Adaptive synchronization of a class of fractional order chaotic system with uncertain parameters

In this manuscript, the adaptive synchronization of a class of fractional order chaotic system with uncertain parameters is studied. Firstly, the local stability of the fractional order chaotic system is analyzed using fractional stability criterion. Then, based on the J function criterion, suitable adaptive synchronization controller and parameter identification rules of the unknown parameters are investigated. Finally, the numerical simulations are presented to verify the effectiveness and robustness of the proposed control scheme.


INTRODUCTION
Fractional calculus is a 300-year-old topic. Although it has a long mathematical history, the applications of fractional calculus to physics and engineering are just a recent focus of interest. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Compared with the classical integer order models, fractional order derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes.
Synchronization of chaos is a phenomenon that can occur when two or more chaotic systems are coupled or one chaotic system drives the other. In the last few decades, chaos synchronization have received lots of attention from the scientists and researchers, due to its potential applications in chemical reactions, biological systems, information processing, secure communications, etc [1][2][3][4][5] . The pioneering work of Pecore and Carrol [6] introduce a method about synchronization between identical and non identical systems with different initial conditional. And to date many methods and techniques have been developed, such as complete synchronization, anti-synchronization, phase synchronization, lag synchronization, intermittent lag synchronization, generalized synchronization, intermittent generalized synchronization, time scale synchronization, projective synchronization, and so on [7][8][9][10][11][12][13][14][15] .
In 2002, S.Chen [16] presented a novel parameter identification and synchronization method for non-identical chaotic systems with unknown parameters based on adaptive control method. In 2006, H.Zhang [17] proposed a method for adaptive synchronization of two chaotic systems with different structure and unknown parameters. In 2007, Yan and Li [18] presented chaos synchronization of fractional order Lorenz, Rossler and Chen systems taking one as master and second one as slave.
In 2008, Zhou and Cheng [19] synchronized between different fractional order chaotic systems.
In this paper the adaptive synchronization of a class of fractional order chaotic system with uncertain parameters is investigated. A reasonable controller and parameter identification rules are designed and proved by J function criterion. At last, numerical simulation coincide with the theoretical analysis.

Preliminaries, problem description 2.1 Fractional calculus
Fractional calculus is a generalization of integration and differentiation to a non-integer order integro-differential operator q t a D , which is defined by Where q is the fractional order which can be a complex number, ) (q R denotes the real part of q and t a  , a is the fixed lower terminal ＆ t is the moving upper terminal.
There are some definitions for fractional derivative.

Definition 1. A real function
, and it is said to be in the space (.)  is the well-known gamma function.
The important reason of choosing Caputo derivatives for solving initial value fractional order differential equation is that the Caputo derivative of a constant is zero, whereas the Riemann-Liouville fractional derivative of constant is not equal to zero.

Numerical approximation method
Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). We only derive the predictor-corrector scheme for drive-response systems. This scheme is the generalization of Adams-Bashforth-Moulton one. We interpret the approximate solution of non-linear fractional-order differential equations by means of this algorithm in the following way.
can be described as follows:

problem description
Consider a master system in the form of are the state vectors, where . is the Euclidean norm.

Theorem 1. Stability theorem of fractional-order system
Consider the following nonlinear system of fractional differential equation.
System (5) According to theorem 1, we know   

. Adaptive synchronization and parameter identification
In this section, adaptive synchronization of the fractional order chaotic system with uncertain parameters will be studied. Theorem 2 [20] . J function criterion.
For one fractional order system, x .Then the fractional order system asymptotically stable.
For this, we define the drive system (11) and the response system (12)  In order to facilitate the following analysis, the error functions are defined as 3 3 3 x y e    e  d  P  e  e  e  e  e  e  e  J  ]  ,  ,  ,  ,  ,  ,  [  ]  ,  ,  , is the real constants, ,so according to theorem 2, the response system (12) is synchronization to the drive system (11) asymptotically.

Numerical simulation
In the numerical simulations, we set the parameters of the system (11) Fig. 4 (a) shows that the error vector 1 e converges asymptotically to zero. Fig. 4 (b) shows that the error vector 2 e also converges asymptotically to zero. The curve of synchronization error rector 3 e is shown in Fig. 4 (c). Fig. 4 (a)-(c) clearly exhibit that the response system will be adaptive synchronized with the drive system, and   Fig. 6 (a) shows that the error vector 1 e converges asymptotically to zero. Fig. 6 (b) shows that the error vector 2 e also converges asymptotically to zero. The curve of synchronization error rector 3 e is shown in Fig. 6 (c). Fig. 6 (a)-(c) clearly exhibit that the response system will be adaptive synchronized with the drive system, and figure 7 shows that the estimated parameter vectors converge to the original parameter vectors.

Conclusion
In this letter, the adaptive synchronization of a class of fractional order chaotic system with uncertain parameters is presented, and based on the fractional stability criterion, the local stability of the fractional order chaotic system is analyzed.
Besides, suitable adaptive synchronization controller and parameter identification rules of the unknown parameters are investigated by using the J function criterion. Numerical simulations are performed to verify these results.