{"id":115410,"date":"2018-03-11T10:43:53","date_gmt":"2018-03-11T10:43:53","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/ecuaciones-en-diferencias-racionales\/"},"modified":"2018-03-11T10:43:53","modified_gmt":"2018-03-11T10:43:53","slug":"ecuaciones-en-diferencias-racionales","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/murcia\/ecuaciones-en-diferencias-racionales\/","title":{"rendered":"Ecuaciones en diferencias racionales"},"content":{"rendered":"<h2>Tesis doctoral de <strong> Antonio Cascales Vicente <\/strong><\/h2>\n<p>Resumen de la tesis &#8211; castellano  t\u00edtulo: ecuaciones en diferencias racionales autor: antonio cascales vicente director: francisco balibrea gallego  objetivos &quot;\tdemostrar la siguiente conjetura: la sucesi\u00f3n x_{-2}=x_{-1}=x_0=1, x_{k+1}=1\/(x_{k}+x_{k-2}) es asint\u00f3ticamente dos peri\u00f3dica prima. &quot;\t estudiar el concepto de irreducibilidad de una ecuaci\u00f3n en diferencias racional, y aplicarlo a la resoluci\u00f3n de un problema sobre el car\u00e1cter no finalmente positivo de las soluciones de ciertas ecuaciones en diferencias racionales. &quot;\tgeneralizar las definiciones de caos en el sentido de li-yorke y marotto, enunciados en el caso diferenciable, al caso no continuo de las ecuaciones en diferencias racionales. Extender la condici\u00f3n suficiente de marotto para la existencia de caos (presencia de un repulsor de retorno finito) a ecuaciones en diferencias racionales. &quot;\trealizar an\u00e1lisis num\u00e9ricos que justifiquen la siguiente conjetura: los teoremas sobre la presencia de repulsores de retorno finito y \u00f3rbitas homocl\u00ednicas en ecuaciones en diferencias perturbadas son tambi\u00e9n v\u00e1lidos en el contexto racional. &quot;\testudiar los conjuntos prohibidos de ecuaciones en diferencias racionales.  metodolog\u00eda el contenido de la memoria se ha desarrollado en las siguientes etapas:  1.\tUn extenso trabajo de documentaci\u00f3n en el que se ha consultado la literatura sobre los objetivos antes expuestos, de la cual destacamos los siguientes textos:  a)\te. Camouzis y g. Ladas. Dynamics of third-order rational difference equations with open problems and conjectures. Advances in discrete mathematics and applications, v.5. Taylor &amp; francis, 2007.  b)\tf.R. Marotto. Snapback repellers imply chaos in rn . Journal of mathematical analysis and applications, 63:199-223, 1978.  c)\tf.R. Marotto. Perturbation of stable and chaotic difference equations. J. Math. Anal. Appl, (72):716-729, 1979.  d)\tf. J. Palladino. On invariants and forbidden sets. Xiv:1203.2170v2, 2012.  e)\tr. Azizi. Global behaviour of the rational riccati difference equation of order two: the general case. Journal of difference equations and applications, 18:947-961, 2012.  2.\tSimulaci\u00f3n num\u00e9rica utilizando los sistemas algebraicos computacionales wxmaxima 11.08.0 y mathematica 5.0  3.\tUn trabajo creativo en el que se han ido obteniendo los teoremas y conjeturas de la memoria.  4.\tLa presentaci\u00f3n de partes de dicho trabajo en diversas publicaciones y comunicaciones en congresos, como por ejemplo:  a)\tthe difference equation x n?1=1 \/? X n ?X n?2 ? . Nolineal 2010 &#8211; cartagena (spain), june 2010.  b)\ton the difference equation x n?1=1 \/? X n ?X n?2 ? . Css workshop on discrete dynamical systems &#8211; la manga (spain), september 2010  c)\ton solutions of rational difference equations with non positive initial conditions. Vcds, bansk\u00e1 bystrica (slovakia), july 2011  d)\tsnap-back repellers in rational difference equations. Icdea &#8211; barcelona (spain), 22th &#8211; 27th july 2012.  5.\tLa revisi\u00f3n del trabajo por el director mediante medios telem\u00e1ticos y entrevistas frecuentes, durante un periodo de siete a\u00f1os.  conclusiones en la memoria se han conseguido los siguientes resultados: &quot;\tla demostraci\u00f3n de la conjetura sobre la sucesi\u00f3n citada en el objetivo 1. &quot;\tla extensi\u00f3n de la conjetura a condiciones iniciales m\u00e1s generales y la determinaci\u00f3n de la cuenca de atracci\u00f3n del equilibrium positivo de la ecuaci\u00f3n x_{k+1}=1\/(x_k+x_{k-2}) &quot;\tla irreducibilidad de ciertas ecuaciones en diferencias racionales. Tambi\u00e9n se ha encontrado un ejemplo de ecuaci\u00f3n no irreducible que puede ser estudiada mediante una ecuaci\u00f3n irreducible asociada a ella. &quot;\tun teorema general sobre ecuaciones en diferencias racionales no uniformemente finalmente positivas. &quot;\tla extensi\u00f3n, al contexto de las ecuaciones en diferencias racionales, del teorema de marotto sobre caos y repulsores de retorno finito. &quot;\tse han propuesto diversas conjeturas sobre ecuaciones en diferencias racionales obtenidas mediante perturbaciones de modelos unidimensionales ca\u00f3ticos. \u00e9stas se podr\u00edan aplicar a la generalizaci\u00f3n del modelo de competici\u00f3n de especies de hassell  y comins. &quot;\tdichas conjeturas se han apoyado en simulaciones num\u00e9rico-gr\u00e1ficas sobre los modelos anteriores. &quot;\tse ha recopilado la literatura reciente m\u00e1s relevante sobre los conjuntos prohibidos de ecuaciones en diferencias racionales. &quot;\tse han investigado, tanto anal\u00edtica como num\u00e9ricamente, los conjuntos prohibidos de las ecuaciones en diferencias estudiadas en el resto de la memoria.  adem\u00e1s durante el desarrollo del trabajo han ido surgiendo diversas conjeturas y problemas abiertos  recopilados en el \u00faltimo cap\u00edtulo de la memoria.  archena, a 27 de enero de 2014  dissertation summary &#8211; english version  title: rational difference equations author: antonio cascales vicente director: francisco balibrea gallego  objectives the main objectives of the dissertation are: &quot;\tto prove the following conjecture: the sequence x_{-2}=x_{-1}=x_0=1, x_{k+1}=1\/(x_{k}+x_{k-2}) is asymptotically a period-two sequence (with prime period). &quot;\tto study the concept of irreducible rational difference equation, applying it to the resolution of a problem about the non eventually positive character of the solutions of some rational difference equations. &quot;\tto generalize the definitions of li-yorke chaos and marotto chaos to the non continuous framework of rational difference equations, and to extend marotto&apos;s snap-back repeller criterion to rational difference equations. &quot;\tto make numerical analysis in the aim of justify some conjectures about the presence of snap-back repellers or homoclinic orbits in rational difference equations constructed by a perturbation of a one-dimensional model having such trajectories. &quot;\tto study the forbidden sets of rational difference equations.  metodology the steps of our work were the following:  1.\tA wide work of documentation. We have read several books and papers related to the former objectives. For example:  a)\te. Camouzis y g. Ladas. Dynamics of third-order rational difference equations with open problems and conjectures. Advances in discrete mathematics and applications, v.5. Taylor &amp; francis, 2007.  b)\tf.R. Marotto. Snapback repellers imply chaos in rn . Journal of mathematical analysis and applications, 63:199-223, 1978.  c)\tf.R. Marotto. Perturbation of stable and chaotic difference equations. J. Math. Anal. Appl, (72):716-729, 1979.  d)\tf. J. Palladino. On invariants and forbidden sets. Xiv:1203.2170v2, 2012.  e)\tr. Azizi. Global behaviour of the rational riccati difference equation of order two: the general case. Journal of difference equations and applications, 18:947-961, 2012.  2.\tA numerical analysis using the computer algebra systems wxmaxima 11.08.0 and mathematica 5.0  3.\tA creative work to obtain the theorems and conjectures of the dissertation.  4.\tThe presentation of the former work in several articles and congress communications. For example:  a)\tthe difference equation x n?1=1 \/? X n ?X n?2 ? . Nolineal 2010 &#8211; cartagena (spain), june 2010.  b)\ton the difference equation x n?1=1 \/? X n ?X n?2 ? . Css workshop on discrete dynamical systems &#8211; la manga (spain), september 2010  c)\ton solutions of rational difference equations with non positive initial conditions. Vcds, bansk\u00e1 bystrica (slovakia), july 2011  d)\tsnap-back repellers in rational difference equations. Icdea &#8211; barcelona (spain), 22th &#8211; 27th july 2012.  5.\tA seven years review work of the dissertation director, using telematic resources and periodic interviews.   conclusions in the dissertation we have achieved the following results: &quot;\tthe proof of the conjecture cited in objective 1. &quot;\tthe extension of the conjecture to more general initial conditions and the determination of the basin of attraction of the positive equilibrium in equation x_{k+1}=1\/(x_k+x_{k-2}) &quot;\tseveral examples of irredutible rational difference equations and a non-irreducible rational difference equation with and associated reducible one. &quot;\ta general theorem about non uniformly eventually positive rational difference equations. &quot;\tthe extension to rational difference equations of the li-yorke and marotto chaos definitions. The extension of marotto&apos;s snap-back repeller rule to detect chaos in a rational difference equation. &quot;\twe have conjectured several results about the existence of snap-back repellers or homoclinic orbits in rational difference equations obtained by perturbing one-dimensional models. The results could be applied to the hassell-comins species competition model. &quot;\tthe former conjectures have been made subsequent to several numerical analysis of those rational difference equations. &quot;\twe have compiled the recent literature about forbidden sets in rational difference equations, providing a summary of the different techniques for the study of such sets. &quot;\twe have studied analytically and numerically the forbidden sets of the equations in the former chapters of the dissertation.  finally we have compiled in the last chapter the open problems and conjectures produced along the dissertation.  archena, january 27, 2014<\/p>\n<p>&nbsp;<\/p>\n<h3>Datos acad\u00e9micos de la tesis doctoral \u00ab<strong>Ecuaciones en diferencias racionales<\/strong>\u00ab<\/h3>\n<ul>\n<li><strong>T\u00edtulo de la tesis:<\/strong>\u00a0 Ecuaciones en diferencias racionales <\/li>\n<li><strong>Autor:<\/strong>\u00a0 Antonio Cascales Vicente <\/li>\n<li><strong>Universidad:<\/strong>\u00a0 Murcia<\/li>\n<li><strong>Fecha de lectura de la tesis:<\/strong>\u00a0 21\/03\/2014<\/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>Francisco Balibrea Gallego<\/li>\n<\/ul>\n<\/li>\n<li><strong>Tribunal<\/strong>\n<ul>\n<li>Presidente del tribunal: victor manuel Jimenez lopez <\/li>\n<li>v\u00edctor Ma\u00f1osa fernandez (vocal)<\/li>\n<li>Ana Mar\u00eda Cima mollet (vocal)<\/li>\n<li>eduardo Liz marz\u00e1n (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tesis doctoral de Antonio Cascales Vicente Resumen de la tesis &#8211; castellano t\u00edtulo: ecuaciones en diferencias racionales autor: antonio cascales [&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 center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"footnotes":""},"categories":[42614,8235],"tags":[228003,228323,32611,72861,107289,162992],"class_list":["post-115410","post","type-post","status-publish","format-standard","hentry","category-ecuaciones-en-diferencias","category-murcia","tag-ana-maria-cima-mollet","tag-antonio-cascales-vicente","tag-eduardo-liz-marzan","tag-francisco-balibrea-gallego","tag-victor-manosa-fernandez","tag-victor-manuel-jimenez-lopez"],"_links":{"self":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/115410","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/comments?post=115410"}],"version-history":[{"count":0,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/115410\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/media?parent=115410"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/categories?post=115410"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/tags?post=115410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}