{"id":98541,"date":"2010-12-01T00:00:00","date_gmt":"2010-12-01T00:00:00","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/analytic-and-numerical-tools-for-the-study-of-quasi-periodic-motions-in-hamiltonian-systems\/"},"modified":"2010-12-01T00:00:00","modified_gmt":"2010-12-01T00:00:00","slug":"analytic-and-numerical-tools-for-the-study-of-quasi-periodic-motions-in-hamiltonian-systems","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/analisis-numerico\/analytic-and-numerical-tools-for-the-study-of-quasi-periodic-motions-in-hamiltonian-systems\/","title":{"rendered":"Analytic and numerical tools for the study of quasi-periodic motions in hamiltonian systems"},"content":{"rendered":"<h2>Tesis doctoral de <strong> Alejandro Luque Jim\u00e9nez <\/strong><\/h2>\n<p>It is well-known that quasi-periodic solutions play a relevant role in order to understand the dynamics of  problems with hamiltonian formulation, which appear in a wide set of applications in astrodynamics, molecular  dynamics, beam\/plasma physics or celestial mechanics.    roughly speaking, we can say that kam theory gathers a collection of techniques and methodologies to study  quasi-periodic solutions (that is, functions depending on a set of frequencies) of differential equations typically  with hamiltonian formulation. Although kam theory is well-known (see [1]), classical methods present  shorcomings and difficulties in order to apply the abstract results to concret examples or models. Nevertheless,  in [2] a new method was developed, without using action-angle variables, which allows us avoid most of the  shortcomings of classical methods. This method was introduced for tori of maximal dimension and there is a  current interest in extending it to other contexts, such us the study of non-twist tori in [4] or normally hyperbolic  tori in [3]. One of the goals of this thesis has been to adapt this method to deal with elliptic lower dimensional  tori. The  additional technical difficulties are related to resonances between the basic frequencies of the tori and the  oscillations  in  the normal directions,  which are characterized by means of reducibility in order to obtain the  geometric properties that we require in the proof.    furthermore, in order to study quasi-periodic invariant tori, valuable information is obtained from the frequency  vector that characterizes the motion. Part of the work in this thesis has been to develop efficient numerical  methods for the study of one dimensional quasi-periodic motions in a wide set of contexts. Our methodology is  an extension of a recently developed approach to compute rotation numbers of circle maps (see [5]) based on  suitable averages of iterates of the map. On the one hand, the ideas of [5] have been adapted to compute  derivatives of the rotation number for parametric families of circle diffeomorphisms, thus obtaining powerful  tools (for example, we can implement newton-like methods) for the study of arnold tongues and invariant  curves for twist maps, if we can build a circle map using suitable coordinates.  On the other hand, we have  developed a solidly justified method that allows us to avoid the practical difficulty of looking for these  coordinates, thus extending the methods to more general contexts such as non-twist maps or quasi-periodic  signals.    [1] r. De la llave. A tutorial on kam theory.  In smooth ergodic theory and its applications, volume 69 of proc.  sympos. Pure math., Pages 175-292. Amer.  Math. Soc., 2001.    [2] r. De la llave, a. Gonz\u00ed\u00a0lez, \u00ed\u00a0. Jorba, and j. Villanueva.  Kam theory without action-angle variables.  nonlinearity, 18(2):855-895, 2005.    [3] e. Fontich, r. De la llave, and y. Sire. Construction of invariant whiskered tori by a parametrization method.  part i: maps and flows in finite dimensions. J. Differential equations, 246:3136-3213, 2009.    [4] r. De la llave , a. Gonz\u00e1lez and a haro. Non-twist kam theory. In preparation.    [5] t.M. Seara and j. Villanueva. On the numerical computation of diophantine rotation numbers of analytic  circle maps. Phys. D, 217(2):107-120, 2006.<\/p>\n<p>&nbsp;<\/p>\n<h3>Datos acad\u00e9micos de la tesis doctoral \u00ab<strong>Analytic and numerical tools for the study of quasi-periodic motions in hamiltonian systems<\/strong>\u00ab<\/h3>\n<ul>\n<li><strong>T\u00edtulo de la tesis:<\/strong>\u00a0 Analytic and numerical tools for the study of quasi-periodic motions in hamiltonian systems <\/li>\n<li><strong>Autor:<\/strong>\u00a0 Alejandro Luque Jim\u00e9nez <\/li>\n<li><strong>Universidad:<\/strong>\u00a0 Polit\u00e9cnica de catalunya<\/li>\n<li><strong>Fecha de lectura de la tesis:<\/strong>\u00a0 12\/01\/2010<\/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>Jordi Villanueva Castelltort<\/li>\n<\/ul>\n<\/li>\n<li><strong>Tribunal<\/strong>\n<ul>\n<li>Presidente del tribunal: carles Sim\u00f3 torres <\/li>\n<li>angel Jorba montes (vocal)<\/li>\n<li>kaloshin Vadim yu (vocal)<\/li>\n<li>Rafael De la llave canaso (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tesis doctoral de Alejandro Luque Jim\u00e9nez It is well-known that quasi-periodic solutions play a relevant role in order to understand 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