INSTABILITIES IN DYNAMICAL SYSTEMS

在这个项目中，将研究动力系统中的不稳定性。 在动力系统中，人们研究的是一个遵循某些固定规则的系统随时间演化的方式。确定性动力系统的长期行为可能是混沌的和不可预测的。 这种行为被称为“不稳定性”。一个例子是众所周知的蝴蝶效应，其中存在对初始条件的敏感依赖性，其中系统初始状态的微小变化可能导致后期状态的巨大差异。 拟议的研究将研究经典力学和广义相对论中出现的系统的不稳定性。我们的目标是了解不稳定性是如何发生的，并量化其属性。另一个目标是基于对不稳定性机制的新理解来发现未知现象。 第一个项目是用辛方法研究哈密顿系统。这将是以前工作的延续。 它包括在非凸Hamilton系统中寻找满足某些拓扑约束的周期轨道和同宿或异宿轨道。 下一个项目是尝试使用证明Arnold扩散的方法来研究广义相对论中的系统。 我们的兴趣是在显示Arnold扩散扰动Kerr-de Sitter度规和Arnold扩散的物理意义，在这种设置是从黑洞的能量和角动量提取的彭罗斯过程。在最后一个项目中，PI和他的合作者将使用一个没有Hamilton结构的一般动力系统，在小的随机扰动的帮助下，在具体系统中产生正的Lyapunov指数。 特别是，对于两个维度的随机扰动的标准映射的方法显示出积极的李雅普诺夫指数。

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.