communication. Recently, Runzi et al. presented a new type of synchronization with two drive systems and one response system [13]. Then, Sun et al. extended multi-to-one system to multi-to-two systems and reported a new type of synchronization, namely, combination-combination synchronization, where synchronization is achieved between two drive systems and two response systems [14]. The type of synchronization can improve the security of communication; for instance, we can split the transmitted signals into several parts, then load each part in different drive systems, and then restore it to the original signals by combining the received signals of different response systems correctly.Notice that the mentioned literatures mainly investigated the asymptotic synchronization of chaotic systems. However, in the view of practical application, optimizing the synchronization time is more important than achieving synchronization asymptotically [15–19]. Recently, based on the step-by-step control method, Wang et al. realized the finite-time synchronization of two chaotic systems by designing a proper controller [15]. The method has the ability to achieve global stability in finite time. In addition, the step-by-step technique has the advantage of reducing controller complexity.Motivated by the previous discussion, this paper aims to study the finite-time synchronization between a combination of two drive systems and a combination of two response systems in drive-response synchronization scheme. We have applied the finite-time stability theory to our analysis to achieve finite-time combination-combination synchronization. The step-by-step control method and nonlinear control technique are adopted to synchronize four different hyperchaotic systems. Numerical simulations are presented to verify the theoretical findings.


## 2. The Finite-Time Combination-Combination Synchronization Scheme

Consider the drive systems and response systems as follows:(1)drivesystem1x˙1=f1(x1),(2)drivesystem2x˙2=f2(x2),(3)responsesystem1y˙1=g1(y1)+u,(4)responsesystem2y˙2=g2(y2)+u*,
where x1=(x1,x2,…,xn)T,  x2=(y1,y2,…,yn)T, y1=(z1,z2,…,zn)T, and y2=(w1,w2,…,wn)T are the state vectors of systems (1), (2), (3), and (4), respectively. f1,f2,g1,g2:Rn→Rn are four continuous functions, and  u,u*:Rn×Rn×Rn×Rn→Rn   are two controllers of the response systems (3) and (4) which will be designed, respectively.Definition 1.
If there exist four constant matricesA,B,C,D∈Rn  and  C≠0, D≠0  such that limt→ts∥Ax1+Bx2-Cy1-Dy2∥=0 and ∥Ax1+Bx2-Cy1-Dy2∥≡0 for t≥ts, where  x1=(x1,x2,…,xn)T, x2=(y1,y2,…,yn)T, y1=(z1,z2,…,zn)T, and y2=(w1,w2,…,wn)T, one gets that the drive systems (1) and (2) are realized as finite-time combination-combination synchronization with the response systems (3) and (4), where ∥·∥ represents the matrix norm.Lemma 2 (see [15]).
Assume that a continuous, positive-definite functionV(t) satisfies the following differential inequality:
(5)V˙(t)≤-cVη(t),∀t≥t0,V(t0)≥0,
where  c>0and  0<η<1  are constants. Then, for any initial time t0, V(t) satisfies
(6)V1-η(t)≤V1-η(t0)-c(1-η)(t-t0),t0≤t≤t1,V(t)≡0,∀t≥t1,
with t1 given by
(7)t1=t0+V1-η(t0)c(1-η).Thus, for any initial valueV(t0), the system (5) has V(t)=0 in t1=t0+V1-η(t0)/c(1-η); that is, the system can achieve global stability in finite time.


## 3. Problem Statement and Control Scheme

### 3.1. Problem Statement

Consider two identity hyperchaotic Chen systems as the drive systems [20]:
(8)x˙1=a(y1-x1)+w1,y˙1=dx1-x1z1+cy1,z˙1=x1y1-bz1,w˙1=y1z1+rw1,(9)x˙2=a(y2-x2)+w2,y˙2=dx2-x2z2+cy2,z˙2=x2y2-bz2,w˙2=y2z2+rw2.Consider two identity hyperchaotic Lorenz systems as the response systems [21]:
(10)x˙3=a1(y3-x3)+w3+u1,y˙3=c1x3-y3-x3z3+u2,z˙3=x3y3-b1z3+u3,w˙3=-y3z3+r1w3+u4,(11)x˙4=a1(y4-x4)+w4+u1*,y˙4=c1x4-y4-x4z4+u2*,z˙4=x4y4-b1z4+u3*,w˙4=-y4z4+r1w4+u4*,
where u=(u1,u2,u3,u4)T and u*=(u1*,u2*,u3*,u4*)T are two controllers of the response systems (10) and (11) which will be designed, respectively.Without loss of generality, we chooseA=diag(α1,α2,α3,α4), B=diag(β1,β2,β3,β4),   C=diag(γ1,γ2,γ3,γ4), and D=diag(1,1,1,1).The objective of the synchronization scheme is to design a suitable controllerU(x,y,z,w) such that limt→ts∥Ax1+Bx2-Cy1-Dy2∥=0 and ∥Ax1+Bx2-Cy1-Dy2∥≡0  for  t≥ts, where x1=(x1,x2,…,xn)T,  x2=(y1,y2,…,yn)T,  y1=(z1,z2,…,zn)T, and y2=(w1,w2,…,wn)T; that is, the drive systems (8) and (9) are realized as finite-time combination-combination synchronization with the response systems (10) and (11).


### 3.2. The Control Scheme

Lete1=α1x1+β1x2-γ1x3-x4, e2=α2y1+β2y2-γ2y3-y4,   e3=α3z1+β3z2-γ3z3-z4, and   e4=α4w1+β4w2-γ4w3-w4. The controller to be designed is  U1=γ1u1+u1*,  U2=γ2u2+u2*,   U3=γ3u3+u3*, and   U4=γ4u4+u4*.Thus, we can get the error system as follows:(12)e˙1=-ae1+e4+a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3-U1,e˙2=c1e1+ce2+(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)-U2,e˙3=-be3+(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)-U3,e˙4=re4+(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)-U4.Our aim is to design a suitable controller, such that the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time. Then, the problem is changed to design a suitable controller, such that the error system (12) achieves the finite-time stability at the origin.The design plan and its steps are as follows.Step 1.
Choose(13)U4=(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)+k4e4+e4α,
where
(14)u4=1γ4[(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)],u4*=k4e4+e4α,k4>r,α=qp
is a proper rational number, andp is a positive odd number,p>q.
SubstitutingU4 into the fourth equation of (12), we get
(15)e˙4=(r-k4)e4-e4α.
Choose a candidate Lyapunov function(16)V4=12e42.
Thus, the derivative ofV4 along the solution of error equation (15) is
(17)V˙4=(r-k4)e42-e4α+1≤-e4α+1=-2(α+1)/2(12e42)(α+1)/2=-2(α+1)/2V4(α+1)/2.
According to Lemma2, the system (15) is finite-time stability, which implies that there exists   T1>0, such that e4≡0 for t≥T1.Step 2.
Choose(18)U1=a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3+k1e1+e1β,
where
(19)u1=1γ1[a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3],u1*=k1e1+e1β,k1>-a,0<β<1.
Fort>T1, substituting U1 into the first equation of (12), we get
(20)e˙1=-(k1+a)e1-e1β.
Choose a candidate Lyapunov function(21)V1=12e12.
Thus, the derivative ofV1 along the solution of error equation (20) is
(22)V˙1=-(k1+a)e12-e1β+1≤-e1β+1=-2(β+1)/2(12e12)(β+1)/2=-2(β+1)/2V1(β+1)/2.
According to Lemma2, the system (20) is finite-time stability, which implies that there exists T2>0, such that e1≡0 for t≥T2.Step 3.
Choose(23)U2=(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)+k2e2+e2γ,
where
(24)u2=1γ2[(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)]u2*=k2e2+e2γ,k2>c,0<γ<1.
Fort>T2, substituting U2 into the second equation of (12), we get
(25)e˙2=(-k2+c)e2-e2γ.
Choose a candidate Lyapunov function(26)V2=12e22.
Thus, the derivative ofV2 along the solution of error equation (25) is
(27)V˙2=(-k2+c)e22-e2γ+1≤-e2γ+1=-2(γ+1)/2(12e22)(γ+1)/2=-2(γ+1)/2V2(γ+1)/2.
According to Lemma2, the system (25) is finite-time stability, which implies that there exists T3>0, such that e2≡0 for t≥T3.Step 4.
Choose(28)U3=(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)+k3e3+e3δ,
where
(29)u3=1γ3[(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)],u3*=k3e3+e3δ,k3>-b,0<δ<1.
Fort>T3, substituting U3 into the third equation of (12), we get
(30)e˙3=-(b+k3)e3-e3δ.
Choose a candidate Lyapunov function(31)V3=12e32.
Thus, the derivative ofV3 along the solution of error equation (30) is
(32)V˙3=-(b+k3)e32-e3δ+1≤-e3δ+1=-2(δ+1)/2(12e32)(δ+1)/2=-2(δ+1)/2V3(δ+1)/2.
According to Lemma2, the system (30) is finite-time stability, which implies that there exists T4>0, such that e3≡0 for t≥T4.The controller is designed as follows:(33)U1=a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3+k1e1+e1β,U2=(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)+k2e2+e2γ,U3=(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)+k3e3+e3δ,U4=(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)+k4e4+e4α.According to what we discussed previously, we can obtain this conclusion that the error system (12) achieves finite-time stability under the control of the controller (33). Furthermore, the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time T≤Ts, where Ts=T1+T2+T3+T4.


## 3.1. Problem Statement

Consider two identity hyperchaotic Chen systems as the drive systems [20]:
(8)x˙1=a(y1-x1)+w1,y˙1=dx1-x1z1+cy1,z˙1=x1y1-bz1,w˙1=y1z1+rw1,(9)x˙2=a(y2-x2)+w2,y˙2=dx2-x2z2+cy2,z˙2=x2y2-bz2,w˙2=y2z2+rw2.Consider two identity hyperchaotic Lorenz systems as the response systems [21]:
(10)x˙3=a1(y3-x3)+w3+u1,y˙3=c1x3-y3-x3z3+u2,z˙3=x3y3-b1z3+u3,w˙3=-y3z3+r1w3+u4,(11)x˙4=a1(y4-x4)+w4+u1*,y˙4=c1x4-y4-x4z4+u2*,z˙4=x4y4-b1z4+u3*,w˙4=-y4z4+r1w4+u4*,
where u=(u1,u2,u3,u4)T and u*=(u1*,u2*,u3*,u4*)T are two controllers of the response systems (10) and (11) which will be designed, respectively.Without loss of generality, we chooseA=diag(α1,α2,α3,α4), B=diag(β1,β2,β3,β4),   C=diag(γ1,γ2,γ3,γ4), and D=diag(1,1,1,1).The objective of the synchronization scheme is to design a suitable controllerU(x,y,z,w) such that limt→ts∥Ax1+Bx2-Cy1-Dy2∥=0 and ∥Ax1+Bx2-Cy1-Dy2∥≡0  for  t≥ts, where x1=(x1,x2,…,xn)T,  x2=(y1,y2,…,yn)T,  y1=(z1,z2,…,zn)T, and y2=(w1,w2,…,wn)T; that is, the drive systems (8) and (9) are realized as finite-time combination-combination synchronization with the response systems (10) and (11).


## 3.2. The Control Scheme

Lete1=α1x1+β1x2-γ1x3-x4, e2=α2y1+β2y2-γ2y3-y4,   e3=α3z1+β3z2-γ3z3-z4, and   e4=α4w1+β4w2-γ4w3-w4. The controller to be designed is  U1=γ1u1+u1*,  U2=γ2u2+u2*,   U3=γ3u3+u3*, and   U4=γ4u4+u4*.Thus, we can get the error system as follows:(12)e˙1=-ae1+e4+a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3-U1,e˙2=c1e1+ce2+(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)-U2,e˙3=-be3+(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)-U3,e˙4=re4+(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)-U4.Our aim is to design a suitable controller, such that the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time. Then, the problem is changed to design a suitable controller, such that the error system (12) achieves the finite-time stability at the origin.The design plan and its steps are as follows.Step 1.
Choose(13)U4=(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)+k4e4+e4α,
where
(14)u4=1γ4[(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)],u4*=k4e4+e4α,k4>r,α=qp
is a proper rational number, andp is a positive odd number,p>q.
SubstitutingU4 into the fourth equation of (12), we get
(15)e˙4=(r-k4)e4-e4α.
Choose a candidate Lyapunov function(16)V4=12e42.
Thus, the derivative ofV4 along the solution of error equation (15) is
(17)V˙4=(r-k4)e42-e4α+1≤-e4α+1=-2(α+1)/2(12e42)(α+1)/2=-2(α+1)/2V4(α+1)/2.
According to Lemma2, the system (15) is finite-time stability, which implies that there exists   T1>0, such that e4≡0 for t≥T1.Step 2.
Choose(18)U1=a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3+k1e1+e1β,
where
(19)u1=1γ1[a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3],u1*=k1e1+e1β,k1>-a,0<β<1.
Fort>T1, substituting U1 into the first equation of (12), we get
(20)e˙1=-(k1+a)e1-e1β.
Choose a candidate Lyapunov function(21)V1=12e12.
Thus, the derivative ofV1 along the solution of error equation (20) is
(22)V˙1=-(k1+a)e12-e1β+1≤-e1β+1=-2(β+1)/2(12e12)(β+1)/2=-2(β+1)/2V1(β+1)/2.
According to Lemma2, the system (20) is finite-time stability, which implies that there exists T2>0, such that e1≡0 for t≥T2.Step 3.
Choose(23)U2=(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)+k2e2+e2γ,
where
(24)u2=1γ2[(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)]u2*=k2e2+e2γ,k2>c,0<γ<1.
Fort>T2, substituting U2 into the second equation of (12), we get
(25)e˙2=(-k2+c)e2-e2γ.
Choose a candidate Lyapunov function(26)V2=12e22.
Thus, the derivative ofV2 along the solution of error equation (25) is
(27)V˙2=(-k2+c)e22-e2γ+1≤-e2γ+1=-2(γ+1)/2(12e22)(γ+1)/2=-2(γ+1)/2V2(γ+1)/2.
According to Lemma2, the system (25) is finite-time stability, which implies that there exists T3>0, such that e2≡0 for t≥T3.Step 4.
Choose(28)U3=(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)+k3e3+e3δ,
where
(29)u3=1γ3[(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)],u3*=k3e3+e3δ,k3>-b,0<δ<1.
Fort>T3, substituting U3 into the third equation of (12), we get
(30)e˙3=-(b+k3)e3-e3δ.
Choose a candidate Lyapunov function(31)V3=12e32.
Thus, the derivative ofV3 along the solution of error equation (30) is
(32)V˙3=-(b+k3)e32-e3δ+1≤-e3δ+1=-2(δ+1)/2(12e32)(δ+1)/2=-2(δ+1)/2V3(δ+1)/2.
According to Lemma2, the system (30) is finite-time stability, which implies that there exists T4>0, such that e3≡0 for t≥T4.The controller is designed as follows:(33)U1=a(α1y1+β1y2-γ1x3-x4)-a1(γ1y3-γ1x3+y4-x4)+(α1-α4)w1+(β1-β4)w2-(γ1-γ4)w3+k1e1+e1β,U2=(c+1)(γ2y3+y4)+d(α2x1+β2x2)-c1[α1x1+β1x2+(γ2-γ1)x3]-(α2x1z1+β2x2z2-γ2x3z3-x4z4)+k2e2+e2γ,U3=(b1-b)(γ3z3+z4)+(α3x1y1+β3x2y2-γ3x3y3-x4y4)+k3e3+e3δ,U4=(r-r1)(γ4w3+w4)+(α4y1z1+β4y2z2+γ4y3z3+y4z4)+k4e4+e4α.According to what we discussed previously, we can obtain this conclusion that the error system (12) achieves finite-time stability under the control of the controller (33). Furthermore, the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time T≤Ts, where Ts=T1+T2+T3+T4.


## 4. Numerical Simulation

To verify the effectiveness of the proposed finite-time synchronization method, we consider the hyperchaotic Chen system with the parametersa=35,b=3, c=12, d=7, and r=0.5.   The hyperchaotic attractor of the system is shown in Figure 1.Hyperchaotic attractors of the hyperchaotic Chen system witha=35,  b=3,  c=12,  d=7,  and r=0.5.
(a)
(b)
(c)
(d)Consider the hyperchaotic Lorenz system with the parametersa1=10,  b1=8/3, c1=28, and  r1=-1. The hyperchaotic attractor of the system is shown in Figure 2.Hyperchaotic attractors of the hyperchaotic Lorenz system witha1=10, b1=8/3, c1=28, and r1=-1.
(a)
(b)
(c)
(d)In the following simulation, we assumeα1=α2=α3=α4=1, β1=β2=β3=β4=1, and   γ1=γ2=γ3=γ4=1,   and the initial states for the drive systems and response systems are arbitrarily given by (x1(0),y1(0),z1(0),w1(0))T=(0.1,0.5,1,0.2)T,  (x2(0),y2(0),z2(0),w2(0))T=(0.2,1,0.3,2)T,(x3(0),y3(0),z3(0),w3(0))T=(-1,1.5,3,-2)T, and  (x4(0),y4(0),z4(0),w4(0))T=(-3,5,1,2.5)T. Then, and we choose the synchronization controller with k1=0,k2=13,k3=0,k4=36,α=β=γ=δ=2/3. The synchronization evolution for this controller is shown in Figure 3.The errors between drive systems (8) and (9) and response systems (10) and (11).
(a)
(b)
(c)
(d)The error vector is achieved to zero which implies that systems (8), (9) and (10), (11) have achieved finite-time combination-combination synchronization.


## 5. Conclusion

In this paper, the problem of finite-time combination-combination synchronization with two drive systems and two response systems was investigated. Based on the finite-time stability theory, the step-by-step control and nonlinear control approach, a suitable controller was introduced. The simulation results demonstrated that the proposed controller works well for synchronizing four hyperchaotic systems in finite time.



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