Stability and Instability in Conservative Dynamical Systems

动力系统理论试图描述系统随时间演化的行为，例如行星的运动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许多应用。它们的应用范围正在扩大，并且将继续扩大，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括三类，称为椭圆型，抛物线型和双曲型，这取决于系统对初始条件的日益敏感。首席研究员将研究这种分类，特别关注这三个类别之间的相互联系以及它们与数学和其他科学各个领域的相互作用。项目的大跨度为研究生提供了丰富的训练机会。此外，在动力系统理论中，有可能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周期性行为，它处于混沌动力学的另一端。具有稳定渐近行为的系统，最好由环面上的准周期运动来表示，并且出现在例如KAM理论中，属于椭圆动力学。但由于所谓的刘维尔现象，在椭圆动力学中也可能存在不稳定性，其中快速周期近似的存在可能是非常复杂遍历行为的来源。由于混沌动力学，最好的代表是双曲动力系统，与轨道复杂性的指数增长有关，人们可以认为这种复杂性的各种特征的缓慢增长是椭圆动力学的标志。在椭圆世界和双曲世界之间存在着所谓的抛物动力系统，它最好是用齐次空间上的幂偶作用来表示。这些流的多项式剪切或轨道之间的分离率使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。主要方向有：KAM稳定性结果超越经典理论；解析哈密顿动力学的鲁棒不稳定行为用超出通常极限的共轭法逼近；拉特纳理论在非代数抛物流中的推广发展了高阶抛物作用的KAM刚度理论；发展高阶双曲作用的极限律，并在丢番图近似理论的系统方法中加以利用；从统计的角度计算问题。该奖项反映了美国国家科学基金会的法定使命，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。

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.