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EUCLIDEAN EXTENSIONS OF DYNAMICAL SYSTEMS
by
Srideep Musuvathy
A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF ARTS
(MATHEMATICS)
May 2012
Copyright 2012 Srideep Musuvathy
Object Description
| Title | Euclidean extensions of dynamical systems |
| Author | Musuvathy, Srideep S. |
| Author email | srideep.musuvathy@gmail.com;musuvathy@gmail.com |
| Degree | Master of Arts |
| Document type | Thesis |
| Degree program | Mathematics |
| School | College of Letters, Arts And Sciences |
| Date defended/completed | 2011-11-07 |
| Date submitted | 2012-02-06 |
| Date approved | 2012-02-07 |
| Restricted until | 2012-02-07 |
| Date published | 2012-02-07 |
| Advisor (committee chair) | Bonahon, Francis |
| Advisor (committee member) |
Haydn, Nicolai T. A. Jonckheere, Edmond A. |
| Abstract | This thesis studies the special Euclidean group extension of discrete dynamical systems. Given a dynamical system (X, f), the extended dynamics is constructed as the pair (X×Γ, T) where Γ is the special Euclidean group and T is a map on this extended space. The special Euclidean extension is viewed as an Rⁿ extension of a SO(n) extension of the base dynamics. ❧ After a review of dynamical systems theory, dynamical systems are classified based on whether they are periodic or chaotic. Chaos is defined as exponential mixing (equivalent to decay of correlations on observables). The behavior of typical trajectories in the extended dynamics are then analyzed for these classes of dynamical system. If the base dynamics is periodic, then typical trajectories on the extended space are bounded. For chaotic dynamics these trajectories are typically unbounded. ❧ Finally these results are numerically verified for various examples of dynamical systems. The plots of typical trajectories are shown for SE(2) extensions and the growth rates for various extensions are shown and the results discussed. Despite building a more complex map on a larger space (compared to the original dynamics) extensions of dynamical systems allow us to simplify detecting behavior that is difficult to show in the original dynamics. |
| Keyword | chaos; dynamical system; group extensions; mixing dynamics; special Euclidean group |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Contributing entity | University of Southern California |
| Rights | Musuvathy, Srideep S. |
| Physical access | The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@lib.usc.edu |
| Archival file | uscthesesreloadpub_Volume7/etd-MusuvathyS-470.pdf |
Description
| Title | Page 1 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | EUCLIDEAN EXTENSIONS OF DYNAMICAL SYSTEMS by Srideep Musuvathy A Thesis Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (MATHEMATICS) May 2012 Copyright 2012 Srideep Musuvathy |
