Symmetry-breaking and bifurcation diagrams of fractional-order maps

Marius-F. Danca

Full text: http://dx.doi.org/10.1016/j.cnsns.2022.106760

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Abstract

In this paper two important aspects related to Caputo’s fractional-order discrete
variant of a class of maps defined on the complex plane, are analytically and numer-
ically 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

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