# Adaptive Modified Function Projective Synchronization between Two Different Hyperchaotic Dynamical Systems

Authors: M. M. El-Dessoky; M. T. Yassen; E. Saleh
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This work investigates modified function projective synchronization between two different hyperchaotic dynamical systems, namely, hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between two diffierent hyperchaotic dynamical systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.


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## Body

## 1. Introduction

During the last three decades, synchronization of chaotic systems has attracted increasing attention from scientists and engineers and has been explored intensively both theoretically and experimentally. Since Pecora and Carrol [1] introduced a method to synchronize two identical systems with different initial conditions, many approaches have been proposed for the synchronization of chaotic or hyperchaotic systems such as complete synchronization [1], phase synchronization [2], generalized synchronization [3], lag synchronization [4], intermittent lag synchronization [5], time-scale synchronization [6], intermittent generalized synchronization [7], projective synchronization [8], modified projective synchronization [9, 10], and function projective synchronization [11, 12]. Most of them are based on exactly knowing the system structure and parameters, but in practice, some or all of the system's parameters are unknown. Moreover, these parameters change from time to time. A lot of works have been done to solve this problem using adaptive synchronization [13–16]. Most of the methods mentioned above synchronize two identical chaotic systems. Hyperchaotic system is usually classified as a chaotic system with more than one positive Lyapunov exponent, indicating that the chaotic dynamics of the system are expanded in more than one direction giving rise to a more complex attractor. The method of the synchronization of two different hyperchaotic systems is far from being straightforward. There is little work about this challenging problem because it consists of different structures and parameter mismatch of the two hyperchaotic systems. Complete synchronization is characterized by the equality of state variables while evolving in time. Antisynchronization is characterized by the vanishing of the sum of relevant variables. Projective synchronization occurs when the drive and response system could be synchronized up to a scaling factor. Function projective synchronization is the most general definition of projective synchronization. It means that the derive and response systems could be synchronized up to a scaling function.A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems. Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant. Recently, Huang proposed a simple adaptive feedback control method, which neednot to estimate or find feedback constant, to effectively synchronize two almost arbitrary identical chaotic systems in his series paper [17–19].In this work, we investigate modified function projective synchronization (MFPS) between hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. This work is organized as follows. In Section2 the modified function projective synchronization (MFPS) scheme is presented. Section 3 briefly describes hyperchaotic Lorenz system and hyperchaotic Chen system. Section 4 proposes adaptive control laws and parameter update rules for the modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Chen system. In Section 5, numerical examples are given to demonstrate the effectiveness of the proposed method. Finally, the conclusions are given in Section 6.


## 2. Adaptive Modified Function Projective Synchronization (MFPS) Scheme

Consider the following master and slave system:(2.1)ẋ=f(x,t),(2.2)ẏ=g(y,t)+u(x,y,t),
where x,y∈Rn are the state vector of the system (2.1) and (2.2), respectively; f,g:Rn→Rn are two continuous nonlinear vector functions, u(x,y,t) is the vector controller. We define the error dynamical system as
(2.3)e(t)=y-Mh(t)x,
where M is a constant diagonal matrix M=diag{m1,m2,…,mn}∈Rn×n and h(t) a continuous differentiable function with h(t)≠0 for all t.The system (2.1) and (2.2) i said to be in modified function projective synchronization if there exists a constant diagonal matrix M and function h(t), such that Limt→∞∥e(t)∥=0.It is easy to see that the definition of modified function projective synchronization encompasses function projective synchronization when the scaling matrixM equals I.


## 3. System Description

The hyperchaotic Lorenz system is described as follows [20–22]:
(3.1)ẋ=α(y-x),ẏ=βx+y-xz-w,ż=xy-bz,ẇ=ryz,
where x, y, z, and w are state variables and α, β, γ, and r are real constant parameters. In [21, 22], it has been shown that the system (3.1) has two positive Lyapunov exponents when α=10,β=28,γ=8/3, and r=0.1, the system (3.1) exhibits hyperchaotic behavior, see Figure 1.Figure 1
The attractor of hyperchaotic Lorenz dynamical system atα=10,β=28,γ=8/3, and r=0.1 in x, y, z subspace.Hyperchaotic Chen system is described as [23, 24]
(3.2)ẋ=a(y-x)+w,ẏ=dx+cy-xz,ż=xy-bz,ẇ=lw+yz,
where x, y, z and w are state variables and a,b,c,d and h are real constant parameters. When a=35,b=3,c=12,d=7,0.798≤l≤0.9, system (3.2) is periodic, when a=35,b=3,c=12,d=7,0≤l≤0.085, system (3.2) is chaotic; when a=35,b=3,c=12,d=7,0.085≤l≤0.798, system (3.2) exhibits hyperchaotic behavior see Figure 2.Figure 2
The attractor of hyperchaotic Chen dynamical system ata=35, b=3, c=12, d=7, and l=0.5 in x, y, z subspace.


## 4. Adaptive MFPS between Hyperchaotic Lorenz System and Chen  System

In order to achieve the synchronization behavior between hyperchaotic Lorenz system and hyperchaotic Chen system, we assume that hyperchaotic Lorenz system is the drive system whose four variables are denoted by subscript 1 and hyperchaotic Chen system is the response system whose variables are denoted by subscript 2. The drive and response systems are described by the following equations, respectively,(4.1)ẋ1=α(y1-x1),ẏ1=βx1+y1-x1z1-w1,ż1=x1y1-γz1,ẇ1=ry1z1,(4.2)ẋ2=a(y2-x2)+w2+u1,ẏ2=dx2+cy2-x2z2+u2,ż2=x2y2-bz2+u3,ẇ2=lw2+y2z2+u4,
where U=[u1,u2,u3,u4]T is the nonlinear controller functions which are to be determined later. The term “synchronization,” in general, means that the signal effect upon the synchronized system is very small in comparison with the amplitude of proper oscillations of the system but, nevertheless, it is enough to change the system behavior and to “impose” the rhythm of external influence to it. The two hyperchaotic dynamical systems can be synchronized in the sense that
(4.3)lim⁡t→∞|x2-m1h(t)x1|=0,lim⁡t→∞|y2-m2h(t)y1|=0,lim⁡t→∞|z2-m3h(t)z1|=0,lim⁡t→∞|w2-m4h(t)w1|=0,
where mi, (i=1,2,3,4) is the scaling factor and h(t) the scaling function.We have the following error dynamical system:(4.4)ėx=ẋ2-m1h(t)ẋ1-m1ḣ(t)x1,ėy=ẏ2-m2h(t)ẏ1-m2ḣ(t)y1,ėz=ż2-m3h(t)ż1-m3ḣ(t)z1,ėw=ẇ2-m4h(t)ẇ1-m4ḣ(t)w1,
where ex=x2-m1h(t)x1, ey=y2-m1h(t)y1, ez=z2-m1h(t)z1, and ew=w2-m1h(t)w1.Substitution of (4.1) and (4.2) in (4.4) yields following error dynamical system
(4.5)ėx=a(y2-x2)+w2+u1-m1h(t)α(y1-x1)-m1ḣ(t)x1,ėy=dx2+cy2-x2z2+u2-m2h(t)(βx1+y1-x1z1-w1)-m2ḣ(t)y1,ėz=x2y2-bz2+u3-m3h(t)(x1y1-γz1)-m3ḣ(t)z1,ėw=lw2+y2z2+u4-m4h(t)ry1z1-m4ḣ(t)w1.Our aim is to find control lawsui(i=1,2,3,4) for stabilizing the error variables of the system at the origin. For this end, we propose following control law:(4.6)u1=m1h(t)α1(y1-x1)+m1ḣ(t)x1-a1(x2-y2)-w2-k1ex,u2=m2h(t)(y1+β1x1-w1-x1z1)+x2z2+m2ḣ(t)y1-d1x2-c1y2-k2ey,u3=m3h(t)(x1y1-γ1z1)+m3ḣ(t)z1+b1z2-x2y2-k3ez,u4=m4h(t)r1y1z1+m4ḣ(t)w1-l1w2-y2z2-k4ew,
and the update laws for the unknown parameters α1, β1, γ1, r1, a1, b1, c1, d1, and l1 are
(4.7)α̇1=(x1-y1)m1h(t)ex-k5(α1-α),β̇1=-x1m2h(t)ey-k6(β1-β),γ̇1=z1m3h(t)ez-k7(γ1-γ),ṙ1=-y1z1m4h(t)ew-k8(r1-r),ȧ1=(y2-x2)ex-k9(a1-a),ḃ1=-z2ez-k10(b1-b),ċ1=y2ey-k11(c1-c),ḋ1=x2ey-k12(d1-d),l̇1=w2ew-k13(l1-l),
where ki>0(i=1,2,3,…,13).Theorem 4.1.
For given constant scaling matrixM and scaling function h(t), the MFPS between two systems (4.1) and (4.2) will occur by the control law (4.6) and update law (4.7), and satisfy
(4.8)lim⁡t→∞|α1-α|=lim⁡t→∞|β1-β|=lim⁡t→∞|γ1-γ|=lim⁡t→∞|r1-r|=lim⁡t→∞|a1-a|=lim⁡t→∞|b1-b|=lim⁡t→∞|c1-c|=lim⁡t→∞|d1-d|=lim⁡t→∞|l1-l|=0.Proof.
Define a Lyapunov function,(4.9)V(e)=12(ex2+ey2+ez2+ew2+eα2+eβ2+eγ2+er2+ea2+eb2+ec2+ed2+el2),
where
(4.10)eα=α1-α,eβ=β1-β,eγ=γ1-γ,er=r1-r,ea=a1-a,eb=b1-b,ec=c1-c,ed=d1-d,el=l1-l.
The time derivative of the Lyapunov function along the trajectory of error system (4.9) is
(4.11)dV(e)dt=exėx+eyėy+ezėz+ewėw+eαėα+eβėβ+eγėγ+erėr+eaėa+ebėb+eceċ+edėd+elėl=exėx+eyėy+ezėz+ewėw+eαα̇1+eββ̇1+eγγ̇1+erṙ1+eaȧ1+ebḃ1+ecċ1+edḋ1+ell̇1.
Inserting (4.6) and (4.7) into (4.11) yields the following:
(4.12)dV(e)dt=-k1ex2-k2ey2-k3ez2-k4ew2-k5eα2-k6eβ2-k7eγ2-k8er2-k9ea2-k10eb2-k11ec2-k12ed2-k13el2=-Ke2,
where e=(ex,ey,ez,ew,eα,eβ,eγ,er,ea,eb,ec,ed,el)T and K=diag(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12,k13)T.
SincedV(e)/dt≤0, we have ex,ey,ez,ew,eα,eβ,eγ,er,ea,eb,ec,ed,el→0 as t→∞,limt→∞∥e∥=0.
Therefore, the drive system (4.1) synchronizes the response system (4.2) in the sense of MFPS.Remark 4.2.
Note that complete synchronization and antisynchronization between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling functionh(t)=1 and the scaling factors mi=1 and mi=-1(i=1,2,3,4), respectively.Remark 4.3.
Note that function projective synchronization (FPS) between two different hyperchaotic dynamical systems is special case of MFPS with the scaling factorsmi=1(i=1,2,3,4), and the scaling function h(t) is chosen later.Remark 4.4.
Note that generalized projective synchronization (GPS) and modified generalized projective synchronization (MGPS) between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling functionh(t)=1 and the scaling factors mi are equal and mi are not equal (i=1,2,3,4), respectively.By suitable choosing forh(t), we can achieve modified function projective synchronization, complete synchronization, antisynchronization, function projective synchronization, generalized projective synchronization, modified generalized projective synchronization, between two different hyperchaotic systems (see Examples 5.1–5.6).


## 5. Numerical Results

In this section, numerical examples are used to demonstrate the effectiveness of the proposed method. By using Maple 12 to solve the systems of differential equations (4.1), (4.2), (4.6), and (4.7), we assume that the initial conditions of the drive system are x1(0)=2, y1(0)=2, z1(0)=3, and w1(0)=1, and the initial conditions of the response system are x2(0)=6, y2(0)=5, z2(0)=3, andw2(0)=3. The initial conditions of the estimated parameters are chosen as α1(0)=0, β1(0)=0, γ1(0)=0, r1(0)=0, a1(0)=0, b1(0)=0, c1(0)=0, d1(0)=0 and l1(0)=0.Let the scaling function beh(t)=sin(0.1πt) and the scaling factors are chosen as m1=2, m2=3, m3=5, and m4=0.5. The simulation of the error dynamical system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions is shown in Figure 3(a)  displays the ex=x2-m1h(t)x1, Figure 3(b) displays the ey=y2-m1h(t)y1, Figure 3(c) displays the ez=z2-m1h(t)z1, and Figure 3(d) displays the ew=w2-m1h(t)w1.The behavior of the trajectoriesex, ey, ez, and ew  of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions.
(a)(b)(c)(d)Example 5.1.
Let the scaling function beh(t)=sin(0.1πt) and the scaling factors are chosen as m1=2,m2=3,m3=5, and m4=0.5. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 4 displays the MFPS between systems (4.1) and (4.2). Figure 5 show that the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 6 show that the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35, b=3, c=12, d=7, and  l=0.5  as t→∞.Figure 4
The behavior of the trajectoriesex,  ey,  ez,  and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MFPS.Figure 5
The estimatesα1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converges to α=10,β=28,γ=8/3, and r=0.1 as t tends to 3.Figure 6
The estimatesa1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and  l=0.5  as t tends to 3.Example 5.2.
Let the scaling function beh(t)=1 and the scaling factors are chosen as m1=m2=m3=m4=1. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 7(a) displays the complete synchronization between systems (4.1) and (4.2). Figure 7(b) show that the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 7(c) show that the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and l=0.5 as t→∞.(a) The behavior of the trajectoriesex,ey,ez,and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for complete synchronization. (b) the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t tends to 3. (c) the estimates a1(t), b1(t), c1(t), d1(t), l1(t) of the unknown parameters converges to a=35,b=3,c=12,d=7, and l=0.5 as t tends to 3.
(a)(b)(c)Example 5.3.
Let the scaling function beh(t)=1 and the scaling factors are chosen as m1=m2=m3=m4=-1. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 8(a) displays the anti-synchronization between systems (4.1) and (4.2). Figure 8(b) shows the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 8(c) shows the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and l=0.5as t→∞.(a) The behaviour of the trajectoriesex,  ey,  ez and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for anti-synchronization. (b): Show the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converges to α=10,β=28,γ=8/3 and r=0.1as t tends to 3. (c): Show the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converges to a=35,b=3,c=12,d=7 and l=0.5 as t tends to 3.
(a)(b)(c)Example 5.4.
Let the scaling function beh(t)=sin(0.1πt) and the scaling factors are chosen as m1=m2=m3=m4=1. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 9(a) displays the FPS between systems (4.1) and (4.2). Figure 9(b) shows the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 9(c) shows the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and l=0.5as t→∞.(a) The behavior of the trajectoriesex, ey, ez,and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for FPS. (b) the estimates α1(t),  β1(t), γ1(t), r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1 as t tends to 3. (c) the estimates a1(t), b1(t), c1(t), d1(t), l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and l=0.5 as t tends to 3.
(a)(b)(c)Example 5.5.
Let the scaling function beh(t)=1 and the scaling factors are chosen as m1=m2=m3=m4=0.5. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 10(a) displays the GPS between systems (4.1) and (4.2). Figure 10(b) shows the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 10(c) shows the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converges to a=35,b=3,c=12,d=7, and l=0.5 as t→∞.(a) The behavior of the trajectoriesex,  ey,  ez,  and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for GPS. (b) the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converges to α=10,β=28,γ=8/3, and r=0.1 as t tends to 3. (c) the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converges to a=35,b=3,c=12,d=7, and l=0.5 as t tends to 3.
(a)(b)(c)Example 5.6.
Let the scaling function beh(t)=1 and the scaling factors are chosen as m1=2,m2=3,m3=-2, and m4=0.1. Furthermore, the control gains are chosen as k1=k2=k3=k4=3,k5=k6=k7=k8=k9=k10=k11=k12=h13=2. Figure 11(a) displays the MGPS between systems (4.1) and (4.2). Figure 11(b) show that the estimates α1(t),  β1(t),  γ1(t),  r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1as t→∞. Figure 11(c) show that the estimates a1(t),  b1(t),  c1(t),  d1(t),  l1(t) of the unknown parameters converge to a=35,b=3,c=12,d=7, and l=0.5 as t→∞.(a) The behaviour of the trajectoriesex, ey, ez, and ew of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MGPS. (b) the estimates α1(t), β1(t), γ1(t), r1(t) of the unknown parameters converge to α=10,β=28,γ=8/3, and r=0.1 as t tends to 3. (c) the estimates a1(t), b1(t), c1(t), d1(t), l1(t) of the unknown parameters converge to a=35, b=3, c=12, d=7, and l=0.5 as t tends to 3.
(a)(b)(c)


## 6. Conclusions

This work investigated modified function projective synchronization between the hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, we design adaptive synchronization controllers with corresponding parameter update laws to synchronize the two systems. The MFPS includes complete synchronization, antisynchronization, function projective synchronization (FPS), generalized projective synchronization (GPS), and modified generalized projective synchronization (MGPS). All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the proposed synchronization schemes. Thus, our synchronization method is successful for some systems with two positive Lyapunov exponents.



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