Stability And Synchronization Of Fractional Order Generalized Sprott C System

Fractional order system is an extension of the integer order system. By contrast, it can describe the dynamic behavior of the system more appropriately. Also it has been widely used in the physical, biological, chemical, engineering, and many other fields. In recent years, with the rapid development of computer technology and the gradually perfect theory of fractional order calculus, fractional order calculus is valued for many scholars, and has become one of the hot issues in mathematics.The dissertation have four chapters, mainly focusing on the dynamic behavior of a new fractional order nonlinear system. Some sufficient conditions for local asymptotical stability of equilibria are given, and the synchronization between such systems is realized. The dissertation is organized as follows.In Chapter 1, the thesis present the background of fractional calculus, purpose and significance of study, and introduce the development and recent results of stability theory of fractional order system. Besides, the main work of this dissertation is introduced.In Chapter 2, some preliminary knowledge which will be used in this paper is intro-duced. Firstly, two basic functions of the Gamma function and Beta function are presented, which are commonly used in the fractional calculus. Secondly, the thesis introduced three basic definitions of the fractional calculus, which are the Griinward-Letnikov fractional or-der derivative, Riemann-Liouville fractional order derivative and Caputo fractional order derivative. Finally, some related theorems are presented.In Chapter 3, the dissertation studied the dynamic behavior of a new fractional or-der generalized Sprott C system, which has only two stable equilibria. Firstly, by using Routh-Hurwitz criterion and the stability theorem of nonlinear fractional order system, the dynamic behavior of the fractional order generalized Sprott C system is analysed. Some sufficient conditions for local stability of equilibria are given. And it is confirmed by numerical simulation that chaotic attractors coexist with two stable equilibria. Fur-thermore, the fractional bifurcation is investigated. It is found that the system admits bifurcations with varying different bifurcation parameters. Numerical simulation is given to illustrate and verify the results.In Chapter 4, synchronization between such systems is analyzed. Through the Lya-punov stability theory, synchronization is realized,while the error system is still kept to be nonlinear. Numerical simulation is performed to verify the theoretical results. Then, the adaptive synchronization with unknown parameters is realized. Finally, numerical simulations are presented to verify the results.