The Onset of Chaos via Asymptotically Period-Doubling Cascade in Fractional Order Lorenz System

Abstract

Little seems to be known about the chaotification problem in the framework of fractional order nonlinear systems. Based on the negative damping instability mechanism and fractional calculus technique, this paper reports the onset of chaos in fractional order Lorenz system with periodic system parameters via asymptotically period-doubling cascade. To further understand the complex dynamics of the system, some basic properties such as the largest Lyapunov exponents, bifurcation diagram, routes to chaos, asymptotically periodic windows, possible chaotic and asymptotically periodic window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations. Of particular interest is a striking finding that fractional derivative can chaotify the globally stable periodic system without feedback control.