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Stability and Instability in Conservative Dynamical Systems

保守动力系统的稳定性和不稳定性

基本信息

批准号：
2101464
负责人：
Bassam Fayad
金额：
$ 31.24万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101464&HistoricalAwards=false
关键词：
Stability Instability Conservative Dynamical Systems

项目摘要

The theory of dynamical systems seeks to describe the behavior of systems that evolve with time, such as the motion of planets or of gas particles in the air. Dynamical systems play an important role in mathematics, as well as having numerous applications in physics, biology, computer science and other sciences. Their range of application is growing, and will continue to grow given the widespread use of mathematical models by scientists and engineers. A modern view of dynamical systems contains three classes, called elliptic, parabolic and hyperbolic, depending on the increasing sensitivity to initial conditions of the system. The principal investigator will study this classification, with particular interest paid to the interconnections between the three classes and their interactions with various areas of mathematics and other sciences. The large span of the project provides rich training opportunities for graduate students. In addition, there are likely to be successful transfers of methodologies from one field to another in the theory of dynamical systems. Elliptic dynamics often refers to recurrent behavior in dynamical systems that is at the other end of the spectrum from chaotic dynamics. The systems with stable asymptotic behavior that are best represented by quasi-periodic motions on tori, and which appear for example in KAM theory, are within elliptic dynamics. But instability is also possible in elliptic dynamics, due to the so-called Liouville phenomena, where the existence of fast periodic approximations may be the source of very complex ergodic behavior. Since chaotic dynamics, that is best represented by hyperbolic dynamical systems, is associated with exponential growth of orbit complexity, one may consider slow growth of various characteristics of such complexity as a hallmark of elliptic dynamics. In between the elliptic and the hyperbolic world lie the so-called parabolic dynamical systems, that are best represented by unipotent actions on homogeneous spaces. The polynomial shear or rate of separation between orbits of these flows makes their ergodic theory very special. The goal of this project is to push forward the study of these three paradigms, as well as to explore the interconnections between them. Some main directions are: KAM stability results beyond the classical theory; robust unstable behavior in analytic Hamiltonian dynamics; approximation by the conjugation method beyond its usual limits; extension of Ratner theory to non-algebraic parabolic flows; developing a KAM rigidity theory for higher rank parabolic actions; developing limit laws for higher rank hyperbolic actions and exploiting them in a systematic approach to Diophantine approximation theory; counting problems from a statistical point of view.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动力系统理论试图描述随时间演化的系统的行为，例如行星或空气中气体粒子的运动。动力系统在数学中扮演着重要的角色，在物理学、生物学、计算机科学和其他科学中也有着广泛的应用。它们的应用范围正在扩大，并将继续扩大，因为科学家和工程师广泛使用数学模型。动力系统的现代观点包含三类，称为椭圆，抛物和双曲，这取决于系统对初始条件的敏感性增加。主要研究者将研究这种分类，特别关注这三类之间的相互联系及其与数学和其他科学各个领域的相互作用。项目跨度大，为研究生提供了丰富的培训机会。此外，在动力系统理论中，很可能会成功地将方法从一个领域转移到另一个领域。椭圆动力学通常是指动力系统中的循环行为，它与混沌动力学处于光谱的另一端。具有稳定渐近行为的系统，最好用环面上的准周期运动来表示，例如出现在KAM理论中，都属于椭圆动力学。但不稳定性在椭圆动力学中也是可能的，这是由于所谓的刘维尔现象，其中快速周期近似的存在可能是非常复杂的遍历行为的来源。由于混沌动力学，这是最好的代表双曲动力系统，是与指数增长的轨道复杂性，人们可以考虑缓慢增长的各种特征，这种复杂性作为椭圆动力学的标志。在椭圆和双曲世界之间是所谓的抛物动力系统，它最好用齐次空间上的幂幺作用来表示。这些流的轨道之间的多项式剪切或分离率使得它们的遍历理论非常特殊。本项目的目标是推动这三种范式的研究，并探索它们之间的相互联系。一些主要方向是：KAM稳定性结果超出了经典理论;在解析哈密顿动力学的鲁棒不稳定行为;近似共轭方法超出其通常的限制;拉特纳理论的扩展到非代数抛物流;发展一个KAM刚性理论的高阶抛物行动;发展极限法律的高阶双曲行动和利用他们在一个系统的方法丢番图近似理论;这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。