Unstable Dynamics in Hamiltonian Systems

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系统中的混沌动力学，该系统结合了规则和混沌分量，因为与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。第一个是所谓的“阿诺德扩散”，是与长期尺度上的行动变量的不稳定性。第二个被称为“正度量熵猜想”，并指出混沌运动是物理相关的，即，它们占据了正Lebesgue测度的一个子集。在研究期间，我将专注于研究具有多个时间尺度的Hamilton系统，因为这个性质经常显式或隐式地存在于方程中。本研究的目的是发展研究动力学不稳定性的数学工具。初步结果表明，我们能够证明存在正常双曲不变对象的慢组件不同的动力学行为。在较长的时间尺度上，动力系统对这个族的约束可能可以用慢变量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一个重要的联系：确定性哈密顿系统理论和随机微分方程，这两个领域目前被认为是不相关的。这些结果的推广将为费米加速理论提供新的见解。与Mather（Princeton）宣布的Arnold扩散的变分方法的比较表明，我们的机制可以用来解决长期存在的问题，在近可积哈密顿系统中的Arnold扩散的通用性。在这个方向上的进展应该需要从根本上改进用于检测与各种不变对象相关联的横向同宿轨迹的方法，即，在我拥有丰富的技术专长的领域。作为总结，下面的技术主题列表将首先讨论：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，随着研究的进展，新的研究方向将出现部分由理论的发展，部分由来自该研究将在沃里克大学数学研究所进行，并将与英国的几个小组合作，并监督。

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.