The important phenomenon of "stickiness" of chaotic orbits in low dimensional dynamical systems has been investigated for several decades, in view of its applications to various areas of physics, such as classical and statistical mechanics, celestial mechanics and accelerator dynamics. Most of the work to date has focused on two-degree of freedom Hamiltonian models often represented by two-dimensional (2D) area preserving maps. In this paper, we extend earlier results using a 4-dimensional extension of the 2D MacMillan map, and show that a symplectic model of two coupled MacMillan maps also exhibits stickiness phenomena in limited regions of phase space. To this end, we employ probability distributions in the sense of the Central Limit Theorem to demonstrate that, as in the 2D case, sticky regions near the origin are also characterized by "weak" chaos and Tsallis entropy, in sharp contrast to the "strong" chaos that extends over much wider domains and is described by Boltzmann Gibbs statistics. Remarkably, similar stickiness phenomena have been observed in higher dimensional Hamiltonian systems around unstable simple periodic orbits at various values of the total energy of the system.

翻译：低维动力系统中混沌轨道的“粘滞”现象因其在经典力学、统计力学、天体力学及加速器动力学等物理领域的应用，已历经数十年研究。现有工作主要聚焦于二维保面积映射所表征的二自由度哈密顿模型。本文通过将二维MacMillan映射推广至四维，证明双耦合MacMillan映射的辛模型在相空间有限区域同样呈现粘滞现象。为此，我们运用中心极限定理意义的概率分布表明：与二维情形类似，原点附近粘滞区域以“弱”混沌和Tsallis熵为特征，这与覆盖更广区域、服从玻尔兹曼-吉布斯统计的“强”混沌形成鲜明对比。值得注意的是，在更高维哈密顿系统中，围绕不稳定简单周期轨道，于系统总能量不同取值处亦观察到类似的粘滞现象。