Symmetry-breaking and bifurcation diagrams of fractional-order maps

TL;DR

This paper investigates how fractional-order modifications affect symmetry and bifurcation diagrams in complex plane maps, revealing symmetry loss and highlighting challenges in bifurcation analysis.

Contribution

It analytically and numerically demonstrates symmetry-breaking in fractional-order maps and discusses the complexity of bifurcation diagram determination.

Findings

Fractional-order maps lose symmetry present in integer-order counterparts.

Symmetry-breaking is analytically and numerically confirmed.

Bifurcation diagrams of fractional-order maps are difficult to determine.

Abstract

In this paper two important aspects related to Caputo fractional-order discrete variant of a class of maps defined on the complex plane, are analytically and numerically revealed: attractors symmetry-broken induced by the fractional-order and the sensible problem of determining the right bifurcation diagram of discrete systems of fractional-order. It is proved that maps of integer order with dihedral symmetry or cycle symmetry loose their symmetry once they are transformed in fractional-order maps. Also, it is conjectured that, contrarily to integer-order maps, determining the bifurcation diagrams of fractional-order maps is far from being a clarified problem. Two examples are considered: dihedral logistic map and cyclic logistic map.