Parameter Dependence in Weakly Hyperbolic Dynamical Systems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Abstract: DMS-0245359 PI: D DOLGOPYAT, U Maryland Smooth ergodic theory studies the long-time asymptotics of those autonomous systems which can be described by a finite number of parameters. Both analytic and numerical results suggest that most of systems fall in one of the following classes: elliptic (quasiperiodic) and hyperbolic (exhibiting sensitive dependence on initial conditions). Systems close to elliptic systems are described by the famous Kolmogorov--Arnold--Moser theory. By contrast small perturbations of hyperbolic systems have been analyzed only in a few special cases. This proposal aims at the general perturbation theory for hyperbolic systems. This includes both the description of the perturbed dynamics up to the certain specific times as well as computing equilibrium characteristics of the perturbed system. Because there is sensitive dependence on initial conditions, it is hopeless to deal with perturbations of individual orbits. Rather, using the fact that hyperbolic systems usually have strong stochastic properties, I will study parameter dependence of frequencies of visits to a given domain in the phase space. My previous work on statistical properties of hyperbolic systems is essential to this project. During the recent years it has been understood that many systems we encounter have chaotic behavior. On the other hand there are several models of chaotic systems which can be exactly solved, which can be analyzed mathematically. However it can be asked how much one could trust predictions made on the basis of these models since the specific features which allow us to precisely solve these models might make them in some sense exceptional. In particular since chaotic systems are very sensitive to initial conditions it is conceivable that a slight change in the parameters of the system could drastically change their behavior. The goal of the current project is to describe the conditions when this drastic change does not happen. The main idea is that most chaotic systems are so complicated that changing parameters does not lead to an appearance of a new pattern of behavior but rather to changing slightly probabilities of already existing patterns.