This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

中文翻译：

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在分数差异逻辑图上：动态分析和同步控制

本文研究了在离散分数微积分框架中从差分方程导出的逻辑图。通过庞加莱图和Julia集，研究了地图的混沌和分形特征，并与拟提出的二次图进行了比较。这些动力学反映了分数差异图的记忆效应，并结合定量分析给出了一些合理的解释。设计了耦合控制器，以实现分数差分逻辑映射和分数差分二次映射之间的同步。