Parameter conditions for the existence of homoclinic orbits in the Lorenz equations.

Persistent Link:
http://hdl.handle.net/10150/186966
Title:
Parameter conditions for the existence of homoclinic orbits in the Lorenz equations.
Author:
Yang, Chun-Woo.
Issue Date:
1994
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:
In this paper, we suggest some sufficient conditions for the existence of homoclinic orbits of the origin in the Lorenz equations. Such conditions are given by countably many bifurcation curves in the parameter space. For this purpose, we use some rigorous perturbation methods for the equivalent Lorenz system x' = k(y-x), y' = x(1-z) – εy, z' = xy – εbz; (k > 0, b > 0, 0 < ε ≪ 1) where we vary ε → 0 while k and b being fixed. For a local analysis of the unstable orbits, we study a cylindrical pendulum system and apply the "continuous dependence on initial data" theorem. For a global analysis of the solutions of Lorenz system, we use a perturbation theory for nonlinear systems of differential equations. The general procedure of perturbation theory is to insert a small parameter δ, 0 ≤ δ ≤ 1, such that when δ = 0 the problem is soluble. In this process, standard Bessel equation and Bessel functions will play some important roles for the representation of the solutions of the Lorenz system. For the asymptotic solutions of the linearized Lorenz system, WKB-theory will be useful.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Rychlik, Marek R.

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleParameter conditions for the existence of homoclinic orbits in the Lorenz equations.en_US
dc.creatorYang, Chun-Woo.en_US
dc.contributor.authorYang, Chun-Woo.en_US
dc.date.issued1994en_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.abstractIn this paper, we suggest some sufficient conditions for the existence of homoclinic orbits of the origin in the Lorenz equations. Such conditions are given by countably many bifurcation curves in the parameter space. For this purpose, we use some rigorous perturbation methods for the equivalent Lorenz system x' = k(y-x), y' = x(1-z) – εy, z' = xy – εbz; (k > 0, b > 0, 0 < ε ≪ 1) where we vary ε → 0 while k and b being fixed. For a local analysis of the unstable orbits, we study a cylindrical pendulum system and apply the "continuous dependence on initial data" theorem. For a global analysis of the solutions of Lorenz system, we use a perturbation theory for nonlinear systems of differential equations. The general procedure of perturbation theory is to insert a small parameter δ, 0 ≤ δ ≤ 1, such that when δ = 0 the problem is soluble. In this process, standard Bessel equation and Bessel functions will play some important roles for the representation of the solutions of the Lorenz system. For the asymptotic solutions of the linearized Lorenz system, WKB-theory will be useful.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairRychlik, Marek R.en_US
dc.contributor.committeememberTabor, Michaelen_US
dc.contributor.committeememberWojtkowski, Maciej P.en_US
dc.identifier.proquest9517577en_US
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