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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.




      May Issue                                            Page 8 of 53                                       ISSN 2229 5208
                                                                            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




        May Issue                                              Page 9 of 53                                      ISSN 2229 5208
                                                                            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




       May Issue                                              Page 10 of 53                                         ISSN 2229 5208
                                                                          International Journal of Computer Information Systems,
                                                                                                               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



       May Issue                                             Page 11 of 53                                     ISSN 2229 5208
                                                                                          International Journal of Computer Information Systems,
                                                                                                                               Vol. 2, No.5, 2011
                                                                                 [8]    K. Murali and M. Lakshmanan, “Secure communication using a
                                                                                        compound signal using sampled-data feedback,” Applied Mathematics
    The initial conditions of the master and slave systems are                          and Mechanics, vol. 11, pp. 1309-1315, 2003.
chosen as x(0)  (25,10,8) and y(0)  (7, 4,30).                                 [9]    L. Huang, R. Feng and M. Wang, “Synchronization of chaotic systems
  Figure 5 shows the synchronization of the master system                               via nonlinear control,” Physics Letters A, vol. 320, pp. 271-275, 2004.
(24) and slave system (25).                                                      [10]   H.K. Chen, “Global chaos synchronization of new chaotic systems via
                                                                                        nonlinear contorl,” Chaos, Solitons and Fractals, vol. 23, pp. 1245-1251,
                                                                                        2005.
                                                                                 [11]   V. Sundarapandian, “Global chaos synchronization of Lorenz and
                                                                                        Pehlivan chaotic systems by nonlinear control,” International Journal of
                                                                                        Advances in Science and Technology, vol. 2, no. 3, pp. 19-28, 2011.
                                                                                 [12]   E. Ott, C. Grebogi and J.A. Yorke, “Controlling chaos,” Physical
                                                                                        Review Letters, vol. 64, pp. 1196-1199, 1990.
                                                                                 [13]   J. Zhao and J. Lu, “Using sampled-data feedback control and linear
                                                                                        feedback synchronization in a new hyperchaotic system,” Chaos,
                                                                                        Solitons and Fractals, vol. 35, no. 2, pp. 376-382, 2008.
                                                                                 [14]   N. Li, Y. Zhang, J. Hu and Z. Nie, “Synchronization for general
                                                                                        complex dynamical networks with sasmpled-data”, Neurocomputing,
                                                                                        vol. 74, no. 5, pp. 805-811, 2011.
                                                                                 [15]   J.H. Park and O.M. Kwon, “A novel criterion for delayed feedback
                                                                                        control of time-delay chaotic systems,” Chaos, Solitons and Fractals,
                                                                                        vol. 17, pp. 709-716, 2003.
                                                                                 [16]   T.L. Liao and S.H. Tsai, “Adaptive synchronization of chaotic systems
                                                                                        and its applications to secure communications”, Chaos, Solitons and
                                                                                        Fractals, vol. 11, pp. 1387-1396, 2000.
                                                                                 [17]   Y.G. Yu and S.C. Zhang, “Adaptive backstepping synchronization of
                                                                                        uncertain chaotic systems,” Chaos, Solitons and Fractals, vol. 27, pp.
  Figure 5. Synchronization of the Liu and Harb Systems                                 1369-1375, 2006.
                                                                                 [18]   J.H. Park, S.M. Lee and O.M. Kwon, “Adaptive synchronization of
                           VI.    CONCLUSIONS                                           Genesio-Tesi chaotic system via a novel feedback control,” Physics
                                                                                        Letters A, vol. 371, pp. 263-270, 2007.
    In this paper, we have used active control method and
                                                                                 [19]   X. Wu and J. Lü, “Parameter identification and backstepping control of
Lyapunov stability theory to achieve global chaos                                       uncertain Lü sysetm,” Chaos, Solitons and Fractals, vol. 18, pp. 721-
synchronization for the following three cases of chaotic                                729, 2003.
systems:                                                                         [20]   H.T. Yau, “Design of adaptive sliding mode controller for chaos
     (A) Identical Liu chaotic systems (2004)                                           synchronization with uncertaintities,” Chaos, Solitons and Fractals, vol.
     (B) Identical Harb chaotic systems (2002)                                          22, pp. 341-347, 2004.
     (C) Non-Identical Liu and Harb chaotic systems.                             [21]   C. Liu, T. Liu, L. Liu and K. Liu, “A new chaotic attractor,” Choas,
                                                                                        Solitons and Fractals, vol. 22, pp. 1031-1038, 2004.
   Since the Lyapunov exponents are not needed for these
                                                                                 [22]    A.M. Harb and M.M. Zohdy, “Chaos and bifurcation control using
calculations, the nonlinear control method is very effective                            nonlinear recursive controller,” Nonlinear Analysis: Modelling and
and convenient to achieve global chaos synchronization for the                          Control, vol. 7, pp. 37-43, 2002.
three cases of chaotic systems studied in this paper. Numerical                  [23]   W. Hahn, The Stability of Motion, Springer, Berlin, 1967.
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.




        May Issue                                                       Page 12 of 53                                              ISSN 2229 5208

				
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