Universal Fractional Map and Cascade of Bifurcations Type Attractors

TL;DR

This paper introduces a universal fractional map family based on fractional derivatives, revealing new cascade bifurcation attractors in fractional systems that extend classical dynamics.

Contribution

It develops the Universal α-Family of Maps depending on fractional order α, and identifies new cascade bifurcation attractors in fractional dynamics.

Findings

Fractional maps exhibit cascade of bifurcations type attractors.

New attractors appear for fractional α<2 during period doubling.

Fractional systems show unique transition to chaos not seen in classical maps.

Abstract

We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal $\alpha$-Family of Maps depending on a single parameter $\alpha > 0$ which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular $\alpha$-families corresponding to the Standard and Logistic Maps. For fractional $\alpha<2$ in the area of parameter values of the transition through the period doubling cascade of bifurcations from regular to chaotic motion in regular dynamics corresponding fractional systems demonstrate a new type of attractors - cascade of bifurcations type trajectories.