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 McMillan map, and show that a symplectic model of two coupled McMillan 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“粘滞”现象的研究已持续数十年，其应用涉及经典力学与统计力学、天体力学及加速器动力学等多个物理学领域。现有工作主要聚焦于通常由二维（2D）保面积映射表征的两自由度哈密顿模型。本文通过将二维McMillan映射推广至四维，证明两个耦合McMillan映射构成的辛模型在相空间有限区域同样存在粘滞现象。为此，我们采用中心极限定理意义上的概率分布来验证：与二维情形一致，原点附近的粘滞区域呈现“弱”混沌特征并服从Tsallis熵，这与由Boltzmann Gibbs统计描述的广域“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，当系统总能量取不同值时，不稳定的简单周期轨道周围也观测到类似的粘滞现象。