Controlling memory chaos and synchronization in real order nonlinear systems

Abstract

Nonlinear systems associated with real orders are known to produce complicated dynamics that can be distinguished from their integer counterparts. In applications of interest, the inclusion of real orders introduces memory to the system, and the settling of initial time can give rise to new representations of real-order systems that may not be similar to their integer version counterparts. Controlling memory chaos and synchronization in real-world systems poses new complex issues due to initial time-dependent nature of fractional derivatives. All the existing literature has not considered an important factor of ultimate random initial time, when it comes to the problem of control and synchronization of real-order systems. This paper uses the active control method and proposes principles of control and synchronization for random initial-time incommensurate real-order systems in the sense of Caputo fractional derivative. First, by using the real-order linear time-varying theory, it is shown that memory chaos in an incommensurate food chain nonlinear system can be controlled when the system starts at a non-zero initial time. Then, we show that the chaotic states of two identical models of such systems can be synchronized whenever the initial time becomes non-zero. Numerical simulations are presented to illustrate the importance of theoretical analysis.