Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste, 63-35121 Padova, Italy.
Chaos. 2013 Jun;23(2):023124. doi: 10.1063/1.4807097.
The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasi-integrable Hamiltonian systems. In particular, long-term instability of the action variables can be observed only in the so-called Arnold web, which is the complement in the phase-space of all KAM invariant tori, and only on the very long times which depend exponentially on an inverse power of the perturbation parameter. Though the structure of the Arnold's web was clearly explained already on Arnold's 1963 article, its numerical detection with a precision sufficient to reveal exponentially slow diffusion of the actions through the web itself has become possible only in the last decade with the extensive computation of dynamical indicators. In this paper, we first review the detection method that allowed us to compute the Arnold web, and then we discuss its use to study the long-term diffusion through the web itself. We also show that the Arnold web of a quasi-integrable Hamiltonian system is useful to track the diffusion of orbits of weakly dissipative perturbations of the same Hamiltonian system.
著名的 KAM 和 Nekhoroshev 定理提供了关于拟线性哈密顿系统的长期动力学的重要信息。特别是，作用变量的长期不稳定性只能在所谓的 Arnold 网中观察到，这是在相空间中所有 KAM 不变环面的补集，并且只在非常长的时间内，这些时间随微扰参数的逆幂次呈指数增长。尽管 Arnold 在 1963 年的文章中已经清楚地解释了 Arnold 网的结构，但只有在过去十年中，通过广泛计算动力学指标，才有可能以足够精确的数值检测到该结构，从而揭示作用变量通过该网的指数级缓慢扩散。在本文中，我们首先回顾了允许我们计算 Arnold 网的检测方法，然后讨论了它在研究通过该网本身的长期扩散中的应用。我们还表明，拟线性哈密顿系统的 Arnold 网可用于跟踪同一哈密顿系统的弱耗散微扰轨道的扩散。
