Analytic and numerical tools for the study of quasi-periodic motions in hamiltonian systems

• Autores: Alejandro Luque Jiménez
• Directores de la Tesis: Jordi Villanueva Castelltort (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2010
• Idioma: inglés
• Tribunal Calificador de la Tesis: Carles Simó (presid.) , Amadeu Delshams i Valdés (secret.) , Rafael de la Llave Canosa (voc.) , Àngel Jorba i Monte (voc.) , Kaloshin Vadim Yu (voc.)
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • 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. Gonzàlez, À. 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. González 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.