Quasi-periodic motions in families of dynamical systems
Quasi-periodic motions in families of dynamical systems
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DOI：
10.1007/978-3-540-49613-7
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
H. Broer;G. Huitema;M. Sevryuk
中科院分区：
文献类型：
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作者：
H. Broer;G. Huitema;M. Sevryuk
本书致力于研究动力系统中的准周期运动现象。相空间中的这种运动密集地填充了不变的环面。这种现象在哈密顿动力学中最为常见。哈密顿系统因其用于模拟与无摩擦力学相关的动力学（包括行星和月球运动）而闻名。在这种情况下，总体情况如下。一方面，哈密顿系统是完全有序的：这些是可积系统，其中所有运动都局限于不变环面。另一方面，存在的系统在每个能级上都是完全混乱的。在这两者之间，我们知道系统，由于可积系统的扰动足够小，表现出秩序（携带准周期动力学的不变环面）和混沌（所谓的随机层）的共存。准周期运动的柯尔莫哥洛夫-阿诺德-莫泽（KAM）理论告诉我们，这种运动的发生在所有哈密顿系统中都是开放的：换句话说，它是在小哈密顿扰动下持续存在的现象。此外，一般来说，对于任何这样的系统，相空间中的准周期环面的并集都是无处密集的正勒贝格测度集，即所谓的康托族。这一事实意味着存在非遍历的哈密顿系统的开类。本书的主要目的是研究在考虑其他类别的系统或环境时该图景的变化。
This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems-or contexts-are considered.