Symmetry-breaking and bifurcation diagrams of fractional-order maps

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 b...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2023-01, Vol.116, p.106760, Article 106760
Main Author: Danca, Marius-F.
Format: Article
Language:English
Subjects:
Bifurcation diagram
Caputo fractional differences
Fractional-order logistic map
Symmetry breaking
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Summary: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. •Discrete fractional order systems with Caputo forward differences.•Caputo forward differences destroy symmetry of discrete systems.•Bifurcation diagram of fractional order systems, differs from his integer order counterpart.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2022.106760