Chaotic Characteristics Analysis and Circuit Implementation for a Fractional-Order System

Abstract

Based on fractional transfer function approximation in frequency domain, the paper firstly discusses the fractional form of a four-wing autonomous integral-order chaotic system, and finds some chaotic attractors in the different orders of the fractional-order system. The chaotic attractors can be found when the order of the fractional-order system is varied from 1.5 to 2.7, especially in the fractional-order system of the order as low as 2.7, and various chaotic behaviors occur when varying four different system parameters. Some bifurcation diagrams and phase portraits including the four-wing chaotic attractors, the two-wing chaotic attractors, and some periodic orbits are given to verify the chaotic behaviors of the fractional-order system. Then, based on the method of topological horseshoe analysis adopted in many integer-order chaotic systems, a topological horseshoe is found to prove the existence of chaos in the fractional-order system of the order as low as 2.7. At last, an analog circuit is designed to confirm the chaotic dynamics of fractional-order system.