Mathematical Analysis of Peculiar Behavior by Chaotic, Fractional and Strange Multiwing Attractors

Abstract

Strange attractors are fascinating not only for the fractal structure most of them display, but also for the freakish chaotic and multiscroll trajectories they portray. In this paper, Haar wavelet numerical method is used to address the solvability of the 3D Lu–Chen model combined with the classical Caputo fractional differentiation. The full model, named Caputo–Lu–Chen model is solved analytically and a complete error analysis is performed to test the convergence of the method. It reveals that the error made using such a technique is negligible. Numerical simulations, performed in pure fractional and standard integer cases, show that the Caputo–Lu–Chen system maintains its status of strange, chaotic and multiscroll attractor. The graphics in both cases present similar features characterized by attractors with many scrolls. Hence, the Caputo derivative order when introduced as parameter in the Lu–Chen system does not remove the multiscroll property of generated attractors; it even becomes an important control parameter able to change the dynamics of the whole system. This result can be very important in control theory.