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Unstable Dynamics in Hamiltonian Systems

哈密顿系统中的不稳定动力学

基本信息

批准号：
EP/J003948/1
负责人：
Vasily Gelfreykh
金额：
$ 104.62万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ003948%2F1
关键词：
Unstable Dynamics Hamiltonian Systems

项目摘要

The key challenge of the modern theory of Hamiltonian Dynamical Systems is to provide adequate mathematical tools for describing chaotic dynamics in a system which combines both regular and chaotic components as an overwhelming majority of models relevant for applications of the theory fell into this category. In this area two major problems have resisted the efforts of scientists for decades. The first one is called "Arnold Diffusion" and is related to instability of action variables on long time scales. The second one is known as "Positive Metric Entropy Conjecture" and states that chaotic motions are physically relevant, i.e., they occupy a subset of positive Lebesgue measure.In the period of the fellowship I will concentrate on the study of Hamiltonian systems with multiple time scales, as this property is often present in the equations either explicitly or implicitly. The aim of this study is to develop mathematical tools for studying instabilities of dynamics. Preliminary results show that we are able to prove existence of normally hyperbolic invariant objects with different dynamical behaviour of slow components. It is probable that on longer time scales the restriction of the dynamical system on this family can be approximated by a stochastic ordinary differential equation in the slow variables. If confirmed, it will establish an important connection between two different fields of Mathematics: the theory of deterministic Hamiltonian systems and stochastic differential equations, which are considered mostly unrelated at the present. An extension of these results should provide a new insight on the theory of the Fermi acceleration. A comparison with variational approach to Arnold Diffusion announced by Mather (Princeton) suggests that our mechanism could be used to solve the long-standing problem of genericity of Arnold Diffusion in near-integrable Hamiltonian systems. The progress in this direction should require radical improvements of methods for detection of transversal homoclinic trajectories associated with various invariant objects, i.e., in the area where I have a substantial technical expertise.As a summary, the following list of technical topics will be addressed initially: exponentially small splitting of invariant manifolds in higher dimension, stochastic description of slow dynamics in slow-fast systems with chaotic fast component, Fermi acceleration, Arnold Diffusion, positive metric entropy conjecture.I expect that as the fellowship advances new research directions will arise partially motivated by the development of the theory and partially by questions coming from its applications.The research will be curried out at the Mathematics Institute, University of Warwick, and will involve collaboration with several groups in the UK and oversees.

现代哈密顿动力系统理论的关键挑战是提供足够的数学工具来描述结合了常规成分和混沌成分的系统中的混沌动力学，因为与该理论应用相关的绝大多数模型都属于这一类。在这一领域，有两个主要问题几十年来一直阻碍着科学家的努力。第一个称为“阿诺德扩散”，与长时间尺度上动作变量的不稳定性有关。第二个被称为“正度量熵猜想”，它指出混沌运动是物理相关的，即它们占据正勒贝格测度的子集。在奖学金期间，我将集中研究具有多个时间尺度的哈密顿系统，因为这个属性通常显式或隐式地出现在方程中。本研究的目的是开发用于研究动力学不稳定性的数学工具。初步结果表明，我们能够证明具有不同慢分量动力学行为的正双曲不变物体的存在。在较长的时间尺度上，动力系统对该族的限制很可能可以通过慢变量中的随机常微分方程来近似。如果得到证实，它将在两个不同的数学领域之间建立重要的联系：确定性哈密顿系统理论和随机微分方程，目前这两个领域被认为大多不相关。这些结果的扩展应该为费米加速理论提供新的见解。与 Mather（普林斯顿）宣布的 Arnold 扩散变分方法的比较表明，我们的机制可用于解决近可积哈密顿系统中 Arnold 扩散的通用性问题。这个方向的进展应该需要对与各种不变物体相关的横向同宿轨迹的检测方法进行根本性的改进，即在我拥有大量技术专长的领域。总而言之，首先将解决以下技术主题列表：高维不变流形的指数小分裂、具有混沌快分量的慢快系统中慢动力学的随机描述、费米加速、阿诺德扩散、正度量熵猜想。我预计，随着该奖学金的进展，新的研究方向将会出现，部分是由理论的发展推动的，部分是由其应用产生的问题推动的。这项研究将在华威大学数学研究所进行，并将涉及与英国和监督机构的几个小组的合作。