Off-campus UMass Amherst users: To download campus access dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.

Non-UMass Amherst users: Please talk to your librarian about requesting this dissertation through interlibrary loan.

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

Author ORCID Identifier

https://orcid.org/0000-0002-6015-9174

AccessType

Open Access Dissertation

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2021

Month Degree Awarded

September

First Advisor

HongKun Zhang

Subject Categories

Dynamical Systems

Abstract

In this thesis, we address some questions about certain chaotic dynamical systems. In particular, the objects of our studies are chaotic billiards. A billiard is a dynamical system that describe the motions of point particles in a table where the particles collide elastically with the boundary and with each other.

Among the dynamical systems, billiards have a very important position. They are models for many problems in acoustics, optics, classical and quantum mechanics, etc.. Despite of the rather simple description, billiards of different shapes of tables exhibit a wide range of dynamical properties from being complete integrable to chaotic. A very important and also very interesting type of billiards is chaotic (or hyperbolic) billiards. In a hyperbolic billiard system, two nearby trajectories in the phase space can be separated exponentially fast in future.

In the first two Chapters, we prove the Central Limit Theorem and the Almost Sure Invariance Principle for a class of billiard systems with flat points. They are two among the important statistical properties for chaotic systems. In the last chapter, we introduce a random perturbation to a wide class of billiards and prove that even if the original system is completely integrable, the perturbed system can be chaotic even under arbitrarily small random perturbation.

DOI

https://doi.org/10.7275/24405389

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Share

COinS