{"id":41428,"date":"1999-01-01T00:00:00","date_gmt":"1999-01-01T00:00:00","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/index-of-vector-fields-on-manifolds-and-isochronicity-for-planar-hamiltonian-differential-systems\/"},"modified":"1999-01-01T00:00:00","modified_gmt":"1999-01-01T00:00:00","slug":"index-of-vector-fields-on-manifolds-and-isochronicity-for-planar-hamiltonian-differential-systems","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/matematicas\/index-of-vector-fields-on-manifolds-and-isochronicity-for-planar-hamiltonian-differential-systems\/","title":{"rendered":"Index of vector fields on manifolds and isochronicity for planar hamiltonian differential systems."},"content":{"rendered":"<h2>Tesis doctoral de <strong> Jordi Villadelprat Yague <\/strong><\/h2>\n<p>Los contenidos de la memoria se enmarcan dentro de la teor\u00eda cualitativa de las ecuaciones diferenciales. el trabajo se divide esencialmente en dos partes que pueden leerse de manera independiente. 6a primera, formada por los cap\u00edtulos 1, 2 y 3, trata cuestiones relacionadas con el \u00edndice de campos vectoriales sobre variedades de dimensi\u00f3n arbitraria. en la segunda, formada por los cap\u00edtulos 4, 5, 6 y 7, se estudian los centros is\u00f3cronos de los sistemas diferenciales hamiltonianos en el plano  en el cap\u00edtulo 1, dado un campo vectorial x sobre una variedad, estudiamos el flujo de x = x(x) y consideramos un atractor compacto k con regi\u00f3n de atracci\u00f3n a. Probamos que el \u00edndice de x en k depende \u00fanicamente de la topolog\u00eda de a y esto nos permite dar la generalizaci\u00f3n natural, desde el punto de vista din\u00e1mico, del teorema de poincar\u00e9-hopf para variedades no compactas. En el cap\u00edtulo 3 obtenemos una formula que permite calcular el \u00edndice de un campo vectorial sobre cualquier superficie compacta.  en la segunda parte de la memoria nos dedicamos a estudiar la isocron\u00eda en los sistemas diferenciales hamiltonianos anal\u00edticos en el plano. Es decir, sistemas de la forma  con h anal\u00edtica. En el cap\u00edtulo 4 probamos que todo centro no degenerado de (1) tiene una normalizaci\u00f3n can\u00f3nica. Como consecuencia obtenemos que un centro es is\u00f3crono si y s\u00f3lo si tiene una linealizaci\u00f3n can\u00f3nica. En el cap\u00edtulo 5 consideramos sistemas potenciales, es decir del tipo (1) con . Mostramos que la isocron\u00eda de un centro esta muy relacionada con la geometr\u00eda de su period annulus y con las involuciones en r. En el cap\u00edtulo 6 estudiamos el sistema (1) con , que es una familia que contiene los sistemas potenciales. Obtenemos caracterizaciones de la isocron\u00eda que son especialmente relevantes en el caso polinomial. Damos tambi\u00e9n los primeros ejemplos de sistemas hamiltonianos polinomiales con uncentro is\u00f3crono no global. finalm<\/p>\n<p>&nbsp;<\/p>\n<h3>Datos acad\u00e9micos de la tesis doctoral \u00ab<strong>Index of vector fields on manifolds and isochronicity for planar hamiltonian differential systems.<\/strong>\u00ab<\/h3>\n<ul>\n<li><strong>T\u00edtulo de la tesis:<\/strong>\u00a0 Index of vector fields on manifolds and isochronicity for planar hamiltonian differential systems. <\/li>\n<li><strong>Autor:<\/strong>\u00a0 Jordi Villadelprat Yague <\/li>\n<li><strong>Universidad:<\/strong>\u00a0 Aut\u00f3noma de barcelona<\/li>\n<li><strong>Fecha de lectura de la tesis:<\/strong>\u00a0 01\/01\/1999<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>Direcci\u00f3n y tribunal<\/h3>\n<ul>\n<li><strong>Director de la tesis<\/strong>\n<ul>\n<li>Anna Cima Mollet<\/li>\n<\/ul>\n<\/li>\n<li><strong>Tribunal<\/strong>\n<ul>\n<li>Presidente del tribunal: jaume Llibre salo <\/li>\n<li>Rafael Ortega rios (vocal)<\/li>\n<li>amadeu Delshams vald\u00e9s (vocal)<\/li>\n<li>Jos\u00e9 angel Rodr\u00edguez m\u00e9ndez (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tesis doctoral de Jordi Villadelprat Yague Los contenidos de la memoria se enmarcan dentro de la teor\u00eda cualitativa de las [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center 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