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Paper-2

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									                                                                International Journal of Advances in Science and Technology,
                                                                                                           Vol. 2, No.4, 2011

Global Chaos Synchronization of Shimizu-Morioka and
Liu-Chen Chaotic Systems by Active Nonlinear Control
                                               Dr. V. Sundarapandian1
               1
                R & D Centre, Vel Tech Dr. RR & Dr. SR Technical University, Avadi-600 062, Chennai, INDIA
                                                 sundarvtu@gmail.com


                                                       Abstract

        This paper investigates the global chaos synchronization of identical Shimizu-Morioka systems
      (1980), identical Liu-Chen systems (2004) and non-identical Shimizu-Morioka and Liu-Chen
      systems. In this paper, active nonlinear control has been deployed to synchronize the identical and
      non-identical Shimizu-Morioka and Liu-Chen chaotic systems and our stability results have been
      established using Lyapunov stability theory. Since the Lyapunov exponents are not required for
      these calculations, the active nonlinear control method is effective and convenient to synchronize
      Shimizu-Morioka and Liu-Chen chaotic systems. Numerical simulations are shown to demonstrate
      the effectiveness of the synchronization schemes derived in this paper for Shimizu-Morioka and Liu-
      Chen chaotic systems.


      Keywords: chaos synchronization, active control, Shimizu-Morioka system, Liu-Chen system.

      1. Introduction

           Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This
      sensitivity is popularly known as the butterfly effect [1].
          Chaos synchronization problem was first described by Fujisaka and Yemada [2] in 1983. This
      problem did not receive great attention until Pecora and Carroll [3-4] published their results on chaos
      synchronization in early 1990s. From then on, chaos synchronization has been extensively and
      intensively studied in the last three decades [3-22]. Chaos theory has been explored in a variety of
      fields including physical systems [5], chemical systems [6] and ecological systems [7], secure
      communications [8-10] etc.
          Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic
      oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator. Because of the
      butterfly effect which causes the exponential divergence of the trajectories of two identical chaotic
      systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly
      a very challenging problem.
          In most of the chaos synchronization approaches, the master-slave or drive-response formalism is
      used. If a particular chaotic system is called the master or drive system and another chaotic system is
      called the slave or response system, then the idea of the synchronization is to use the output of the
      master system to control the slave system so that the output of the slave system tracks the output of the
      master system asymptotically.
          Since the seminal work by Pecora and Carroll [3-4], a variety of impressive approaches have been
      proposed for the synchronization for the chaotic systems such as PC method [3-4], the sampled-data
      feedback synchronization method [10-11], OGY method [12], time-delay feedback approach [13],
      backstepping design method [14], adaptive design method [15-19], sliding mode control method [20],
      Lyapunov stability theory method [21], hyperchaos [22], etc.
          In this paper, we derive new results for the global chaos synchronization for identical and different
      Shimizu-Morioka and Liu-Chen chaotic systems using active nonlinear control. Explicitly, using active
      control and Lyapunov stability theory, we achieve global chaos synchronization for identical Shimizu-
      Morioka, identical Liu-Chen systems and non-identical Shimizu-Morioka and Liu-Chen systems.




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        This paper has been organized as follows. In Section 2, we give the problem statement and our
     methodology. In Section 3, we discuss the chaos synchronization of two identical Shimizu-Morioka
     chaotic systems ([23], 1980). In Section 4, we discuss the chaos synchronization of two identical Liu-
     Chen chaotic systems ([24], 2004). In Section 5, we discuss the chaos synchronization of Shimizu-
     Morioka and Liu-Chen chaotic systems. In Section 6, we summarize the main results of this paper.

     2. Problem Statement and Our Methodology

        Consider the chaotic system described by the dynamics
               x  Ax  f ( x)                                                          (1)
     where x  R is the state of the system, A is the n  n matrix of the system parameters and
                 n


      f : Rn  Rn is the nonlinear part of the system.
       We consider the system (1) as the master or drive system.
       As the slave or response system, we consider the following chaotic system described by the
     dynamics
              y  By  g ( y)  u                                                         (2)
     where y  R is the state of the system, B is the n  n matrix of the system parameters,
                     n


     g : Rn  Rn is the nonlinear part of the system and u  Rn is the controller of the slave system.
       If A  B and f  g , then x and y are the states of two identical chaotic systems. If A  B and
     f  g , then x and y are the states of two different chaotic systems.
       In the nonlinear feedback control approach, we design a feedback controller u , which synchronizes
     the states of the master system (1) and the slave system (2) for all initial conditions x(0), z (0)  R .
                                                                                                                  n

        If we define the synchronization error as
                 e  y  x,                                                                            (3)
     then the synchronization error dynamics is obtained as
                e  By  Ax  g ( y)  f ( x)  u                                                      (4)
        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)  Rn .                                                           (5)
              t 
        We take as a candidate Lyapunov function
                V (e)  eT Pe,
     where P is a positive definite matrix.
       Note that
                 V : Rn  R
     is a positive definite function by 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
                V (e)  eT Qe,
     where Q is a positive definite matrix, then
                V : Rn  R
     is a negative definite function.
         Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable and
     hence the condition (5) will be satisfied. Hence, the states of the master system (1) and the slave
     system (2) will be globally and exponentially synchronized.




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     3. Synchronization of Identical Shimizu-Morioka Systems

     3.1 Theoretical Results

        In this section, we apply the nonlinear control technique for the synchronization of two identical
     Shimizu-Morioka systems ([23], 1980). The Shimizu-Morioka system is a well-known three-
     dimensional chaotic system.
        Thus, the master system is described by the Shimizu-Morioka dynamics
              x1  x2
              x2  x1   x2  x1 x3                                                                          (6)
              x3   x3  x12
     where x1 , x2 , x3 are the states of the system and     0,   0    are constant parameters of the system.
        The slave system is described by the controlled Shimizu-Morioka dynamics
               y1  y2  u1
               y2  y1   y2  y1 y3  u2                                                                   (7)
               y3   y3  y12  u3
     where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be
     designed.
        The Shimizu-Morioka system (6) is chaotic when
                 0.75 and   0.45.
        Figure 1 depicts the chaotic state portrait of the Shimizu-Morioka system (6).




                                 Figure 1. State Orbits of the Shimizu-Morioka System
        The synchronization error     e is defined by
                ei  yi  xi ,      (i  1, 2,3)                                                                (8)




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        The error dynamics is obtained as
              e1  e2  u1
              e2  e1   e2  y1 y3  x1 x3  u2                                                              (9)
              e3   e3  y12  x12  u3
         We choose the nonlinear controller as
               u1  e1  e2
               u2  e1  y1 y3  x1 x3                                                                       (10)
               u3   y12  x12
        Substituting (10) into (9), we obtain the linear system
                e1  e1 , e2  e2 , e3   e3                                                              (11)
        We consider the quadratic Lyapunov function defined by

              V (e)       e1  e2  e3  ,
                        1 2 2 2
                        2
                                                                                                               (12)

     which is a positive definite function on R .
                                               3

       Differentiating (12) along the trajectories of (11), we get
              V (e)  e12  e2   e3 ,
                               2      2
                                                                                                                 (13)
     which is a negative definite function on R .
                                                   3

        Thus, the error dynamics (11) is globally exponentially stable.
        Hence, we arrive at the following result.
     Theorem 1. The identical Shimizu-Morioka systems (6) and (7) are exponentially and globally
     synchronized for all initial conditions by the active nonlinear controller defined by (10). 

      3.2 Numerical Results
                                                                                             6
        For simulations, the fourth-order Runge-Kutta method with time-step h  10 is used to solve the
     differential equations (6) and (7) with the active nonlinear controller (10).




                      Figure 2. Synchronization of the Identical Shimizu-Morioka Systems




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        The parameters of the Shimizu-Morioka chaotic systems are chosen as   0.75 and               0.45.
        The initial conditions of the master and slave systems are chosen as
              x(0)  (20,36,14) and y(0)  (43,10, 26).
        Figure 2 shows the synchronization of the master system (6) and slave system (7).

     4. Synchronization of Identical Liu-Chen Systems

     4.1 Theoretical Results

        In this section, we apply the nonlinear control technique for the synchronization of two identical
     Liu-Chen systems ([24], 2004). The Liu-Chen system is a well-known three-dimensional chaotic
     system, known as a four-scroll chaotic attractor.
        Thus, the master system is described by the Liu-Chen dynamics
              x1  ax1  x2 x3
              x2  bx2  x1 x3                                                                            (14)
            x3  cx3  x1 x2
     where x1 , x2 , x3 are the states of the system and a, b, c are constant, positive parameters of the system
     such that b  c  a.
        The slave system is described by the controlled Liu-Chen dynamics
               y1  ay1  y2 y3  u1
               y2  by2  y1 y3  u2                                                                      (15)
             y3  cy3  y1 y2  u3
     where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be
     designed.
        The Liu-Chen system (14) is chaotic when
               a  0.4, b  12 and c  5




                            Figure 3. State Orbits of the Liu-Chen Four-Scroll System




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        Figure 3 depicts the state orbits of the Liu-Chen four-scroll system (14).
        The synchronization error e is defined by
               e1  y1  x1
               e2  y2  x2                                                                                     (16)
               e3  y3  x3
        The error dynamics is obtained as
               e1  ae1  y2 y3  x2 x3  u1
               e2  be2  y1 y3  x1 x3  u2                                                                   (17)
               e3  ce3  y1 y2  x1 x2  u3
         We choose the nonlinear controller as
               u1  (a  1)e1  y2 y3  x2 x3
               u2   y1 y3  x1 x3                                                                            (18)
               u3   y1 y2  x1 x2
        Substituting (18) into (17), we obtain the linear system
                e1  e1
                e2  be2                                                                                      (19)
                e3  ce3
        We consider the quadratic Lyapunov function defined by

              V (e)       e1  e2  e3  ,
                        1 2 2 2
                        2
                                                                                                                (20)

     which is a positive definite function on R .
                                                3

       Differentiating (20) along the trajectories of (19), we get
              V (e)  e12  be2  ce3 ,
                               2     2
                                                                                                                 (21)

     which is a negative definite function on R .
                                                    3

        Thus, the error dynamics (19) is globally exponentially stable.
        Hence, we arrive at the following result.
     Theorem 2. The identical Liu-Chen systems (14) and (15) are exponentially and globally synchronized
     for all initial conditions by the active nonlinear controller defined by (18). 

     4.2 Numerical Results
                                                                                              6
        For simulations, the fourth-order Runge-Kutta method with time-step h  10 is used to solve the
     differential equations (14) and (15) with the active nonlinear controller (18).
        The parameters of the Liu-Chen chaotic systems are chosen as
               a  0.4, b  12 and c  5
         The initial conditions of the master system are chosen as
               x(0)  (16, 22,18)
        The initial conditions of the slave system are chosen as
               y(0)  (30,12, 24)
        Figure 4 shows the synchronization of the master system (14) and slave system (15).




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                           Figure 4. Synchronization of the Identical Liu-Chen Systems

     5. Synchronization of Shimizu-Morioka and Liu-Chen Systems

     5.1 Theoretical Results

        In this section, we apply the active control method for the synchronization of non-identical chaotic
     systems, viz. Shimizu-Morioka system ([22], 1980) and Liu-Chen system ([23], 2004). We take the
     Shimizu-Morioka system as the master system and Liu-Chen system as the slave system for the global
     chaos synchronization.
        Thus, the master system is described by the Shimizu-Morioka dynamics
                 x1  x2
                 x2  x1   x2  x1 x3                                                                         (22)
               x3   x3  x12
     where x1 , x2 , x3 are the states of the system and  ,  are constant, positive parameters of the system.
        The slave system is described by the controlled Liu-Chen dynamics
              y1  ay1  y2 y3  u1
              y2  by2  y1 y3  u2                                                                          (23)
             y3  cy3  y1 y2  u3
     where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be
     designed.




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         The synchronization error   e is defined by
               e1  y1  x1
               e2  y2  x2                                                                                        (24)
               e3  y3  x3
        The error dynamics is obtained as
               e1  ae1  ax1  x2  y2 y3  u1
               e2  be2  x1  (  b) x2  y1 y3  x1 x3  u2                                                    (25)
               e3  ce3  (c   ) x3  x  y1 y2  u3
                                               2
                                               1

         We choose the nonlinear controller as
               u1  (a  1)e1  ax1  x2  y2 y3
               u2  x1  (  b) x2  y1 y3  x1 x3                                                               (26)
               u3  x  (c   ) x3  y1 y2
                        2
                        1

        Substituting (26) into (25), we obtain the linear system
                e1  e1
                e2  be2                                                                                         (27)
                e3  ce3
        We consider the quadratic Lyapunov function defined by

              V (e)       e1  e2  e3  ,
                        1 2 2 2
                        2
                                                                                                                   (28)

     which is a positive definite function on R .
                                                   3

       Differentiating (28) along the trajectories of (27), we get
              V (e)  e12  be2  ce3 ,
                               2     2
                                                                                                                    (29)

     which is a negative definite function on R .
                                                       3

        Thus, the error dynamics (27) is globally exponentially stable.
        Hence, we arrive at the following result.
     Theorem 3. The Shimizu-Morioka system (22) and the Liu-Chen system (23) are exponentially and
     globally synchronized for all initial conditions with the nonlinear controller defined by (26). 

     5.2 Numerical Results
                                                                                                 6
        For simulations, the fourth-order Runge-Kutta method with time-step h  10 is used to solve the
     differential equations (22) and (23) with the active nonlinear controller (26).
        The parameters of the Shimizu-Morioka and Liu-Chen chaotic systems are chosen as
                0.75,   0.45, a  0.4, b  12                and c  5
         The initial conditions of the master system are chosen as
               x(0)  (33, 21,16)
        The initial conditions of the slave system are chosen as
               y(0)  (14, 26,11)
        Figure 5 shows the synchronization of the master system (22) and slave system (23).




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                   Figure 5. Synchronization of the Shimizu-Morioka and Liu-Chen Systems

     6. Conclusions

        In this paper, we have used nonlinear control method and Lyapunov stability theory to achieve
     global chaos synchronization for the following three cases of chaotic systems:
          (A) Identical Shimizu-Morioka chaotic systems (1980)
          (B) Identical Liu-Chen chaotic systems (2004)
          (C) Non-Identical Shimizu-Morioka and Liu-Chen systems.
        Since the Lyapunov exponents are not needed for these calculations, the nonlinear control method is
     very effective and convenient to achieve global chaos synchronization for the three cases of chaotic
     systems studied in this paper. Numerical simulations have been shown to illustrate the effectiveness of
     the synchronization schemes derived in this paper.

     7. References

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     [5] M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization,
         World Scientific, Singapore, 1996.




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     Authors Profile


                                Dr. V. Sundarapandian received his D.Sc. degree from the Department of
                                Electrical and Systems Engineering, Washington University, Saint Louis, USA in
                                1996. He is currently working as Professor (Systems & Control Engineering) at the
                                Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
                                University, Chennai. He has published two text-books (Numerical Linear Algebra,
                                Probability, Statistics and Queueing Theory) with Prentice Hall of India, New Delhi.
                                He has published over 100 refereed international journal papers and presented over
                                140 papers in National and International Conferences.




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