Chaos Control and Synchronization in Fractional-Order Lorenz-Like System

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.


Introduction
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 1 .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 2 , backstepping design method 3 , differential geometric method 4 , inverse optimal control 5 , sampled-data feedback control 6 , adaptive control 7 , and so on.One simple approach is the linear feedback control 8 .Linear feedback controllers are easy to implement, they can perform the job automatically, and stabilize the overall control system efficiently 9 .
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 International Journal of Differential Equations 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 14 and for developing safe and reliable cryptographic systems 15 .
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 19 , viscoelasticity 20 , diffusion 21-25 , turbulence, electromagnetism, signal processing 26, 27 , and bioengineering 28 .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 31 .Further, many systems such as Li and Peng 32 , Lu 33 , Li and Chen 34 , Daftardar-Gejji and Bhalekar 35 , and unified system 36 were investigated in this regard.Effect of delay on chaotic solutions in fractional order dynamical system is investigated by the present author 37 .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 42 .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.

Fractional Calculus
Few definitions of fractional derivatives are known 16-18 .Probably the best known is the Riemann-Liouville formulation.
The Riemann-Liouville integral of order μ, μ > 0 is given by 2.1 An alternative definition was introduced by Caputo.Caputo's derivative is defined as International Journal of Differential Equations 3 where m ∈ N. 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 m − 1 < μ ≤ m, m ∈ N, 2.3

2.4
The IVP 2.4 is equivalent to the Volterra integral equation Consider the uniform grid {t n nh/n 0, 1, . . ., N} for some integer N and h : T/N.Let y h t n be approximation to y t n .Assume that we have already calculated approximations y h t j , j 1, 2, . . ., n, and we want to obtain y h t n 1 by means of the equation where

2.7
International Journal of Differential Equations The preliminary approximation y P h t n 1 is called predictor and is given by where Error in this method is where p min 2, 1 α .

Asymptotic Stability of the Fractional-Ordered System
Consider the following fractional-ordered dynamical system:

2.11
Let p ≡ x * 1 , x * 2 , x * 3 be an equilibrium point of the system 2.11 that is, f i p 0, 1 ≤ i ≤ 3 and ξ i x i − x * i a small disturbance from a fixed point.Then

2.14
Consider the linear autonomous system where J is n × n matrix and 0 < α < 1 is asymptotically stable if and only if | arg λ | > απ/2 for all eigenvalues λ of J.In this case, each component of solution ξ t decays towards 0 like t −α 29, 46 .This shows that if | arg λ | > απ/2 for all eigenvalues λ of J then the solution ξ i t of the system 2.13 tends to 0 as t → ∞.Thus, the equilibrium point p of the system is asymptotically stable if | arg λ | > απ/2, for all eigenvalues λ of J, that is, if 2.16

Fractional Lorenz-Like System
In where α ∈ 0, 1 .The equilibrium points of the system 3.1 and the eigenvalues of corresponding Jacobian matrix   It is clear from Table 1 that the equilibrium points E 1 and E 2 are saddle points of index two; hence, there exists a two-scroll attractor 47 in the system 3.2 .
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 α < 0.96345.This result is supported by numerical experiments.Figure 1 shows phase portrait in xy-plane for α 0.96.It is observed that the system shows chaotic behavior for α ≥ 0.97.For α 0.97 xz-phase portrait is shown in Figure 2. Figures 3 and 4 show xy-and yz-phase portraits, respectively, for α 0.98.The phase portraits in xy-and xz-plane are drawn for α 0.99 in Figures 5 and 6 respectively.Thus, the minimum effective dimension of the system is 0.97 × 3 2.91.
International Journal of Differential Equations

Control of Chaos
In this section, we control the chaos in system 3.2 .Consider where u is the linear feedback control term.We set u kx, where k is a parameter to be determined so that the system 4.1 is stable.Equilibrium points of system 4. O 0, 0, 0 , The points P 1 and P 2 decide.The stability of the system.The Jacobian matrix is given by One eigenvalue e 0 of the matrix J .

4.3
The eigenvalue e 0 is having negative real part for all k ≥ 0 and hence stable for all 0 ≤ α ≤ 1. .

4.4
The stability of eigenvalues λ ,− depends on k.We have plotted the curves | arg λ | and απ/2 for α 0.97, 0.98, 0.99, 1 in Figure 7.The intersection points of the curve | arg λ | with the curves 0.97π/2, 0.98π/2, 0.99π/2 and π/2 are k ≈ 3.5, k ≈ 8.5, k ≈ 14 and k ≈ 21, respectively.Following stability condition 2.16 , it is clear that the chaos in the system can be controlled if we take the value of k greater than the corresponding intersection point.Figure 8 shows controlled time series for α 1 and k 22.Similarly, Figures 9,10,and 11 show controlled time series for α 0.99, 0.98, 0.97, and k 20, 15, 7, respectively.

Synchronization
Present section deals with synchronization of proposed fractional-order system.Consider the master system 5.1 and the slave system

5.2
The unknown terms u 1 , u 2 , u 3 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

5.4
The control terms u i are chosen so that the system 5.4 becomes stable.There is not a unique choice for such functions.We choose

5.5
International Journal of Differential Equations

5.6
The eigenvalues of the coefficient matrix of linear system 5.6 are −10, −1, and −2.5.Hence, the stability condition 2.16 is satisfied for 0 ≤ α ≤ 1 and the errors e i t tend to zero as t → ∞.Thus, we achieve the required synchronization.The simulation results in case α 0.99 are summarized in

Conclusion
In the present work, we demonstrate the fractional order Lorenz-like system.We have observed that the system is chaotic for the fractional order α ≥ 0.97, that is, the minimum effective dimension of the system is 2.91.We have used simple linear feedback controller International Journal of Differential Equations u kx, k > 0 and given sufficient condition on k to control the chaos in the proposed system.Further, we have synchronized the system using feedback control.

Figures 12 - 15 .
Synchronization is shown in Figure 12 signals x 1 , x 2 , Figure 13 signals y 1 , y 2 , and Figure 14 signals z 1 , z 2 .Note that the master systems are shown by solid lines whereas slave systems are shown by dashed lines.The errors e 1 t solid line , e 2 t dashed line and e 3 t dot-dashed line in the synchronization are shown in Figure 15.We have studied other cases of α namely, 0.97, and 0.98 but the results are omitted.
3are given in Table1.An equilibrium point p of the system 3.1 is called as saddle point if the Jacobian matrix at p 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.

Table 1 :
Equilibrium points and corresponding eigenvalues.
1 at point P 1 is Other two complex-conjugate eigenvalues λ ,− are given by