A robust method for new fractional hybrid chaos synchronization

Abstract

In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer-order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used.

Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems

“…, antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist in the synchronization of the master system (7) and the slave system (8) if there exist controllers u i (1 ≤ i ≤ 4) and real numbers (α 31 , α 32 , α 33 , α 34 ) such that the synchronization errors e 1 (t) = x 1 (t) -y 1 (t), e 2 (t) = x 2 (t) --y 2 (t) , and (9)…”

“…Very few methods for synchronizing nonidentical fractional-order chaotic systems have been illustrated [28][29][30]. Additionally, the topic related to the coexistence of different synchronization types between fractionalorder systems is almost unexplored [31,32].…”

“…The approach presents the remarkable feature of being both rigorous and applicable to a wide class of commensurate and incommensurate fractional-order systems of different dimensions and different orders. It is worth noting that the proposed approach is more general than those illustrated in [31,32] because, among other things, it guarantees the coexistence of three different synchronization types (instead of only two types, as in [31,32]). This increased complexity provides a deeper insight into the synchronization phenomena between systems described by fractional differential equations.…”

Chaos Synchronization of Nonlinear Fractional Discrete Dynamical Systems via Linear Control

“…There are many synchronization schemes for fractional differential systems, such as synchronization via the linear control technique [20], synchronization via the adaptive sliding mode [21], projective synchronization via single sinusoidal coupling [22], hybrid chaos synchronization with a robust method [23], synchronization with activation feedback control [24], synchronization via a scalar transmitted signal [25], adaptive synchronization via a single driving variables [26], synchronization via novel active pinning controls [27]. In fact, most mentioned synchronization schemes of fractional differential systems can be used in the synchronization of fractional discrete dynamical systems.…”

A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control

“…If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the response of the slave system so that the slave system tracks the output of the master system asymptotically [105,106,107,108].…”