Control Chaos in System with Fractional Order

In this paper, by utilizing the fractional calculus theory and computer simulations, dynamics of the fractional order system is studied. Further, we have extended the nonlinear feedback control in ODE systems to fractional order systems, in order to eliminate the chaotic behavior. The results are proved analytically by stability condition for fractional order system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.


Introduction
Fractional calculus is a classical mathematical concept, with a history as long as calculus itself.It is a generalization of ordinary differentiation and integration to arbitrary order, and is the fundamental theories of fractional order dynamical systems.Fractional-order differential/integral has been applied in physics and engineering, such as viscoelastic system [1], dielectric polarization [2], electrode-electrolyte polarization [3] and electromagnetic wave [4], and so on.
The fractional order system and its potential application in engineering field become promising and attractive due to the development of the fractional order calculus.Typically, chaotic systems remain chaotic when their equations become fractional.For example, it has been shown that the fractional order Chua's circuit with an appropriate cubic nonlinearity and with an order as low as 2.7 can produce a chaotic attractor [5].
However, there are essential differences between ordinary differential equation systems and fractional order differential systems.Most properties and conclusions of ordinary differential equation systems cannot be extended to that of the fractional order differential systems.Therefore, the fractional order systems have been paid more attention.Recently, many investigations were devoted to the chaotic dynamics and chaotic control of fractional order systems [6][7][8][9][10][11][12].
In this paper, practical scheme is proposed to eliminate the chaotic behaviors in fractional order system by extending the nonlinear feedback control in ODE systems to fractional-order systems.This paper is organized as follows.In Section 2, the numerical algorithm for the fractional order system is briefly introduced.In Section 3, Dynamics of the fractional order system is numerically studied.In Section 4, general approach to feedback control scheme is given, and then we have extended this control scheme to fractional order system, numerical results are shown.Finally, in Section 5, concluding comments are given.

Fractional Derivative and Numerical Algorithm
There are two approximation methods for solving fractional differential equations.The first one is an improved version of the Adams-Bashforth-Moulton algorithm, and the rest one is the frequency domain approximation.The Caputo derivative definition involves a time-domain computation in which nonhomogenous initial conditions are needed, and those values are readily determined.In this paper, the Caputo fractional derivative defined in [13] is often described by is the first integer that is not less than ,  is the α-order Riemann-Liouville integral operator which defined by where  is the Gamma function, .0 < 1

 
Now we consider the fractional order system [14] which is given by where is the fractional order, i By exploiting the Adams-Bashforth-Moulton scheme [15], the fractional order system (1) can be discretized as followings:

Dynamic Analysis of the Fractional Order System
Theorem 1: The fractional linear autonomous system is locally asymptotically stable if and only if be an equilibrium point of a fractional nonlinear system If the eigenvalues of the Jacobian matrix then the system is locally asymptotically stable at the equilibrium point x x   The system (1) has five equilibrium points: and

S S S S S
, q q 3 q 3 According to Theorem 2, we can easily conclude that the equilibrium 0 of system (1) is unstable when and are all greater than zero.

 
.50000 1 We can obtain and 3 According to Theorem 2, we can easily conclude that when 1 and 3 are all great than , the equilibrium of system (1) is unstable.
, , S , q q q 0.7834 In sum, there exists at least one stable equilibrium 1 2 3 and 4 of system (1), when 1 2 and 3 are all less than , i.e., the system (1) will be stabilized at one point  1 2 3 4 finally; when 1 2 and 3 are all greater than 0. , all the equilibriums of system (1) are unstable, the system (1) will exhibit a chaotic behaviour; when i the problem will be complicated, the system (1) may be convergent, periodic or chaotic.For example, when 1 2 3 the value of the largest Lyapunov exponent is 0.1653.Obviously, the fractional order system (1) is chaotic.When 1 2 3 the fractional order system (1) is not chaotic, but periodic orbits appear.

Feedback Control
Let us consider the fractional order system where   x t is the system state vector, and   u t the control input vector.Given a reference signal  , the problem is to design a controller in the state feedback form: where   , g x t is the vector-valued function, so that the controlled system can be driven by the feedback control g(x, t) to achieve the goal of target tracking so we must have , be a periodic orbit or fixed point of the given system (2) with , then we obtain the system error where and

   
, F e t g x t t f x t   Theorem 3: If 0 is a fixed poi of the system (2) an f x nt d the eigenvalues of the Jacobian matrix at the equilibrium point 0 satisfies the condition  1), then we have:  we obtain the system error

yz yz u t e e ae zx zx u t e e xy xy u t
We define the control function as follow So the system error (5) becomes The Jacobian matrix of system ( 7) is so we have th


all eigenvalues are real negatives, one has  for all i q satisfies 2, = 1,2,3, q i it follows from Theorem 3 that the The control can be started at any time according to our needs, so we choose to activate the control when in order to make a comparison between the behavior before activation of control and after it.
For 1 2 and q 3 = 0.98, unstable point has been stabilized, as shown in Figure 1, note that The control is activated when and the evolution of     , , x t y t z t = 22.5 t 0 S = 0.93, = 0.95 q q = 0,98, q 1 S = 0.92 q = = 0,97, q q 2 S = = = 0,94, q q q 3 S = = = 0,96, q q q 4 S t 20 is chaotic, then when the control is started at we see that is rapidly stabilized.the unstable point has been stabilized, as shown in Figure 5. sed schem .As menti implemented the improved Adams-Bashforth-lton algorithm for numerical simul on.When is less than , there is a chaotic behavior, the equilibrium point S 0 for q 1 = 0.93, q 2 = 0.95 and q 3 = 0.98. Figure 1.Stabilizing Figure 2. Stabilizing the equilibrium point S 1 for q 1 = 0.93, q 2 = 0.95 and q 3 = 0.98. Figure 3. Stabilizing the equilibrium point S 2 for q 1 = 0.92 and q 2 = q 3 = 0.97. Figure 4. Stabilizing the equilibrium point S 3 for q 1 = q 2 = q 3 = 0.94. Figure 5. Stabilizing the equilibrium point S for q = q = q = 0.96.but when the control is activated at = 20 t , the five 2 3 , S S S S and pidly stabilized.

Conclusions
Chaotic phenomenon makes prediction impossible in the real world; then the deletion of this phenomenon from fractional order system is very useful, the main contribution of this paper is to this end.
In this paper, we investig rder applying the fractiona al ord h the rdback control scheme has been exactional order system.ate the system with fractional l calculus technique.Accord-o ing to the stability theory of the fraction er system, dynamical behaviors of the fractional order system are analyzed, bot oretically and numerically.Furthe more, nonlinear fee tended to control fr proved analytically by stability condition for fractional order system.Numerically the unstable fixed points have been successively stabilized for different values of 1 2 , q q and .