Investigating dynamic properties of discrete chaotic systems with fractional order has been receiving much attention recently. This paper provides a contribution to the topic by presenting a novel version of the fractional Grassi–Miller map, along with improved schemes for controlling and synchronizing its dynamics. By exploiting the Caputo h-difference operator, at first, the chaotic dynamics of the map are analyzed via bifurcation diagrams and phase plots. Then, a novel theorem is proved in order to stabilize the dynamics of the map at the origin by linear control laws. Additionally, two chaotic fractional Grassi–Miller maps are synchronized via linear controllers by utilizing a novel theorem based on a suitable Lyapunov function. Finally, simulation results are reported to show the effectiveness of the approach developed herein.

中文翻译：

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基于Caputo h差算子的分数阶Grassi-Miller映射：用于混沌控制和同步的线性方法

具有分数阶的离散混沌系统的动力学特性研究近来受到了广泛关注。本文通过提出分数Grassi-Miller映射的新版本，以及用于控制和同步其动力学的改进方案，为该主题做出了贡献。通过利用Caputo h-差分算子，首先，通过分叉图和相图分析图的混沌动力学。然后，证明了一个新的定理，以通过线性控制定律使原点的动态图稳定。此外，利用基于合适的Lyapunov函数的新定理，通过线性控制器可以同步两个混沌分数阶Grassi-Miller映射。最后，报告了仿真结果，以显示本文开发的方法的有效性。