Stability, Synchronization Control and Numerical Solution of Fractional Shimizu–Morioka Dynamical System

In this paper we concern with asymptotic stability, synchronization control and numerical solution of incommensurate order fractional Shimizu–Morioka dynamical system. Firstly we prove the existence and uniqueness of the solutions via a new theorem. After finding steady–state points, we obtain necessary and sufficient conditions for the asymptotic stability of the Shimizu–Morioka system. We also study the synchronization control where we employ master–slave synchronization scheme. Finally, employing Adams– Bashforth–Moulton’s technique we solve the Shimizu–Morioka system numerically. To the best of our knowledge, there exist not any study about analysis of chaotic dynamics of fractional Shimizu–Morioka system in the literature. In this sense the present paper is going to be a totally new contribution and highly useful research for synthesis of a nonlinear system of fractional equations.