Considerations of the dynamics of the Poincaré map acting on the attractor of the Newton-Leipnik system

Author

Benjamin Arthur Marlin

Abstract

"This dissertation is an exploration of the dynamics of a non-linear system of ordinary differential equations known as the Newton-Leipnik equations. This system exhibits unusual behavior in its phase space which indicates the presence of a pair of strange attractors.

The Ruelle plot is used to analyze the action of the Poincaré map on an invariant set. Through the use of the Unstable Manifold Theorem, we are able to say that such an invariant set will be present in a neighborhood of a periodic orbit for the Poincaré map. By paralleling numerical work done for the Lorenz attractor, we have been able to establish the existence of a period 2 point for the Poincaré map for the parameter value α = 0.187. Some explorations are made for possible periodic orbits for other parameter values. At α = 0.190 it is established that a period 4 point cannot exist in a region where one would appear to be indicated. At α = 0.156 it is shown that our current techniques simply do not suffice to either confirm or deny the existence of the period 6 point which is strongly indicated by the Ruelle plot technique.

A possible model of the dynamics of the Poincaré map restricted to this invariant set has been proposed by LoFaro. We discuss some of the dynamical properties of this model, concentrating on its' bifurcations. A second model -- a piecewise linear symmetric bimodal map with symmetric slopes -- is proposed, and its dynamics are considered. The concentration with this model, however, is the nature of the inverse limits which arise for different parameter values. A result of Holte establishes a correspondence between these inverse limits and the invariant set of the Poincaré map.

Finally, we consider a slight alteration of the piecewise linear symmetric bimodal map with only partial symmetry in the slopes that seems to be more strongly indicated by the conjugacy technique proposed by LoFaro"--Abstract, page iii.

Advisor(s)

Roe, Robert Paul

Committee Member(s)

Hicks, Troy L.
Ingram, W. T. (William Thomas), 1937-
Stanojevic, Caslav V., 1928-2008
Stutts, Daniel S.

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

University of Missouri--Rolla

Publication Date

Summer 2000

Pagination

vii, 82 pages

Note about bibliography

Includes bibliographical references (page 81).

Rights

© 2000 Benjamin Arthur Marlin, All rights reserved.

Document Type

Dissertation - Restricted Access

File Type

text

Language

English

Thesis Number

T 7785

Print OCLC #

45965329

Electronic OCLC #

1112606739

Recommended Citation

Marlin, Benjamin Arthur, "Considerations of the dynamics of the Poincaré map acting on the attractor of the Newton-Leipnik system" (2000). Doctoral Dissertations. 1367.
https://scholarsmine.mst.edu/doctoral_dissertations/1367