Stabilization of Fractional Order Discrete Chaotic Systems

Abstract

Chaos is almost ubiquitous in field of science and engineering. The insurgent and typically unpredictable behavior exhibited by nonlinear systems is seen as chaos. In recent decades, fractional (non-integer) order chaotic systems have also been developed and their applications have invited a lot of interest of research community. Along with fractional order continuous time chaotic systems, researchers have also explored fractional order discrete chaotic systems to some extent. These systems can also be exploited for the same application for which continuous versions are used, thus providing increased flexibility and reliability. Although the mathematics of fractional discrete calculus is still in development phase, still with the help of available knowledge, research community has started giving attention to this emerging field. A number of contributions are available in literature in the area of fractional discrete calculus and its applications in control systems. One can represent linear systems using fractional difference equations in state space domain. Similarly, fractional difference equations can be used to represent nonlinear dynamical systems, especially chaotic systems. Fractional order discrete chaotic systems offer a new domain of exploration to research fraternity. As the work reported is limited, so the need arises to review and consolidate it. The analysis, control and synchronization of fractional order chaotic systems is the aim of this chapter. This chapter initially, gives a brief overview of fractional difference equations and their solution. Thereafter, the results obtained so far in this area are discussed and presented. Chaotic behavior of discrete fractional versions of the famous Logistic map and Henon map is studied first. Further, control of the same class of systems is tackled. The main contribution of the work is to present analysis of control of fractional Henon map using backstepping control which is a well-known technique to researchers in area of nonlinear control. Simulation results are obtained using MATLAB and are presented at the end to validate the results.