INSTABILITIES IN DYNAMICAL SYSTEMS

University Of Chicago, Chicago IL

Investigators

Abstract

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.