International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1001

ISSN 2229-5518

FUNCTION PROJECTIVE SYNCHRONIZATION OF A NEW HYPER CHAOTIC SYSTEM

Ayub Khan1 and Priyamvada Tripathi2

Abstract- In this article a function projective synchronization (FPS)

of two identical new hyper chaotic systems is defined and scheme of

FPS is developed by using Open-Plus-Closed-Looping (OPCL) coupling method. A new hyper chaotic system has been constructed and then response system with parameters perturbation and without perturbation.

Numerical simulations verify the effectiveness of this scheme, which has been performed by mathematica.

Index Term: Function Projective Synchronization, Chaotic systems and Hyper

Chaos, OPCL.

——————————  ——————————

1. Introduction
Chaos is a dynamical regime in which a system becomes extremely sensitive to initial
conditions and reveals an unpredictable and random-like behavior, even though the underlying model of a system exhibiting chaos can be deterministic and very simple. Small differences in initial conditions yield widely diverging outcomes for chaotic systems, rendering long term prediction impossible in general.

1. Professor, Deptt of Mathematics, Jamia Millia Islamia, Delhi-25.

E-mail: ayubkdu@gmail.com

2. Research Scholar, University of

Delhi, Deptt of Mathematics, Delhi-

7, E-mail: dupriyam@gmail.com
Chaotic behavior can be observed in many natural phenomenon such as weather etc. Pecora and Carroll introduced a

paper entitled Synchronization in Chaotic Systems in 1990. By that time, if there was a system challenging the capability of synchronizing that was a chaotic one. They demonstrated that chaotic synchronization could be achieved by driving or replacing one of the variables of a chaotic system with a variable of another

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similar chaotic device. Chaotic synchronization did not attract much attention until Pecora and Carroll [4] introduced a method to synchronize two identical chaotic systems with different initial conditions. From then on, enormous studies have been done by researchers on the synchronization of dynamical systems[1, 2, 3]. In the last two decades considerable research has been done in non-linear dynamical systems and their various properties. One of the
Also, several types of chaos synchronization are well known, which include complete synchronization (CS), antisynchronization (AS), phase synchronization, generalized synchronization (GS), projective synchronization(PS), and modified projective synchronization (MPS). Among all type of synchronizations, projective synchronization (PS) [17, 20, 21, 22] has been extensively considered because it can obtain faster communication.

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most important properties is
synchronization.
Synchronization techniques have been improved in recent years
and many different methods are applied theoretically as well as
experimentally to synchronize the chaotic-systems including
adaptive control [5, 6, 7],
backstepping design [8, 9, 10], active control [11, 12, 13], nonlinear control [14, 15] and observer based control method [16]. Using these methods, numerous synchronization problem of well-known chaotic systems such as Lorenz, Chen, L¨u and R¨ossler system have been worked on by many researchers.
The drive and response system
could be synchronized up to a scaling factor in projective synchronization. In this continuation of study, in order to increase the degree of secrecy for secure communications, function projective synchronization (FPS) [23] is characterized by a scaling function matrix. In this paper, we have constructed a new hyper chaotic system and verified the chaotic behavior of this system by time series analysis and chaotic attractors via mathematica. Hyperchaotic behavior of this system is discovered within some system parameters range, which has not yet been reported previously.

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Since hyperchaotic systems have the characteristics of high
where x = (x1 (t), x2(t),… xm (t))T , y
= (y1(t), y2(t), … ym (t))T ,
capacity, high security and high
efficiency, it has been studied with increasing interest in recent years [19, 20] in the fields of non-linear circuits, secure communications, lasers, control,

U = (u (x, y), u (x, y),… u (x, y))T

is a controller to be determined later.

Denote ei = xi−fi(x)yi ; (i = 1,

2,…m), fi(x); (i = 1; 2;…;m) are functions of x. If
synchronization, and so on. So,
we have studied Function

lim

→∞

e(t)

= 0 ,

Projective Synchronization behavior for this new hyper chaotic systems, which is ofcourse more effective and useful in secure communication
as FPS is more useful in secure

e = (e1; e2; :::; em), then there

exists function projective synchronization (FPS) between these two identical chaotic (hyperchaotic) systems, and we
call f a scaling function matrix.

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communication as compare to
others because of its unpredictability . Here we have used OPCL coupling scheme for
FPS. Numerical simulations have
been done by using Mathematica.

2. Preliminaries

In this section we mention some
definitions and scheme of the
Here we use the OPCL coupling
method for FPS.
2.2. Methodology for FPS via OPCL. Here, we will construct corresponding response system through the OPCL coupling method. Consider the following n-dimensional chaotic system as drive (master) system
main task.

2.1. Function Projective

Synchronization. Function

dx =

where

f (x) + ∆f (x)

x∈ℜn

(1)

and
Projective synchronization

∆f (x)

is the perturbation part of
is defined in the following manner:
Let x˙ = F(x, t) be the drive
chaotic system, and y˙ = F(y, t)+U is the response system,
the parameters. Now, consider
the following n-dimensional chaotic system as responsive system according to coupling method

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dy =

where

f ( y) + D( y, g) , (2)

y∈ℜn . The coupling

equations exhibiting chaotic behavior for certain values of parameters.
function is:
They are named after Mikhail

D( y, g) = .

.( y − g),

f (g) + (H − ∂f (g ))

∂g

Rabinovich and Anatoly
Fabrikant, who described them in
1979 [18]. The equations of system are :
where

∂f (g )

∂g

is the jacobian

 

 x = x (x

−1+ x 2) + γ x ,

matrix of the dynamical system.

 1 2 3 1 1



 

H is an n×n Hurwitz constant



x = x

(3x

+1− x 2) + γ x ,

matrix, whose eigen values are

 2 1 3 1 2

negative and

g = β (t)x

with 

β (t)

as a scaling function which

 x3 = −2

3( 1 2 +α ).

x x x



is continuoIusly JdiffereSntiable. ER

When β (t) = ±1,

system is
where α

and

γ are constant

complete synchronized or antisynchronized accordingly.
parameters that control the
evolution of the system. For
Our goal in this paper is to find
some values of α

and

γ the

out D(y, g) and hence find error dynamics of the system such that
system is chaotic but for other
it tends to a stable periodic orbit.

lim

→∞

e(t) =

y − g = 0

Now, we construct a new hyper
chaotic system by introducing
where . is the Euclidean norm,
then the systems (1) and (2) are said to be Function Projective synchronized.

3. System Description

3.1. Hyper Chaotic

Rabinovich-Fabrikant system.

The
Rabinovich-Fabrikant chaotic system is a set of three coupled ordinary differential
one more differential equation with a new parameter δ in the above system as follows:

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



 

 x =

2( 3 −1+ x 2) + γ x ,

x x

 1 1

 

 x = x

(3x

+1− x 2) + γ x ,

 2 1 3 1 2



 

 x3 = −2

3( 1 2 +α ),

Fig.2 Time series analysis of

x x x



 

y1[t] with α = 0.14, γ = 1:1

 x = −3

3(x x

+δ ) + x2.

and −0.01 ≤ δ≤ 7650.

 4

(4)

x 2 4 4

This new system shows hyper chaotic behavior with some values of parameters and tend to stable periodic orbits with other values of parameters. We have
investigated system’s behavior
for different values of parameters
Fig.1
Chaotic behavior of the system with α = 0.14,γ = 1:1 and −0.01 ≤ δ≤ 7650 tending to
stable periodic orbits.
Fig.3 Time series analysis of y2[t] with α = 0.14,γ = 1:1 and −0.01 ≤ δ≤ 7650.

Fig.4 Time series analysis of y3[t] with α = 0.14,γ = 1:1 and −0.01 ≤ δ≤ 7650

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and δ=1890.
Fig.5 Time series analysis of y4[t] with α = 0.14,γ = 1:1 and −0.01 ≤ δ≤ 7650.

Fig.8 Time series analysis of y2[t] with α = 0:87, γ = 1:1 and δ=1890.

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Fig.9 Time series analysis of
y3[t] with α = 0:87, γ = 1:1
Fig.6 Chaotic Behavior of the

system with α = 0:87, γ = 1:1 and δ=1890.
Fig.7 Time series analysis of y1 [t] with α = 0:87, γ = 1:1

and δ=1890.
Fig.10 Time series analysis of y4[t] with α = 0:87, γ = 1:1 and δ=1890.

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Fig.14 Time series analysis of y3[t] with α = 0:87, γ = 1:1 and δ=-0.2.
Fig.11 Chaotic Behavior of the system with α = 0:87, γ = 1:1 and δ=-0.2.

Fig.15 Time series analysis of y4[t] with α = 0:87, γ = 1:1 and δ=-0.2.

Fig.12 Time series analysis of y1[t] with α = 0:87, γ = 1:1 and δ=-0.2.
Fig.13 Time series analysis of y2[t] with α = 0:87, γ = 1:1 and δ=-0.2.

3.2. Results and Discussions. In this section, we perform function projective synchronization of above described system via OPCL coupling method. Define following system as a drive system with parameters perturbation as

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

x = 2(

3 −1+ x 2) + (γ + ∆γ )x ,

 −γ − ∆γ + 2β 2 x x −1

−β 2 x2 − β x + 1

1 x x

1 1 

 − β

1 2 1 3

+ β 2 2 − −γ − ∆γ −

 3 x3 3

x1 1 1

x = x

(3x

+1− x 2) + (γ + ∆γ )x , = 

2β 2 x x

2 1 3 1 2  2 3 1 3





 −β x

0 3β 2 x x

3 4



x x x 2 0

x3 = −2

3( 1 2 +α + ∆α ),



−3β x 0



x = −3

3(x x

+δ + ∆δ ) + x2.

2α + 2∆α + 2β 2 x x

−1 0 

4 x 2 4 4

3β 2 x x

+ 3δ + 3∆δ

−2β x

+ 3β 2 x x

−1

where

∆α , ∆γ and ∆δ

are the

2 4 4 2 3 

perturbation parts in the parameters. Now construct the corresponding response system via OPCL coupling method.
Therefore, response system after coupling is as follows

 •

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The Jacobian matrix of the above

y f ( y )

f (g ) g

(H ∂f ( g1) )( y g ),

system is

∂f ( x) =

1 = 1 −

1 + 1+ −

∂g1

1 − 1

∂x 

= f ( y

) − f (g

•

) + g

+(H −

∂f ( g 2 )

)( y

− g ),

 γ + ∆γ + 2 x x x2 +

3 −1

2 2 2 2 2

 1 2 1 x



∂g 2

 3 3 − 3x2 + 1

γ + ∆γ

x 1



 − x x

− x x



y3 = f ( y

) − f (g

•

) + g 3 +(H −

∂f (g3 )

)( y3 − g3 ),

2 2 3

2 1 3 3 3

 − x x

∂g3

 0 3 3 4

x  

• ∂f ( g )

2 0 

y = f ( y

) − f (g

) + g

+(H − 4 )( y

− g ).

−2α − 2∆α − 2x1 x2

0  4

0  (6)

4 4 4 ∂g 4 4

− x x

− δ − ∆δ

x − x x 

As error dynamics is defined as

3 2 4

3 3 2 4

3 2 3 

Define Hurwitz matrix H as the unit negative matrix −I (as

g = β (t)x ),

e˙ = y˙ −g˙, so we have final

equation of error dynamics after coupling and putting values of

∂f (g )

then

H − ∂f (g ) =

∂g

f(y), f(g) and

H − ∂g

in above equation as follows

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

e1 = ∆γ 1 +

2 3 +

2 2 + 2β

11 2

synchronization between master and slave system.

e e e e

+β x e2 − e ,

11 1



e x e e

e2 = ∆γ

2 + 3 1 3

e3 + 3β 1 2 − 2,

e ee + 1



x e e

= − ∆α e −

ee e +

β x ee

e3 2

3 2 1 2 3 2

3 1 2

−2β



− 2β

2 1 3

1 2 3 − e ,

(7)

= − ∆δ e −

e e e −

β x e e

e4 3

3 2 4 2 3 3

3 4 2

Fig.16 Convergence of error

−3β

2 4 3 − 3β

− e .

4 2

e1, t∈[0,10]

So, from the above error

dynamics we can conclude that FPS between two identical hyper chaotic system can be achieved.

4. Numerical Simulations

If Perturbation of Parameters of the response system of hyper
chaotic Rabinovich-Fabrikant system are zero and β = 0.5 with
the initial conditions of drive
system [x1(0), x2(0), x3(0), x4(0)]
= [0, 2, 0.5,−0.2] and response systems [y1(0), y2(0), y3(0). y4(0)]
= [0.5, 1,−0.1,−0.15]
respectively.
So, the initial conditions for
[e1(0), e2(0), e3(0), e4(0)] = [0.5,
0,−0.35,−0.05]
diagrams of convergence of errors given below are the witness of achieving
function projective
Fig.17 Convergence error of

e2, t∈[0, 10]

Fig.18 Convergence of error

e3, t∈[0, 10]

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physiology. Nature,
410:277,2001.
[3] Strogatz S. Sync. Hyperion,
2003.
Fig.19 Convergence of error

e4, t∈[0, 10]

5.Conclusion:

In this paper, we have investigated function projective
synchronization behavior of a new hyper chaotic Rabinovich-
Fabrikant system . The results
[4] L. M. Pecora and T. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett.
64(1990): 821–824.
[5] T. L. Liao and S. H Lin. Adaptive control and synchronization of Lorenz systems, J. Franklin
Inst. 336(1999): 925-937.
are validatIed Jby nuSmerical ER
simulations using mathematica. It
has more advantage over other synchronization to enhance security of communicationas
function projective
synchronization is more unpredictable and moreover it is performed for hyperchaotic system, which makes it more useful.

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