Asymptotically periodic points, bifurcations, and transition to chaos in fractional difference maps

TL;DR

This paper derives formulas for asymptotically periodic points in fractional difference maps, analyzes bifurcations and chaos transition, and conjectures that Feigenbaum constants are unchanged from regular maps.

Contribution

It provides analytic expressions for periodic points in fractional difference maps and explores bifurcation behavior, including a conjecture on Feigenbaum constants.

Findings

Derived formulas for asymptotically periodic points.

Generated bifurcation diagrams for fractional logistic maps.

Conjecture that Feigenbaum constant remains the same as in regular maps.

Abstract

In this paper, we derive analytic expressions for coefficients of the equations that allow calculations of asymptotically periodic points in fractional difference maps. Numerical solution of these equations allows us to draw the bifurcation diagram for the fractional difference logistic map. Based on the numerically calculated bifurcation points, we make a conjecture that in fractional maps the value of the Feigenbaum constant $\delta$ is the same as in regular maps, $\delta=4.669...$.