Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.

In this thesis we will consider a nonautonomous linear system which admits nonuniform contraction and a nonlinear perturbation bounded at the origin. We search to establish an equivalence between the solutions of the systems before mentioned, being our main objective to construct a topological equiv...

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Detalles Bibliográficos
Autor: Huerta-Navarro, Ignacio Roberto
Tipo de recurso: tesis doctoral
Estado:Versión publicada
Fecha de publicación:2019
País:Chile
OAI Identifier:oai:repositorio.anid.cl:10533/252989
Acceso en línea:https://hdl.handle.net/10533/252989
Access Level:acceso abierto
Palabra clave:Ciencias Naturales
Matemáticas
Matemáticas Puras
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dc.title.es_CL.fl_str_mv Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
Visión espectral del Teorema de Hartman-Grobman para flujos hiperbólicos no uniforme y no acotados.
title Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
spellingShingle Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
Huerta-Navarro, Ignacio Roberto
Ciencias Naturales
Matemáticas
Matemáticas Puras
title_short Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
title_full Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
title_fullStr Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
title_full_unstemmed Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
title_sort Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
dc.creator.none.fl_str_mv Huerta-Navarro, Ignacio Roberto
author Huerta-Navarro, Ignacio Roberto
author_facet Huerta-Navarro, Ignacio Roberto
author_role author
dc.contributor.advisor.none.fl_str_mv Rivas-Espinosa, Cristobal
Castañeda-González, Álvaro
dc.contributor.institution.es_CL.fl_str_mv UNIVERSIDAD DE SANTIAGO DE CHILE
dc.subject.oecd1n.es_CL.fl_str_mv Ciencias Naturales
topic Ciencias Naturales
Matemáticas
Matemáticas Puras
dc.subject.oecd2n.es_CL.fl_str_mv Matemáticas
dc.subject.oecd3n.es_CL.fl_str_mv Matemáticas Puras
description In this thesis we will consider a nonautonomous linear system which admits nonuniform contraction and a nonlinear perturbation bounded at the origin. We search to establish an equivalence between the solutions of the systems before mentioned, being our main objective to construct a topological equivalence between them. In order to obtain such a result we review the spectral theory associated with the nonuniform hyperbolicity, specifically we consider nonuniform exponential dichotomy of the linear system. In addition we highlight some fundamental results of the spectral theory such as: i) under certain hypotheses the nonuniform spectrum of the linear system can be written as the finite union of compact intervals, ii) the linear system is equivalent by means of a nonuniform kinematic similarity to a new linear system composed of blocks, where the spectrum of each of these blocks corresponds to one of the connected components of the original linear system spectrum. On the other hand and thanks to the aforementioned spectral theory, we will show that the initial linear system is nonuniformly contracted to its spectrum when it is nonuniformly kinemically similar to a linear system composed of the sum of one diagonal matrix, where its elements are functions whose images belong to the spectrum, and a matrix whose norm can be chosen sufficiently small. We will call this property almost nonuniform reducibility. Finally, if the linear system admits nonuniform contraction, we use the previous results combined with Lyapunov's theory of functions to establish the existence of homeomorphism that relates the solutions of both systems.
publishDate 2019
dc.date.issued.es_CL.fl_str_mv 2019
dc.date.accessioned.none.fl_str_mv 2021-10-05T18:37:16Z
2022-08-19T17:11:46Z
dc.date.available.none.fl_str_mv 2021-10-05T18:37:16Z
2022-08-19T17:11:46Z
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dc.type.tesis.none.fl_str_mv Tesis
format doctoralThesis
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spelling UNIVERSIDAD DE SANTIAGO DE CHILEHuerta-Navarro, Ignacio Roberto2019https://hdl.handle.net/10533/252989http://purl.org/coar/access_right/c_abf2Matemáticas PurasMatemáticasCiencias NaturalesSpectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.Rivas-Espinosa, CristobalCastañeda-González, ÁlvaroUNIVERSIDAD DE SANTIAGO DE CHILEChileHuerta-Navarro, Ignacio Roberto2021-10-05T18:37:16Z2022-08-19T17:11:46Z2021-10-05T18:37:16Z2022-08-19T17:11:46Z2019In this thesis we will consider a nonautonomous linear system which admits nonuniform contraction and a nonlinear perturbation bounded at the origin. We search to establish an equivalence between the solutions of the systems before mentioned, being our main objective to construct a topological equivalence between them. In order to obtain such a result we review the spectral theory associated with the nonuniform hyperbolicity, specifically we consider nonuniform exponential dichotomy of the linear system. In addition we highlight some fundamental results of the spectral theory such as: i) under certain hypotheses the nonuniform spectrum of the linear system can be written as the finite union of compact intervals, ii) the linear system is equivalent by means of a nonuniform kinematic similarity to a new linear system composed of blocks, where the spectrum of each of these blocks corresponds to one of the connected components of the original linear system spectrum. On the other hand and thanks to the aforementioned spectral theory, we will show that the initial linear system is nonuniformly contracted to its spectrum when it is nonuniformly kinemically similar to a linear system composed of the sum of one diagonal matrix, where its elements are functions whose images belong to the spectrum, and a matrix whose norm can be chosen sufficiently small. We will call this property almost nonuniform reducibility. Finally, if the linear system admits nonuniform contraction, we use the previous results combined with Lyapunov's theory of functions to establish the existence of homeomorphism that relates the solutions of both systems.En esta tesis consideraremos un sistema lineal no autónomo el cual admite contracción no uniforme y una perturbación no lineal acotada en el origen. Buscamos establecer una equivalencia entre las soluciones de los sistemas anteriormente mencionados, siendo nuestro principal objetivo construir una equivalencia topológica entre ellos. Con la finalidad de obtener tal resultado repasamos la teoría espectral asociada a la hiperbolicidad no uniforme, específicamente la dicotomía exponencial no uniforme del sistema lineal. Además destacamos algunos resultados fundamentales de la teoría espectral tales como: i) bajo ciertas hipótesis el espectro no uniforme del sistema lineal se puede escribir como la unión finita de intervalos compactos, ii) el sistema lineal es equivalente por medio de una similaridad cinemática no uniforme a un nuevo sistema lineal compuesto por bloques, donde el espectro de cada uno de esos bloques corresponde a una de las componentes conexas del espectro del sistema lineal original. Por otro lado y gracias a la teoría espectral anteriormente mencionada, el sistema lineal inicial es no uniformemente contra\'ido a su espectro cuando es no uniformemente cinemáticamente similar a un sistema lineal compuesto por la suma de una matriz diagonal, donde sus elementos son funciones cuyas imágenes pertenecen al espectro, y una matriz cuya norma puede ser escogida suficientemente pequeña. A esta propiedad la llamaremos casi reducibilidad no uniforme. Finalmente, si el sistema lineal admite contracción no uniforme, usamos los resultados anteriores combinado con la teoría de funciones de Lyapunov para establecer la existencia del homeomorfismo que relaciona las soluciones de ambos sistemas.21150270https://hdl.handle.net/10533/252989instname: Conicytreponame: Repositorio Digital RI2.0info:eu-repo/grantAgreement//21150270info:eu-repo/semantics/dataset/hdl.handle.net/10533/93488chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/viewer.html?pdfurl=http%3A%2F%2Fwww.doctmat.dmcc.usach.cl%2Fimages%2FTesis_Ignacio_Huerta.pdf&clen=576360&chunk=trueinfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-nd/3.0/cl/Ciencias NaturalesMatemáticasMatemáticas PurasSpectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.Visión espectral del Teorema de Hartman-Grobman para flujos hiperbólicos no uniforme y no acotados.info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionTesisTesishttps://hdl.handle.net/10533/2529891a586501-9acc-471a-bde5-9db91b75aaf4virtual::64893-11a586501-9acc-471a-bde5-9db91b75aaf4virtual::64893-1THUMBNAILTesis Ignacio Huerta.pdf.jpgIM Thumbnailimage/jpeg3369https://repositorio.anid.cl/bitstreams/907105e5-cdb1-4e72-bab8-0183311e9d70/downloadb45d7bddb3a065ac7cdc0e7b2485387cMD51TEXTTesis Ignacio Huerta.pdf.txtExtracted texttext/plain155653https://repositorio.anid.cl/bitstreams/29aa3b3d-9033-4aed-b2aa-91eee1fc8ba1/download3c0a7bcb77a684ba91e523eca17e4b8fMD52LICENSElicense.txttext/plain1779https://repositorio.anid.cl/bitstreams/20e073b7-e248-478b-8d04-72be5efd3c87/download593a6e7305c66c56041a9f9e15a649c1MD53CC-LICENSElicense_rdfapplication/octet-stream1232https://repositorio.anid.cl/bitstreams/97864020-e19e-4d02-8890-dd91b6108f6e/downloadf97bcfdf58f3e17b5cec231112dab5b1MD54ORIGINALTesis Ignacio Huerta.pdfTesis completaapplication/pdf604346https://repositorio.anid.cl/bitstreams/976cbf9d-7ee2-423e-bd73-d389c348dda1/downloadb165e433de9d00bf686ea859d457a66aMD5510533/252989oai:repositorio.anid.cl:10533/2529892023-07-24 07:37:38.713http://creativecommons.org/licenses/by-nc-nd/3.0/cl/info:eu-repo/semantics/openAccesshttps://repositorio.anid.clRepositorio ANIDaletelier@anid.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