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.

在本文中，从解析和数值上研究了关于定义在复平面上的 Caputo 分数阶离散映射的离散版本的两个重要问题：分数阶导数引起的吸引子对称破坏和确定分岔图的敏感性. 证明了具有二面体对称性或循环对称性的整数阶映射在转换为分数阶映射后可能会失去对称性。此外，据推测，与整数阶映射相反，确定分数阶映射的分岔图远未得到很好的理解。举两个例子来说明：二面逻辑图和循环逻辑图。