INSTABILITIES IN DYNAMICAL SYSTEMS

本课题将研究动力系统的不稳定性。在动力系统中，研究的是一个系统服从某些固定规则而随时间演化的方式。确定性动力系统的长期行为可能是混乱和不可预测的。这种行为被称为“不稳定”。众所周知的蝴蝶效应就是一个例子，它对初始条件有一种敏感的依赖性，在这种依赖性中，系统初始状态的微小变化可能导致后来状态的巨大差异。提出的研究将研究在经典力学和广义相对论中产生的系统的不稳定性。目标是了解不稳定性是如何发生的，并量化其性质。另一个目标是基于对不稳定机制的新理解来发现未知现象。拟议的项目分为以下三个不同的领域。第一个项目是用辛方法研究哈密顿系统。这将是以前工作的继续。它包括在非凸哈密顿系统中寻找满足一定拓扑约束的周期轨道和同斜轨道或异斜轨道。下一个项目是尝试用证明阿诺德扩散的方法来研究广义相对论中的系统。我们感兴趣的是在摄动Kerr-de Sitter度规中展示Arnold扩散，Arnold扩散在这种情况下的物理意义就是从黑洞中提取能量和角动量的Penrose过程。在最后一个项目中，PI和他的合作者将使用一个没有哈密顿结构的一般动力系统，在小随机扰动的帮助下，在具体系统中产生正李雅普诺夫指数。特别地，对于二维的随机摄动的标准映射，该方法显示出正的李雅普诺夫指数。

In this project instabilities in dynamical systems will be studied. In dynamical systems the way in which a system obeying some fixed rules evolves over time is studied. The long term behavior of a deterministic dynamical system may be chaotic and unpredictable. Such behavior is termed "instability". An example is the well-known butterfly effect, where there is a sensitive dependence on the initial conditions in which a small change of the initial state of a system may lead to huge differences in later states. The proposed research will study instabilities of systems arising in classical mechanics and general relativity. The goal is to understand how the instability occurs and to quantify its properties. Another goal is to discover unknown phenomena based on a new understanding of instability mechanisms.The proposed projects fall into the following three different fields. The first project is to study Hamiltonian systems using symplectic methods. This will be a continuation of previous work. It includes finding periodic orbits and homoclinic or heteroclinic orbits satisfying certain topological constraints in non-convex Hamiltonian systems. The next project is to try to use the methods of proving Arnold diffusion to study systems in general relativity. The interest is in showing Arnold diffusion in a perturbed Kerr-de Sitter metric and the physical meaning of Arnold diffusion in this setting is the Penrose process for energy and angular momentum extraction from black hole. In the last project the PI and his collaborators will use a general dynamical system without Hamiltonian structure to produce positive Lyapunov exponents in concrete systems with the help of small random perturbations. In particular, for two dimensions the methods show positive Lyapunov exponents for a randomly perturbed standard map.