Introduction to the fractional-order chaotic system under fractional operator in Caputo sense

Abstract

In this paper, we consider a new fractional-order chaotic system described by the Caputo fractional derivative. This paper’s main objective is to analyze the bifurcation maps to detect the chaotic regions for a new fractional-order chaotic system. We introduce a numerical discretization, which will help us to depict the phase portraits of the new model and to illustrate the influence of the order of the Caputo derivative. The Lyapunov exponents will play an important role in giving the nature of the chaos at different Caputo derivative orders. The theoretical results obtained in this paper will be confirmed by the simulation of the circuit representation of the considered fractional-order chaotic system. As it will be observed, the graphical representations obtained with Matlab using the introduced numerical scheme and the phase portraits in the oscilloscopes will be in good agreement. The local asymptotic stability for the equilibrium points of the new fractional-order chaotic system will help us to delimit the chaotic region according to the variation of the order of the Caputo derivative.