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International Journal of Differential Equations
Volume 2012 (2012), Article ID 623234, 16 pages
Chaos Control and Synchronization in Fractional-Order Lorenz-Like System
Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India
Received 6 May 2012; Accepted 22 June 2012
Academic Editor: Wen Chen
Copyright © 2012 Sachin Bhalekar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006). The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback.
A variety of problems in engineering and natural sciences are modeled using chaotic dynamical systems.A chaotic system is a nonlinear deterministic system possessing complex dynamical behaviors such as being extremely sensitive to tiny variations of initial conditions, unpredictability, and having bounded trajectories in the phase space . Controlling the chaotic behavior in the dynamical systems using some form of control mechanism has recently been the focus of much attention. So many approaches are proposed for chaos control namely, OGY method , backstepping design method , differential geometric method , inverse optimal control , sampled-data feedback control , adaptive control , and so on. One simple approach is the linear feedback control . Linear feedback controllers are easy to implement, they can perform the job automatically, and stabilize the overall control system efficiently .
The controllers can also be used to synchronize two identical or distinct chaotic systems [10–13]. Synchronization of chaos refers to a process wherein two chaotic systems adjust a given property of their motion to a common behavior due to a coupling. Synchronization has many applications in secure communications of analog and digital signals  and for developing safe and reliable cryptographic systems .
Fractional calculus deals with derivatives and integration of arbitrary order [16–18] and has deep and natural connections with many fields of applied mathematics, engineering, and physics. Fractional calculus has wide range of applications in control theory , viscoelasticity , diffusion [21–25], turbulence, electromagnetism, signal processing [26, 27], and bioengineering . Study of chaos in fractional order dynamical systems and related phenomena is receiving growing attention [29, 30]. I. Grigorenko and E. Grigorenko investigated fractional ordered Lorenz system and observed that below a threshold order the chaos disappears . Further, many systems such as Li and Peng , Lu , Li and Chen , Daftardar-Gejji and Bhalekar , and unified system  were investigated in this regard. Effect of delay on chaotic solutions in fractional order dynamical system is investigated by the present author . It is demonstrated that the chaotic systems can be transformed into limit cycles or stable orbits with appropriate choice of delay parameter. Synchronization of fractional order chaotic systems was also studied by many researchers [38–41].
In this paper, we propose fractional version of the Lorenz-like chaotic dynamical system . We investigate minimum effective dimension of the system for chaos to exist. Then we control the chaos using simple linear feedback control. Further, we synchronize the proposed fractional order system using feedback control.
2.1. Fractional Calculus
The Riemann-Liouville integral of order is given by An alternative definition was introduced by Caputo. Caputo's derivative is defined as where . The main advantage of the Caputo's formulation is that the Caputo derivative of a constant is equal to zero, that is not the case for the Riemann-Liouville derivative. Note that for ,
2.2. Numerical Method for Solving Fractional Differential Equations
Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). A modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm et al. in [43–45] to solve FDEs.
Consider for the initial value problem (IVP) The IVP (2.4) is equivalent to the Volterra integral equation Consider the uniform grid for some integer and . Let be approximation to . Assume that we have already calculated approximations , and we want to obtain by means of the equation where The preliminary approximation is called predictor and is given by where Error in this method is where .
2.3. Asymptotic Stability of the Fractional-Ordered System
Consider the following fractional-ordered dynamical system: Let be an equilibrium point of the system (2.11) that is, and a small disturbance from a fixed point. Then System (2.12) can be written as where and Consider the linear autonomous system where is matrix and is asymptotically stable if and only if for all eigenvalues of . In this case, each component of solution decays towards like [29, 46].
This shows that if for all eigenvalues of then the solution of the system (2.13) tends to as . Thus, the equilibrium point of the system is asymptotically stable if , for all eigenvalues of , that is, if
3. Fractional Lorenz-Like System
In , Chongxin et al. proposed novel Lorenz-like chaotic system where and initial conditions ((2.2),(2.3), and (28)). In this paper, we study the corresponding fractional order system where . The equilibrium points of the system (3.1) and the eigenvalues of corresponding Jacobian matrix are given in Table 1. An equilibrium point of the system (3.1) is called as saddle point if the Jacobian matrix at has at least one eigenvalue with negative real part (stable) and one eigenvalue with nonnegative real part (unstable). A saddle point is said to have index one (/two) if there is exactly one (/two) unstable eigenvalue/s. It is established in the literature [47–51] that scrolls are generated only around the saddle points of index two. Saddle points of index one are responsible only for connecting scrolls.
The system (3.2) shows regular behavior if it satisfies (2.16), that is, the system is stable if Thus, the system does not show chaotic behavior for . This result is supported by numerical experiments. Figure 1 shows phase portrait in -plane for . It is observed that the system shows chaotic behavior for . For -phase portrait is shown in Figure 2. Figures 3 and 4 show - and -phase portraits, respectively, for . The phase portraits in - and -plane are drawn for in Figures 5 and 6 respectively. Thus, the minimum effective dimension of the system is .
4. Control of Chaos
In this section, we control the chaos in system (3.2). Consider where is the linear feedback control term. We set , where is a parameter to be determined so that the system (4.1) is stable. Equilibrium points of system (4.1) are , and . The points and decide. The stability of the system. The Jacobian matrix is given by One eigenvalue of the matrix at point is The eigenvalue is having negative real part for all and hence stable for all . Other two complex-conjugate eigenvalues are given by The stability of eigenvalues depends on . We have plotted the curves and for in Figure 7. The intersection points of the curve with the curves , , and are , , and , respectively. Following stability condition (2.16), it is clear that the chaos in the system can be controlled if we take the value of greater than the corresponding intersection point. Figure 8 shows controlled time series for and . Similarly, Figures 9, 10, and 11 show controlled time series for , and , respectively.
Present section deals with synchronization of proposed fractional-order system. Consider the master system and the slave system The unknown terms in (5.2) are control functions to be determined. Define the error functions as Equation (5.3) together with (5.1) and (5.2) yields the error system The control terms are chosen so that the system (5.4) becomes stable. There is not a unique choice for such functions. We choose With the choice of given by (5.5), the error system (5.4) becomes The eigenvalues of the coefficient matrix of linear system (5.6) are , and . Hence, the stability condition (2.16) is satisfied for and the errors tend to zero as . Thus, we achieve the required synchronization. The simulation results in case are summarized in Figures 12–15. Synchronization is shown in Figure 12 (signals ), Figure 13 (signals ), and Figure 14 (signals ). Note that the master systems are shown by solid lines whereas slave systems are shown by dashed lines. The errors (solid line), (dashed line) and (dot-dashed line) in the synchronization are shown in Figure 15. We have studied other cases of namely, , and but the results are omitted.
In the present work, we demonstrate the fractional order Lorenz-like system. We have observed that the system is chaotic for the fractional order , that is, the minimum effective dimension of the system is 2.91. We have used simple linear feedback controller , and given sufficient condition on to control the chaos in the proposed system. Further, we have synchronized the system using feedback control.
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