Analysis of a fractional-order chaotic system in the context of the Caputo fractional derivative via bifurcation and Lyapunov exponents

Abstract

This research focuses on the characterization of the chaotic behaviors, the hyperchaotic behaviors, and the impact of the fractional-order derivative in a class of fractional chaotic system. The numerical scheme, including the discretization of the Riemann–Liouville derivative, will be used to depict the phase portraits of the fractional-order chaotic system when the order of the used fractional-order derivative takes different values. The impact of the fractional-order derivative in the fractional chaotic system will be investigated. The proposed numerical scheme proposes a new alternative to obtain the phase portraits of the fractional-order chaotic systems. The sensitivity of the chaotic systems to the changes in the initial condition and the variation of the parameters of the considered model will be focussed with precision using the bifurcation diagrams and the Lyapunov exponent. The stability of the equilibrium points of the commensurable fractional-order chaotic system will be addressed in the context of fractional calculus. In other words, we will use the standard Matignon criterion to address the problem of stability. The main attraction and novelty of this paper will be the use of the Lyapunov exponent to characterize the nature of chaos and to prove the dissipativity of the considered chaotic system.