Symmetry-breaking and bifurcation diagrams of fractional-order maps

In this paper, two important issues about the discrete version of Caputo’s fractional-order maps defined on complex plane are investigated, both analytically and numerically: attractors symmetry-breaking induced by derivative sensitivity in determining bifurcation diagram. It is proved that integer-order with dihedral symmetry or cycle may lose their once they transformed to maps. Also, it conjectured that, contrarily maps, diagrams far from being well understood. Two examples presented for illustration: logistic map cyclic map.