Synchronization between a Class of Variable-Order Fractional Hyperjerk Chaotic Systems

Abstract

Variable-order fractional derivatives can be considered as a natural and analytical extension of constant fractional-order derivatives. In variable order derivatives, the order can vary continuously as a function of either dependent or independent variables of differentiation, such as time, space, or even independent external variables. The main contribution in this paper is the use of fractional orders that vary in time for a new class of chaotic system. In this paper, we also study the synchronization between a new class of variable-order fractional hyperjerk chaotic systems. The Grünwald-Letnikov's definition of fractional derivative will be implemented to solve variable-order fractional problem, in addition by considering the bifurcation diagram a periodic function was proposed to vary the order of the derivative. The chaos synchronization will be carried out via active control approach. Regarding the results, and focusing on synchronization, it can be observed that the error converges asymptotically to zero. Finally, the theoretical work agrees satisfactorily with the numerical results.