# Chaos Synchronization between Two Different Fractional Systems of Lorenz Family

Authors: A. E. Matouk
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.


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## Body

## 1. Introduction

Fractional calculus has been known since the early 17th century [1, 2]. It has useful applications
in many fields of science like physics [3], engineering [4], mathematical biology
[5, 6], and finance [7, 8].The fractional
order derivatives have many definitions; one of them is the Riemann-Liouville
definition [9] which is given by
(1.1)Dαf(t)=dldtlJl−αf(t),α>0,
where Jθ is the θ-order Riemann-Liouville integral operator
which is given as
(1.2)Jθu(t)=1Γ(θ)∫0t(t−τ)θ−1u(τ)dτ,θ>0.However, the
most common definition is the Caputo definition [10], since it is widely used
in real applications:
(1.3)D*αf(t)=Jl−αf(l)(t),
where f(l) represents the l-order derivative of f(t) and l=[α];
this means that l is the first
integer which is not less than α.
The operator D*α is called “the Caputo differential operator of
order α.” Hence, I choose the Caputo type throughout
this paper.On the other hand, chaos has been studied and developed
with much interest by scientists since the birth of Lorenz chaotic attractor in
1963 [11]. Chen attractor is similar to Lorenz attractor but not
topologically equivalent [12]. Recently, Lü et al.  found a new chaotic system
which connects the Lorenz and Chen attractors, according to the conditions formulated
by Vaněček and  Čelikovský, and it is called
Lü system [13]. Afterwards, chaos in fractional order dynamical systems has
become an interesting topic. In [14] chaotic behaviors of the fractional order
Lorenz system are studied. Moreover, chaotic behaviors have also been found in
the fractional order Chen system [15] and the fractional order Lü system [16].
Furthermore, Chaos synchronization in fractional order chaotic systems starts
to attract increasing attention [16–20]. However, it has been studied very well
in the case of integer order chaotic systems, due to its potential applications
in physical, chemical, and  biological systems [21–24] and secure communications [25].The generalized synchronization between
two different fractional order systems is investigated in [26]. However, in
this paper, I investigate the conditions of chaos synchronization between two
different fractional order chaotic systems of Lorenz family by designing suitable
linear controllers. I give examples to achieve chaos synchronization of two pairs
of different fractional order chaotic systems (fractional Chen & fractional
Lü, fractional Lorenz-like, and fractional Chen) in drive-response
structure. Conditions for achieving chaos synchronization using linear
control method are further discussed using Laplace transform theory.


## 2. Systems Description

The fractional order Chen system isgiven as follows:(2.1)dαxdtα=a(y−x),dαydtα=(c−a)x−xz+cy,dαzdtα=xy−bz.
Here and
throughout, (a,b,c)=(35,3,28) where α is the fractional order. In the following I
choose α=0.9 at which system (2.1) exhibits chaotic attractor (see Figure 1).Figure 1
Chaotic attractor of the fractional order Chen
system (2.1) with α=0.9 and (a,b,c)=(35,3,28).The fractional order Lü system is
given as follows(2.2)dαxdtα=r(y−x),dαydtα=−xz+py,dαzdtα=xy−qz.
Here and
throughout, (r,p,q)=(35,28,3).
By choosing α=0.9,
system (2.2) has chaotic attractor (see Figure 2).Figure 2
Chaotic attractor of the fractional order Lü
system (2.2) with α=0.9 and (r,p,q)=(35,28,3).The fractional
order Lorenz-like system [27] is described by
(2.3)dαxdtα=σ(y−x),dαydtα=ρx−xz+γy,dαzdtα=xy−βz,
which has a chaotic attractor as
shown in Figure 3 when β=2.8,γ=10.6,ρ=14,σ=20, and α=0.9.Figure 3
Chaotic attractor of the fractional order
Lorenz-like system (2.3) with α=0.9 and (β,γ,ρ,σ)=(2.8,10.6,14,20).It should be
also noted that, the systems (2.1), (2.2), and (2.3) are still chaotic at the
fractional order values α=0.95 and α=0.99.


## 3. Synchronization between Two Different Fractional Order Systems

Consider the
master-slave (or drive-response) synchronization scheme of two autonomous
different fractional order chaotic systems:(3.1)dαXdtα=f(X),dαYdtα=g(Y)+U(t),
where α is the fractional order, X∈Rn,Y∈Rn represent the states of the drive and response
systems, respectively, f:Rn→Rn,g:Rn→Rn are the vector fields of the drive and
response systems, respectively. The aim is to choose a suitable linear control
function U(t)=(u1,…,un)T such that the states of the drive and response
systems are synchronized (i.e.,  limt→∞∥X−Y∥=0,
where ∥⋅∥ is the Euclidean norm).


### 3.1. Synchronization between Chen and Lü Fractional Order Systems

In this subsection,
the goal is to achieve chaos synchronization between the fractional order Chen
system and the fractional order Lü system by using the fractional order Chen
system to drive the fractional order Lü system. The drive and response systems
are given as follows:(3.2)dαxmdtα=a(ym−xm),dαymdtα=(c−a)xm−xmzm+cym,dαzmdtα=xmym−bzm,(3.3)dαxsdtα=r(ys−xs)+u1,dαysdtα=−xszs+pys+u2,dαzsdtα=xsys−qzs+u3,
where u1,u2, and u3 are the linear control functions. Define the
error variables as follows:(3.4)e1=xs−xm,e2=ys−ym,e3=zs−zm.By subtracting (3.2) from (3.3) and
using (3.4), we obtain
(3.5)dαe1dtα=r(e2−e1)+(r−a)(ym−xm)+u1,dαe2dtα=pe2−zme1−xme3−e1e3−(c−a)xm+(p−c)ym+u2,dαe3dtα=−qe3+yme1+xme2+e1e2−(q−b)zm+u3.Now, by letting(3.6)u1=(a−r)(ym−xm),u2=(c−a)xm+(c−p)ym−k1(ys−ym),u3=(q−b)zm−k2(zs−zm),
where k1,k2≥0,
then the error system (3.5) is reduced to(3.7)dαe1dtα=r(e2−e1),dαe2dtα=(p−k1)e2−zme1−xme3−e1e3,dαe3dtα=−(q+k2)e3+yme1+xme2+e1e2.By taking the Laplace transform in both sides of (3.7), letting Ei(s)=L{ei(t)} where (i=1,2,3),
and applying L{dαei/dtα}=sαEi(s)−sα−1ei(0),
we obtain(3.8)sαE1(s)−sα−1e1(0)=r(E2(s)−E1(s)),sαE2(s)−sα−1e2(0)=(p−k1)E2(s)−L{xme3}−L{zme1}−E1(s)E3(s),sαE3(s)−sα−1e3(0)=−(q+k2)E3(s)+L{yme1}+L{xme2}+E1(s)E2(s).Proposition 3.1.
IfE1(s),E2(s) are bounded and p−k1≠0,
then the drive and response systems (3.2) and (3.3) will be synchronized under a
suitable choice of k1 and k2.Proof.
Rewrite (3.8) as follows:
(3.9)E1(s)=rE2(s)sα+r+sα−1e1(0)sα+r,E2(s)=−L{zme1}sα−p+k1−L{xme3}sα−p+k1−E1(s)E3(s)sα−p+k1+sα−1e2(0)sα−p+k1,E3(s)=L{yme1}sα+q+k2+L{xme2}sα+q+k2+E1(s)E2(s)sα+q+k2+sα−1e3(0)sα+q+k2.
Using the final value theorem of
the Laplace transform, it follows that(3.10)limt→∞e1(t)=lims→0+sE1(s)=lims→0+sE2(s)=limt→∞e2(t),limt→∞e2(t)=lims→0+sE2(s)=1p−k1lims→0+sL{xme3}+1p−k1lims→0+sL{zme1}+1p−k1limt→∞e1(t)⋅limt→∞e3(t),limt→∞e3(t)=lims→0+sE3(s)=1q+k2lims→0+sL{yme1}+1q+k2lims→0+sL{xme2}+1q+k2limt→∞e1(t)⋅limt→∞e2(t).
SinceE1(s),E2(s) are bounded and p−k1≠0 then limt→∞e1(t)=limt→∞e2(t)=0.
Now, owing to the attractiveness of the attractors of systems (2.1) and (2.2), there
exists η>0 such that |xi(t)|≤η<∞,|yi(t)|≤η<∞, and |zi(t)|≤η<∞ where i refers to the index of the drive or response
variables. Therefore, limt→∞e3(t)=0.
This implies that
(3.11)limt→∞ei(t)=0,i=1,2,3.
Consequently, the synchronization
between the drive and response systems (3.2) and (3.3) is achieved.


#### 3.1.1. Numerical Results

An efficient
method for solving fractional order differential equations is the
predictor-correctors scheme or more precisely, PECE   (Predict, Evaluate, Correct,
Evaluate) technique which has been investigated in [28, 29], and represents a
generalization of the Adams-Bashforth-Moulton algorithm. It is used throughout
this paper.Based on the
above mentioned discretization scheme, the drive and response systems (3.2) and (3.3)
are integrated numerically with the fractional orders α=0.9,0.95,0.99 and using the initial values xm(0)=15,ym(0)=20,zm(0)=29 and xs(0)=10,ys(0)=15,zs(0)=25.
From Figure 4,  it is clear that the synchronization is achieved for all these
values of fractional order  when k1=20 and k2=10.Synchronization errors of
the drive system (3.2) and response system (3.3) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


### 3.2. Synchronization between Lorenz-Like and Chen Fractional Order Systems

In this case it
is assumed that, the fractional order Lorenz-like system drives the fractional
order Chen system. The drive and response systems are defined as follows:(3.12)dαxmdtα=σ(ym−xm),dαymdtα=ρxm−xmzm+γym,dαzmdtα=xmym−βzm,(3.13)dαxsdtα=a(ys−xs)+v1,dαysdtα=(c−a)xs−xszs+cys+v2,dαzsdtα=xsys−bzs+v3,
where v1,v2, and v3 are the linear control functions. The error
variables are given by(3.14)e1=xs−xm,e2=ys−ym,e3=zs−zm.By subtracting (3.12) from (3.13) and
using (3.14), we get(3.15)dαe1dtα=a(e2−e1)+(a−σ)(ym−xm)+v1,dαe2dtα=ce2−zme1−xme3−e1e3+(c−a)xs−ρxm+(c−γ)ym+v2,dαe3dtα=−be3+yme1+xme2+e1e2+(β−b)zm+v3.Now, by choosing(3.16)v1=(σ−a)(ym−xm),v2=ρxm+(a−c)xs+(γ−c)ym−k1e2,v3=(b−β)zm−k2e3,
where k1,k2≥0,
then the error system (3.15) is rewritten as(3.17)dαe1dtα=a(e2−e1),dαe2dtα=(c−k1)e2−zme1−xme3−e1e3,dαe3dtα=−(b+k2)e3+yme1+xme2+e1e2.Take Laplace transform in both sides of (3.17), let Ei(s)=L{ei(t)}, where (i=1,2,3),
and apply L{dαei/dtα}=sαEi(s)−sα−1ei(0).
After that, by doing similar analysis like the previous subsection, we obtain(3.18)limt→∞e1(t)=lims→0+sE1(s)=lims→0+sE2(s)=limt→∞e2(t),limt→∞e2(t)=lims→0+sE2(s)=1c−k1lims→0+sL{xme3}+1c−k1lims→0+sL{zme1}+1c−k1limt→∞e1(t)⋅limt→∞e3(t),limt→∞e3(t)=lims→0+sE3(s)=1b+k2lims→0+sL{yme1}+1b+k2lims→0+sL{xme2}+1b+k2limt→∞e1(t)⋅limt→∞e2(t).If we assume
thatc−k1≠0 and E1(s),E2(s) are bounded, then it follows that limt→∞e1(t)=limt→∞e2(t)=0.
Now, owing to the attractiveness of the attractors of systems (2.1) and (2.3),
there exists ξ>0 such that |xi(t)|≤ξ<∞,|yi(t)|≤ξ<∞, and |zi(t)|≤ξ<∞ where i refers to the index of the drive or response
variables. Therefore, limt→∞e3(t)=0.
Consequently,(3.19)limt→∞ei(t)=0,i=1,2,3.Thus, the states of the drive
system (3.12) are synchronized with the states of the response system (3.13), as
the controllers (3.16) are activated.


#### 3.2.1. Numerical Results

Numerical
simulations are carried out to integrate the drive and response systems (3.12)
and (3.13) using the predictor-correctors scheme, with the fractional orders α=0.9,0.95,0.99 and the initial values xm(0)=10,ym(0)=16,zm(0)=25 and xs(0)=15,ys(0)=20,zs(0)=29.
Thus, the drive and response systems (3.12) and (3.13) are synchronized in such a
successful way for all at the above-mentioned fractional orders values, using
the linear controllers (3.16) with k1=20 and k2=10 (see Figure 5).Synchronization errors
of the drive system (3.12) and response system (3.13) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


## 3.1. Synchronization between Chen and Lü Fractional Order Systems

In this subsection,
the goal is to achieve chaos synchronization between the fractional order Chen
system and the fractional order Lü system by using the fractional order Chen
system to drive the fractional order Lü system. The drive and response systems
are given as follows:(3.2)dαxmdtα=a(ym−xm),dαymdtα=(c−a)xm−xmzm+cym,dαzmdtα=xmym−bzm,(3.3)dαxsdtα=r(ys−xs)+u1,dαysdtα=−xszs+pys+u2,dαzsdtα=xsys−qzs+u3,
where u1,u2, and u3 are the linear control functions. Define the
error variables as follows:(3.4)e1=xs−xm,e2=ys−ym,e3=zs−zm.By subtracting (3.2) from (3.3) and
using (3.4), we obtain
(3.5)dαe1dtα=r(e2−e1)+(r−a)(ym−xm)+u1,dαe2dtα=pe2−zme1−xme3−e1e3−(c−a)xm+(p−c)ym+u2,dαe3dtα=−qe3+yme1+xme2+e1e2−(q−b)zm+u3.Now, by letting(3.6)u1=(a−r)(ym−xm),u2=(c−a)xm+(c−p)ym−k1(ys−ym),u3=(q−b)zm−k2(zs−zm),
where k1,k2≥0,
then the error system (3.5) is reduced to(3.7)dαe1dtα=r(e2−e1),dαe2dtα=(p−k1)e2−zme1−xme3−e1e3,dαe3dtα=−(q+k2)e3+yme1+xme2+e1e2.By taking the Laplace transform in both sides of (3.7), letting Ei(s)=L{ei(t)} where (i=1,2,3),
and applying L{dαei/dtα}=sαEi(s)−sα−1ei(0),
we obtain(3.8)sαE1(s)−sα−1e1(0)=r(E2(s)−E1(s)),sαE2(s)−sα−1e2(0)=(p−k1)E2(s)−L{xme3}−L{zme1}−E1(s)E3(s),sαE3(s)−sα−1e3(0)=−(q+k2)E3(s)+L{yme1}+L{xme2}+E1(s)E2(s).Proposition 3.1.
IfE1(s),E2(s) are bounded and p−k1≠0,
then the drive and response systems (3.2) and (3.3) will be synchronized under a
suitable choice of k1 and k2.Proof.
Rewrite (3.8) as follows:
(3.9)E1(s)=rE2(s)sα+r+sα−1e1(0)sα+r,E2(s)=−L{zme1}sα−p+k1−L{xme3}sα−p+k1−E1(s)E3(s)sα−p+k1+sα−1e2(0)sα−p+k1,E3(s)=L{yme1}sα+q+k2+L{xme2}sα+q+k2+E1(s)E2(s)sα+q+k2+sα−1e3(0)sα+q+k2.
Using the final value theorem of
the Laplace transform, it follows that(3.10)limt→∞e1(t)=lims→0+sE1(s)=lims→0+sE2(s)=limt→∞e2(t),limt→∞e2(t)=lims→0+sE2(s)=1p−k1lims→0+sL{xme3}+1p−k1lims→0+sL{zme1}+1p−k1limt→∞e1(t)⋅limt→∞e3(t),limt→∞e3(t)=lims→0+sE3(s)=1q+k2lims→0+sL{yme1}+1q+k2lims→0+sL{xme2}+1q+k2limt→∞e1(t)⋅limt→∞e2(t).
SinceE1(s),E2(s) are bounded and p−k1≠0 then limt→∞e1(t)=limt→∞e2(t)=0.
Now, owing to the attractiveness of the attractors of systems (2.1) and (2.2), there
exists η>0 such that |xi(t)|≤η<∞,|yi(t)|≤η<∞, and |zi(t)|≤η<∞ where i refers to the index of the drive or response
variables. Therefore, limt→∞e3(t)=0.
This implies that
(3.11)limt→∞ei(t)=0,i=1,2,3.
Consequently, the synchronization
between the drive and response systems (3.2) and (3.3) is achieved.


### 3.1.1. Numerical Results

An efficient
method for solving fractional order differential equations is the
predictor-correctors scheme or more precisely, PECE   (Predict, Evaluate, Correct,
Evaluate) technique which has been investigated in [28, 29], and represents a
generalization of the Adams-Bashforth-Moulton algorithm. It is used throughout
this paper.Based on the
above mentioned discretization scheme, the drive and response systems (3.2) and (3.3)
are integrated numerically with the fractional orders α=0.9,0.95,0.99 and using the initial values xm(0)=15,ym(0)=20,zm(0)=29 and xs(0)=10,ys(0)=15,zs(0)=25.
From Figure 4,  it is clear that the synchronization is achieved for all these
values of fractional order  when k1=20 and k2=10.Synchronization errors of
the drive system (3.2) and response system (3.3) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


## 3.1.1. Numerical Results

An efficient
method for solving fractional order differential equations is the
predictor-correctors scheme or more precisely, PECE   (Predict, Evaluate, Correct,
Evaluate) technique which has been investigated in [28, 29], and represents a
generalization of the Adams-Bashforth-Moulton algorithm. It is used throughout
this paper.Based on the
above mentioned discretization scheme, the drive and response systems (3.2) and (3.3)
are integrated numerically with the fractional orders α=0.9,0.95,0.99 and using the initial values xm(0)=15,ym(0)=20,zm(0)=29 and xs(0)=10,ys(0)=15,zs(0)=25.
From Figure 4,  it is clear that the synchronization is achieved for all these
values of fractional order  when k1=20 and k2=10.Synchronization errors of
the drive system (3.2) and response system (3.3) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


## 3.2. Synchronization between Lorenz-Like and Chen Fractional Order Systems

In this case it
is assumed that, the fractional order Lorenz-like system drives the fractional
order Chen system. The drive and response systems are defined as follows:(3.12)dαxmdtα=σ(ym−xm),dαymdtα=ρxm−xmzm+γym,dαzmdtα=xmym−βzm,(3.13)dαxsdtα=a(ys−xs)+v1,dαysdtα=(c−a)xs−xszs+cys+v2,dαzsdtα=xsys−bzs+v3,
where v1,v2, and v3 are the linear control functions. The error
variables are given by(3.14)e1=xs−xm,e2=ys−ym,e3=zs−zm.By subtracting (3.12) from (3.13) and
using (3.14), we get(3.15)dαe1dtα=a(e2−e1)+(a−σ)(ym−xm)+v1,dαe2dtα=ce2−zme1−xme3−e1e3+(c−a)xs−ρxm+(c−γ)ym+v2,dαe3dtα=−be3+yme1+xme2+e1e2+(β−b)zm+v3.Now, by choosing(3.16)v1=(σ−a)(ym−xm),v2=ρxm+(a−c)xs+(γ−c)ym−k1e2,v3=(b−β)zm−k2e3,
where k1,k2≥0,
then the error system (3.15) is rewritten as(3.17)dαe1dtα=a(e2−e1),dαe2dtα=(c−k1)e2−zme1−xme3−e1e3,dαe3dtα=−(b+k2)e3+yme1+xme2+e1e2.Take Laplace transform in both sides of (3.17), let Ei(s)=L{ei(t)}, where (i=1,2,3),
and apply L{dαei/dtα}=sαEi(s)−sα−1ei(0).
After that, by doing similar analysis like the previous subsection, we obtain(3.18)limt→∞e1(t)=lims→0+sE1(s)=lims→0+sE2(s)=limt→∞e2(t),limt→∞e2(t)=lims→0+sE2(s)=1c−k1lims→0+sL{xme3}+1c−k1lims→0+sL{zme1}+1c−k1limt→∞e1(t)⋅limt→∞e3(t),limt→∞e3(t)=lims→0+sE3(s)=1b+k2lims→0+sL{yme1}+1b+k2lims→0+sL{xme2}+1b+k2limt→∞e1(t)⋅limt→∞e2(t).If we assume
thatc−k1≠0 and E1(s),E2(s) are bounded, then it follows that limt→∞e1(t)=limt→∞e2(t)=0.
Now, owing to the attractiveness of the attractors of systems (2.1) and (2.3),
there exists ξ>0 such that |xi(t)|≤ξ<∞,|yi(t)|≤ξ<∞, and |zi(t)|≤ξ<∞ where i refers to the index of the drive or response
variables. Therefore, limt→∞e3(t)=0.
Consequently,(3.19)limt→∞ei(t)=0,i=1,2,3.Thus, the states of the drive
system (3.12) are synchronized with the states of the response system (3.13), as
the controllers (3.16) are activated.


### 3.2.1. Numerical Results

Numerical
simulations are carried out to integrate the drive and response systems (3.12)
and (3.13) using the predictor-correctors scheme, with the fractional orders α=0.9,0.95,0.99 and the initial values xm(0)=10,ym(0)=16,zm(0)=25 and xs(0)=15,ys(0)=20,zs(0)=29.
Thus, the drive and response systems (3.12) and (3.13) are synchronized in such a
successful way for all at the above-mentioned fractional orders values, using
the linear controllers (3.16) with k1=20 and k2=10 (see Figure 5).Synchronization errors
of the drive system (3.12) and response system (3.13) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


## 3.2.1. Numerical Results

Numerical
simulations are carried out to integrate the drive and response systems (3.12)
and (3.13) using the predictor-correctors scheme, with the fractional orders α=0.9,0.95,0.99 and the initial values xm(0)=10,ym(0)=16,zm(0)=25 and xs(0)=15,ys(0)=20,zs(0)=29.
Thus, the drive and response systems (3.12) and (3.13) are synchronized in such a
successful way for all at the above-mentioned fractional orders values, using
the linear controllers (3.16) with k1=20 and k2=10 (see Figure 5).Synchronization errors
of the drive system (3.12) and response system (3.13) using k1=20,k2=10 and fractional orders: (a) α=0.9,  (b) α=0.95, and (c) α=0.99.
(a)(b)(c)


## 4. Conclusion

Chaos synchronization
between two different fractional order chaotic systems has been studied using
linear control technique. Fractional order Chen system has been used to drive
fractional order Lü system, and fractional order Lorenz-like system has been
used to drive fractional order Chen system. Conditions for chaos
synchronization have been investigated theoretically by using Laplace transform. Numerical simulations have been carried out using different
fractional order values to show the effectiveness of the proposed
synchronization techniques.



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*Source: 572724-2009-05-06.xml*