Symmetry-breaking and bifurcation diagrams of fractional-order maps

摘要

In this paper, two important issues about the discrete version of Caputo's fractional -order discrete maps defined on the complex plane are investigated, both analytically and numerically: attractors symmetry-breaking induced by the fractional-order derivative and the sensitivity in determining the bifurcation diagram. It is proved that integer -order maps with dihedral symmetry or cycle symmetry may lose their symmetry once they are transformed to 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 well understood. Two examples are presented for illustration: dihedral logistic map and cyclic logistic map. (c) 2022 Elsevier B.V. All rights reserved.