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International Journal of Computer Information Systems,
Vol. 2, No.5, 2011
Global Chaos Synchronization of Liu and Harb
Chaotic Systems by Active Nonlinear Control
Dr. V. Sundarapandian
Research and Development Centre
Vel Tech Dr. RR & Dr. SR Technical University
Chennai-600 062, INDIA
sundarvtu@gmail.com
Abstract— This paper investigates the global chaos In the chaos theory literature, variety of impressive
synchronization of identical Liu systems (2004), identical Harb approaches have been proposed for the synchronization for the
systems (2002) and non-identical Liu and Harb systems. chaotic systems such as PC method [1-2], active control
Nonlinear feedback control is the method adopted to achieve the method [9-11], OGY method [12], sampled-data feedback
complete synchronization of the Liu and Harb systems. Our
active synchronization results derived in this paper are
method [13-14], time-delay feedback method [15], adaptive
established using Lyapunov stability theory. Since the Lyapunov control method [16-18], backstepping method [19], sliding
exponents are not required for these calculations, the nonlinear mode control method [20], etc.
control method is effective and convenient to synchronize the In this paper, we derive new results for the global chaos
identical and different Liu and Harb systems. Numerical synchronization for identical and different Liu and Harb
simulations are shown to validate the synchronization results chaotic systems using active nonlinear control. Explicitly,
derived in this paper. using active nonlinear control and Lyapunov stability theory,
Keywords-chaos synchronization; nonlinear control, Liu system,
we achieve global chaos synchronization for identical Liu
Harb system. systems, identical Harb systems and non-identical Liu and
Harb systems.
I. INTRODUCTION This paper has been organized as follows. In Section II, we
give the problem statement and our methodology. In Section
Chaotic systems are dynamical systems that are highly III, we discuss the chaos synchronization of two identical Liu
sensitive to initial conditions. This sensitivity is popularly systems ([21], 2004). In Section IV, we discuss the chaos
known as the butterfly effect [1]. synchronization of two identical Harb systems ([22], 2002). In
Synchronization of chaotic systems is a phenomenon that Section V, we discuss the chaos synchronization of Liu and
may occur when two or more chaotic oscillators are coupled or Harb systems. In Section VI, we summarize the main results
when a chaotic oscillator drives another chaotic oscillator. of this paper.
Because of the butterfly effect which causes the exponential
divergence of the trajectories of two identical chaotic systems II. PROBLEM STATEMENT AND OUR METHODOLOGY
started with nearly the same initial conditions, synchronizing Consider the chaotic system described by the dynamics
two chaotic systems is seemingly a very challenging problem.
x Ax f ( x) (1)
Since the seminal papers by Pecora and Carroll [2-3] in
where x R is the state of the system, A is the n n matrix
n
early 1990s, the chaos synchronization problem has been
extensively and intensively studied in the last three decades of the system parameters and f : R R is the nonlinear
n n
[1-20]. Chaos theory has been explored in a variety of fields
part of the system. We consider the system (1) as the master or
including physical systems [4], chemical systems [5],
drive system.
ecological systems [6], secure communications [7-8] etc.
As the slave or response system, we consider the following
In most of the chaos synchronization approaches, the chaotic system described by the dynamics
master-slave or drive-response formalism is used. If a
particular chaotic system is called the master or drive system
y By g ( y) u (2)
where y R is the state of the system, B is the n n matrix
n
and another chaotic system is called the slave or response
system, then the idea of the synchronization is to use the
of the system parameters, g : R R is the nonlinear part
n n
output of the master system to control the slave system so that
the output of the slave system tracks the output of the master of the system and u R n is the controller of the slave system.
system asymptotically.
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International Journal of Computer Information Systems,
Vol. 2, No.5, 2011
If A B and f g , then x and y are the states of two The slave system is described by the controlled Liu
dynamics
identical chaotic systems. If A B or f g , then x and
y1 a( y2 y1 ) u1
y are the states of two different chaotic systems.
In the nonlinear feedback control approach, we design a y2 by1 y1 y3 u2 (9)
feedback controller u , which synchronizes the states of the
y3 cy3 dy u3 2
1
master system (1) and the slave system (2) for all initial
where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are
conditions x(0), y(0) R .
n
If we define the synchronization error as the active nonlinear controllers to be designed.
e y x, (3) The Liu system (8) is chaotic when
then the synchronization error dynamics is obtained as
a 10, b 40, c 2.5 and d 4.
e By Ax g ( y) f ( x) u (4) Figure 1 depicts the state orbits of the Liu system (8).
Thus, the global synchronization problem is essentially to
find a feedback controller u so as to stabilize the error
dynamics (4) for all initial conditions e(0) R .
n
Hence, we find a feedback controller u so that
lim e(t ) 0 for all e(0) R n . (5)
t
We take as a candidate Lyapunov function
V (e) eT Pe, (6)
where P is a positive definite matrix.
Note that V : R R is a positive definite function by
n
construction.
We assume that the parameters of the master and slave
system are known and that the states of both systems (1) and
(2) are measurable.
If we find a feedback controller u so that
Figure 1. Chaotic State Portrait of the Liu System
V (e) eT Qe, (7) The synchronization error e is defined by
where Q is a positive definite matrix, then V : R R is a
n
ei yi xi , (i 1, 2,3) (10)
negative definite function.
Thus, by Lyapunov stability theory [23], the error The error dynamics is obtained as
dynamics (4) is globally exponentially stable and hence the e1 a(e2 e1 ) u1
condition (5) will be satisfied. Hence, the states of the master
system (1) and the slave system (2) will be globally and e2 be1 y1 y3 x1 x3 u2 (11)
exponentially synchronized. e3 ce3 d ( y12 x12 ) u3
III. SYNCHRONIZATION OF IDENTICAL LIU SYSTEMS We choose the nonlinear controller as
In this section, we apply the nonlinear control technique for u1 ae2
the synchronization of two identical Liu systems ([21], 2004).
u2 be1 e2 y1 y3 x1 x3 (12)
The Liu system is one of the paradigms of the three-
dimensional chaotic systems proposed by the scientists, C. Liu,
T. Liu, L. Liu and K. Liu in 2004.
u3 d y12 x12
Thus, the master system is described by the Liu dynamics Substituting (12) into (11), we obtain the linear system
x1 a ( x2 x1 ) e1 ae1
x2 bx1 x1 x3 (8) e2 e2 (13)
x3 cx3 dx12 e3 ce3
We consider the quadratic Lyapunov function defined by
where x1 , x2 , x3 are the states of the system and a 0,
V (e) eT e e12 e2 e3 ,
1 1 2 2
b 0, c 0 are parameters of the system. (14)
2 2
which is a positive definite function on R .
3
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International Journal of Computer Information Systems,
Vol. 2, No.5, 2011
Differentiating (14) along the trajectories of (13), we get The slave system is also described by the Harb dynamics
V (e) ae12 e2 ce3 ,
2 2
(15) y1 y3 u1
which is a negative definite function on R .
3 y2 y1 y2 u2 (17)
Thus, the error dynamics (13) is globally exponentially
stable and hence we arrive at the following result.
y3 y1 y y3 u3
2
2
Theorem 1. The identical Liu systems (8) and (9) are where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are
exponentially and globally synchronized for all initial the active nonlinear controllers to be designed.
conditions by the active nonlinear controller (12). The Harb system (16) is chaotic when
Numerical Results: 3.1 and 0.5
For simulations, the fourth-order Runge-Kutta method with Figure 3 depicts the state orbits of the Harb system (16).
6
time-step h 10 is used to solve the differential equations
(8) and (9) with the active nonlinear controller (12).
The parameters of the Liuchaotic systems are chosen as
a 10, b 40, c 2.5 and d 4.
The initial conditions of the master and slave systems are
chosen as x(0) (10, 28,32) and y(0) (43,14, 25).
Figure 2 shows the synchronization of the master system (8)
and slave system (9).
Figure 3. State Orbits of the Harb System
The synchronization error e is defined by
ei yi xi , (i 1, 2,3) (18)
The error dynamics is obtained as
e1 e3 u1
e2 e1 e2 u2 (19)
Figure 2. Synchronization of the Identical Liu Systems e3 e1 e3 y2 x2 u3
2 2
IV. SYNCHRONIZATION OF IDENTICAL HARB SYSTEMS We choose the nonlinear controller as
In this section, we apply the nonlinear control technique for u1 e1 e3
the synchronization of two identical Harb systems ([22], 2002). u2 e1 (20)
The Harb system is one of the paradigms of the three-
dimensional chaotic systems proposed by the scientists, A.M. u3 e1 ( 1)e3 y x 2
2
2
2
Harb and M.M. Zohdy (2002). Substituting (20) into (19), we obtain the linear system
Thus, the master system is described by the Harb dynamics
e1 e1
x1 x3
e2 e2 (21)
x2 x1 x2 (16)
e3 e3
x3 x1 x2 x3
2
We consider the candidate Lyapunov function defined by
where x1 , x2 , x3 are the states and 0, 0
e1 e2 e3 ,
are 1 2 2 2
parameters of the system.
V (e) (22)
2
which is a positive definite function on R .
3
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Vol. 2, No.5, 2011
Differentiating (22) along the trajectories of (21), we get The slave system is described by the controlled Harb
V (e) e12 e2 e3 ,
2 2
(23) dynamics
y1 y3 u1
which is a negative definite function on R .
3
Thus, the error dynamics (19) is globally exponentially y2 y1 y2 u2 (25)
stable and hence we arrive at the following result.
Theorem 2. The identical Harb systems (16) and (17) are
y3 y1 y2 y3 u3
2
exponentially and globally synchronized for all initial where y1 , y2 , y3 are the states, 0, 0 are
conditions by the active nonlinear controller (20).
parameters of the system and u1 , u2 , u3 are the active
Numerical Results: nonlinear controllers to be designed.
For simulations, the fourth-order Runge-Kutta method with
6
The synchronization error e is defined by
time-step h 10 is used to solve the differential equations
(16) and (17) by the active nonlinear controller (18).
ei yi xi , (i 1, 2,3) (26)
The parameters of the Harb chaotic systems are chosen as The error dynamics is obtained as
3.1 and 0.5. e1 a(e2 e1 ) a( y2 y1 ) y3 u1
The initial conditions of the master and slave systems are
chosen as x(0) (3,1, 2) and y(0) (4,6,12). e2 e1 e2 (b 1) x1 x2 x1 x3 u2
(27)
Figure 4 shows the synchronization of the master system e3 ce3 y1 ( c) y3
(16) and slave system (17).
y2 dx12 u3
2
We choose the nonlinear controller as
u1 ae2 a( y2 y1 ) y3
u2 e1 (b 1) x1 x2 x1 x3 (28)
u3 y1 ( c) y3 y2 dx12
2
Substituting (28) into (27), we obtain the linear system
e1 ae1
e2 e2 (29)
e3 ce3
We consider the quadratic Lyapunov function defined by
V (e) eT e e12 e2 e3 ,
1 1 2 2
(30)
2 2
which is a positive definite function on R .
3
Differentiating (30) along the trajectories of (29), we get
V (e) ae12 e2 ce3 ,
2 2
(31)
which is a negative definite function on R .
3
Figure 4. Synchronization of the Identical Harb Systems Thus, the error dynamics (29) is globally exponentially
stable and hence we arrive at the following result.
V. SYNCHRONIZATION OF LIU AND HARB SYSTEMS
Theorem 3. The non-identical Liu system (24) and Harb
In this section, we apply the nonlinear control technique for system (25) are exponentially and globally synchronized for
the synchronization of Liu system ([21], 2004) and Harb all initial conditions by the active nonlinear controller (28).
system ([22], 2002).
Thus, the master system is described by the Liu dynamics Numerical Results:
For simulations, the fourth-order Runge-Kutta method with
x1 a ( x2 x1 ) 6
time-step h 10 is used to solve the differential equations
x2 bx1 x1 x3 (24) (24) and (25) with the active nonlinear controller (28).
The parameters of the Liu chaotic system are chosen as
x3 cx3 dx12
a 10, b 40, c 2.5 and d 4.
where x1 , x2 , x3 are the states and a 0, b 0, c 0 are The parameters of the Harb chaotic system are chosen as
parameters of the system (24). 3.1 and 0.5
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International Journal of Computer Information Systems,
Vol. 2, No.5, 2011
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simulations have been shown to illustrate the effectiveness of
the synchronization schemes derived in this paper for identical
and non-identical Liu and Harb chaotic systems. AUTHORS PROFILE
REFERENCES Dr. V. Sundarapandian is currently Professor
[1] K.T. Alligood, T. Sauer and J.A. Yorka, Chaos: An Introduction to (Systems and Control Engineering), Research
Dynamical Systems, Springer, New York, 1997. and Development Centre at Vel Tech Dr. RR &
Dr. SR Technical University, Chennai, Tamil
[2] L.M. Pecora and T.L. Carroll, “Synchronization in chaotic systems,” Nadu, India. He has published two books titled
Physical Review Letters, vol. 64, pp. 821-824, 1990. Numerical Linear Algebra and Probability,
[3] L.M. Pecora and T.L. Carroll, “Synchronizing in chaotic circuits,” IEEE Statistics and Queueing Theory (New Delhi:
Transactions on Circuits and Systems, vol. 38, pp. 453-456, 1991. Prentice Hall of India). He has published 90
[4] M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: papers in National Conferences and 45 papers
Controlling and Synchronization, World Scientific, Singapore, 1996. in International Conferences. He has published
[5] S.K. Han, C. Kerrer and Y. Kuramoto, “D-phasing and bursting in over 110 refereed journal publications. His
coupled neural oscillators,” Physical Review Letters, vol. 75, pp. 3190- research interests are Linear and Nonlinear
3193, 1995. Control Systems, Chaos and Stability. He has
delivered many Key Note Addresses on
[6] B. Blasius, A. Huppert and L. Stone, “Complex dynamics and phase
Modern Control Theory, Linear and Nonlinear Control Systems,
synchronization in spatially extended ecological system,” Nature, Vol.
Mathematical Modelling, Scientific Computing with SCILAB, etc.
399, pp. 354-359, 1999.
[7] L. Kocarev and U. Partliz, “General approach for chaotic
synchronization with applications to communications,” Physical Review
Letters, vol. 74, pp. 5028-5030, 1995.
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