保面积映射中的随机变迁理论
Theory of stochastic transitions in area preserving maps
摘要：
由面积保持映射定义的离散动力系统的一个著名方面是对局部发生的随机跃迁的物理解释，这些随机跃迁通过破坏不变的KAM曲线和局部或全局混沌的发生来表现。尽管许多以前的调查（见特别是奇里科夫，格林，珀西瓦尔，埃斯坎德和Doveil和麦凯）基于不同的方法，几个方面的现象仍然逃避一个完整的理解和严格的描述。特别是格林的方法是基于几个假设，其中之一是，随机过渡导致破坏的最后KAM曲线在标准地图是由于线性不稳定的椭圆点属于一个特殊的家庭的不变量集{I（m，n）}（有理迭代）具有合理的缠绕数和相关的最后KAM曲线。这项工作的目的是分析非线性现象导致的随机过渡的标准地图和他们的影响不稳定的不变集相关的KAM曲线，最终导致的KAM曲线本身的破坏。
英文摘要：
A famous aspect of discrete dynamical systems defined by area-preserving maps is the physical interpretation of stochastic transitions occurring locally which manifest themselves through the destruction of invariant KAM curves and the local or global onset of chaos. Despite numerous previous investigations (see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay) based on different approaches, several aspects of the phenomenon still escape a complete understanding and a rigorous description. In particular Greene's approach is based on several conjectures, one of which is that the stochastic transition leading to the destruction of the last KAM curve in the standard map is due the linear destabilization of the elliptic points belonging to a peculiar family of invariants sets {I(m,n)} (rational iterates) having rational winding numbers and associated to the last KAM curve. Purpose of this work is to analyze the nonlinear phenomena leading to the stochastic transition in the standard map and their effect on the destabilization of the invariant sets associated to the KAM curves, leading, ultimately, to the destruction of the KAM curves themselves.
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