INSTABILITIES IN DYNAMICAL SYSTEMS

在这个项目中，我们将研究动力系统中的不稳定性。在动力系统中，研究了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”。一个例子是众所周知的蝴蝶效应，它对初始条件有敏感的依赖，在这种情况下，系统初始状态的微小变化可能会导致随后状态的巨大差异。这项拟议的研究将研究经典力学和广义相对论中出现的系统不稳定性。我们的目标是了解这种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领域。第一个项目是用辛方法研究哈密顿系统。这将是先前工作的延续。它包括在非凸哈密顿系统中寻找满足某些拓扑约束的周期轨道和同宿轨或异宿轨。下一个项目是尝试使用证明阿诺德扩散的方法来研究广义相对论中的系统。我们感兴趣的是在扰动的Kerr-de Sitter度规中显示Arnold扩散，在这种情况下Arnold扩散的物理意义是从黑洞中提取能量和角动量的彭罗斯过程。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的Lyapunov指数。特别地，对于二维随机扰动的标准映射，该方法显示了正的Lyapunov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.