A classification of the rational integrable generalized standard maps

Persistent Link:
http://hdl.handle.net/10150/282501
Title:
A classification of the rational integrable generalized standard maps
Author:
Torgerson, Mark Dolan, 1964-
Issue Date:
1997
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
This dissertation examines the integrability of certain planar maps. An integral of a planar map is a real-valued function that remains invariant under composition with the map. In particular we classify the generalized standard maps, maps of the form φ(x, y) = (f(x)-y, x), that have a polynomial integral and those that have a rational integral. Chapter 1 brings to light the pertinent definitions. Here we show numerical examples of integrability, non-integrability and questionable integrability. Chapter 1 concludes with a discourse on linear maps, giving a classification of those with a polynomial or rational integral. The meat of this work lies in Chapter 2, which contains the classification of the generalized standard maps. Here we give a short list of the maps that have a polynomial integral as well as an integral for each case. Finally we show that if a generalized standard map has a rational integral then it has a polynomial integral. Deciding if a planar map is integrable or deciding what family the integral lies in is one of the difficult tasks assumed by the field of dynamical systems. Here we have approached the integrability question from an algebraic viewpoint. Some attempt has been made to make the proofs simple yet complete, without using the heavy-duty analytical tools that can often be found in the analysis of a dynamical systems.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Rychlik, Marek R.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleA classification of the rational integrable generalized standard mapsen_US
dc.creatorTorgerson, Mark Dolan, 1964-en_US
dc.contributor.authorTorgerson, Mark Dolan, 1964-en_US
dc.date.issued1997en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis dissertation examines the integrability of certain planar maps. An integral of a planar map is a real-valued function that remains invariant under composition with the map. In particular we classify the generalized standard maps, maps of the form φ(x, y) = (f(x)-y, x), that have a polynomial integral and those that have a rational integral. Chapter 1 brings to light the pertinent definitions. Here we show numerical examples of integrability, non-integrability and questionable integrability. Chapter 1 concludes with a discourse on linear maps, giving a classification of those with a polynomial or rational integral. The meat of this work lies in Chapter 2, which contains the classification of the generalized standard maps. Here we give a short list of the maps that have a polynomial integral as well as an integral for each case. Finally we show that if a generalized standard map has a rational integral then it has a polynomial integral. Deciding if a planar map is integrable or deciding what family the integral lies in is one of the difficult tasks assumed by the field of dynamical systems. Here we have approached the integrability question from an algebraic viewpoint. Some attempt has been made to make the proofs simple yet complete, without using the heavy-duty analytical tools that can often be found in the analysis of a dynamical systems.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorRychlik, Marek R.en_US
dc.identifier.proquest9814388en_US
dc.identifier.bibrecord.b37742036en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.