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dc.contributor.authorYang, Chun-Woo.
dc.creatorYang, Chun-Woo.en_US
dc.date.accessioned2011-10-31T18:25:11Z
dc.date.available2011-10-31T18:25:11Z
dc.date.issued1994en_US
dc.identifier.urihttp://hdl.handle.net/10150/186966
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.
dc.language.isoenen_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.titleParameter conditions for the existence of homoclinic orbits in the Lorenz equations.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.contributor.chairRychlik, Marek R.en_US
thesis.degree.grantorUniversity of Arizonaen_US
thesis.degree.leveldoctoralen_US
dc.contributor.committeememberTabor, Michaelen_US
dc.contributor.committeememberWojtkowski, Maciej P.en_US
dc.identifier.proquest9517577en_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.namePh.D.en_US
dc.description.noteThis item was digitized from a paper original and/or a microfilm copy. If you need higher-resolution images for any content in this item, please contact us at repository@u.library.arizona.edu.
dc.description.admin-noteOriginal file replaced with corrected file November 2023.
refterms.dateFOA2018-08-23T18:14:45Z
html.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.


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