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  • Quasi-periodic frequency analysis using averaging-extrapolation methods

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    SIAM journal on applied dynamical systems
    Date of publication: 2014-01-01
    Journal article

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    We present a new approach to the numerical computation of the basic frequencies of a quasi-periodic signal. Although a complete toolkit for frequency analysis is presented, our methodology is better understood as a refinement process for any of the frequencies, provided we have a rough approximation of the frequency that we wish to compute. The cornerstone of this work is a recently developed method for the computation of Diophantine rotation numbers of circle diffeomorphisms, based on suitable averages of the iterates and Richardson extrapolation. This methodology was successfully extended to compute rotation numbers of quasi-periodic invariant curves of planar maps. In this paper, we address the case of a signal with an arbitrary number of frequencies. The most outstanding aspect of our approach is that frequencies can be calculated with high accuracy at a moderate computational cost, without simultaneously computing the Fourier representation of the signal. The method consists in the construction of a new quasi-periodic signal by appropriate averages of phase-shifted iterates of the original signal. This allows us to define a quasi-periodic orbit on the circle in such a way that the target frequency is the rotation frequency of the iterates. This orbit is well suited for the application of the aforementioned averaging-extrapolation methodology for computing rotation numbers. We illustrate the presented methodology with the study of the vicinity of the Lagrangian equilibrium points of the restricted three body problem (RTBP), and we consider the effect of additional planets using a multicircular model.

  • Premi extraordinari de Doctorat UPC

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
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  • A KAM theorem without action-angle variables for elliptic lower dimensional tori

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Nonlinearity
    Date of publication: 2011-02
    Journal article

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    We study elliptic lower dimensional invariant tori of Hamiltonian systems via parametrizations. The method is based on solving iteratively the functional equations that stand for invariance and reducibility. In contrast with classical methods, we do not assume that the system is close to an integrable one nor that it is written in action-angle variables. We only require an approximation of an invariant torus with a fixed vector of basic frequencies and a basis along the torus that approximately reduces the normal variational equations to constant coefficients. We want to highlight that this approach presents many advantages compared with methods which are built in terms of canonical transformations, e.g., it produces simpler and more constructive proofs that lead to more efficient numerical algorithms for the computation of these objects. Such numerical algorithms are suitable to be adapted in order to perform computer assisted proofs.

  • DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES, ASTRODINÁMICA, NEUROCIENCIA Y OTRAS APLICACIONES

     Blazquez Sanz, David; Olive Farre, Maria del Carme; Gomez-Ullate Oteiza, David; Morales Ruiz, Juan Jose; Canalias Vila, Elisabet; Martinez-seara Alonso, Maria Teresa; Baldoma Barraca, Inmaculada Concepcion; Fedorov, Yury; Garcia Taberner, Laura; Guardia Munarriz, Marcel; Guillamon Grabolosa, Antoni; Gonchenko, Marina; Gutiérrez Serrés, Pere; Lazaro Ochoa, Jose Tomas; Luque Jiménez, Alejandro; Martin De La Torre, Pablo; Masdemont Soler, Josep Joaquim; Olle Torner, Maria Mercedes; Pacha Andujar, Juan Ramon; Pantazi, Chara; Puig Sadurni, Joaquim; Ramirez Ros, Rafael; Roldan Gonzalez, Pablo; Villanueva Castelltort, Jordi; Tamarit, Anna; Olivé Farré, Carme; Morales Ruíz, Juan José; Larreal Barreto, Oswaldo José; Huguet Casades, Gemma; Gómez-Ullate Oteiza, David; Acosta Humanez, Primitivo Belen; Benita Bordes, Jose Manuel; Basak, Inna; Blázquez Sanz, David; de la Llave Canosa, Rafael; de la Rosa Ibarra, Abraham; Delshams i Valdes, Amadeu
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    A KAM theorem without action-angle variables for elliptic lower dimensional tori  Open access

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Date: 2010-02-15
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    We study elliptic lower dimensional invariant tori of Hamiltonian systems via parameterizations. The method is based in solving iteratively the functional equations that stand for invariance and reducibility. In contrast with classical methods, we do not assume that the system is close to integrable nor that is written in action-angle variables. We only require an approximation of an invariant torus of fixed vector of basic frequencies and a basis along the torus that approximately reduces the normal variational equations to constant coefficients. We want to highlight that this approach presents many advantages compared with methods which are built in terms of canonical transformations, e.g., it produces simpler and more constructive proofs that lead to more efficient numerical algorithms for the computation of these objects. Such numerical algorithms are suitable to be adapted in order to perform computer ssisted proofs.

  • Analytic and numerical tools for the study of quasi-periodic motions in Hamiltonian Systems  Open access  awarded activity

     Luque Jiménez, Alejandro
    Defense's date: 2010-01-12
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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    És un fet ben conegut que les solucions quasi-periòdiques juguen un paper rellevant a l'hora d'entendre la dinàmica de problemes amb formulació hamiltoniana, els quals apareixen en una gran quantitat d'aplicacions en astrodinàmica, dinàmica molecular, física de d'acceleradors/plasmes o mecànica celest.De forma imprecisa i imcomplerta, hom pot dir que la teoria KAM recull una serie de tècniques i metodologies per estudiar solucions quasi-periòdiques (és a dir, funcions dependents d'un conjunt de freqüències) d'equacions diferencials típicament amb formulació hamiltoniana. Tot i que la teoria KAM és ben coneguda (veure [1]), els mètodes clàssics presenten inconvenients i dificultats a l'hora d'aplicar els resultats abstractes a exemples o models concrets. Nogensmenys, a [2] es va desenvolupar un nou mètode, sense usar transformacions ni coordenades acció-angle, amb el que es poden superar molts dels inconvenients de les tècniques clàssiques. Aquest mètode fou introduit per a tors de dimensió màxima i, en la actualitat, hom considera de gran interés la seva extensió a altres contextos, com ara l'estudi de tors "sense torsió' a [4] o l'estudi de tors de dimensió inferior normalment hiperbòlics a [3]. Un dels objectius d'aquesta tesi doctoral ha estat adaptar aquests mètodes per demostrar l'existència de tors de dimensió inferior normalment el·liptics i reductibles. Les dificultats tècniques que calen superar deriven de les ressonàncies que tenen lloc entre les freqüències internes del tor i les frequències d'oscil·lació de les "direccions normals', que cal caracteritzar (mitjançant reductibilitat) per tal d'obtenir les propietats geomètriques que es fan servir en la demostració.Per altra banda, a l'hora d'estudiar un tor invariant amb dinàmica quasi-periòdica, hom pot obtenir molta informació coneixent el seu vector de freqüències. És per això que el càlcul numèric d'aquests objectes ha esdevingut un tema de molt interés durant els darrers anys i ha portat al desenvolupament de diversos mètodes. Recentment s'ha desenvolupat a [5] un mètode molt eficient per calcular nombres de rotació per aplicacions del cercle. Hom pot identificar aquest problema amb el càlcul de la freqüència d'un tor unidimensional escrit en unes bones coordenades. Bona part de la recerca realitzada en la meva tesi doctoral continua la linea de treball encetada a [5]. Concretament, donada una família paramètrica de difeomorfismes del cercle, aquesta metodología s'ha adaptat en per a calcular derivades del nombre de rotació respecte de paràmetres. Mitjançant aquesta informació hom pot implementar esquemes tipus Newton per calcular corbes invariants. Com s'ha remarcat abans, hom pot aplicar aquestes tècniques a l'estudi de corbes invariants sempre que es pugui construir una aplicació del cercle amb la mateixa dinàmica. A tal efecte, hem desenvolupat un mètode sòlidament justificat que permet evitar la dificultat pràctica de buscar unes bones coordenades pel tor, extenent així els mètodes a contextes més generals com ara aplicacions "sense torsió" o senyals quasi-periodiques.[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.

    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. Theadditional 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.

  • A dynamical systems approach to Bohmian trajectories in a 2D harmonic oscillator

     Borondo Rodríguez, Florentino; Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Wisniacki, Diego Ariel
    Journal of physics A. Mathematical and theoretical
    Date of publication: 2009-11-25
    Journal article

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    Vortices are known to play a key role in the dynamics of the quantum trajectories defined within the framework of the de Broglie–Bohm formalism of quantum mechanics. It has been rigourously proved that the motion of a vortex in the associated velocity field can induce chaos in these trajectories, and numerical studies have explored the rich variety of behaviors that due to their influence can be observed. In this paper, we go one step further and show how the theory of dynamical systems can be used to construct a general and systematic classification of such dynamical behaviors. This should contribute to establish some firm grounds on which the studies on the intrinsic stochasticity of Bohm’s quantum trajectories can be based. An application to the two-dimensional isotropic harmonic oscillator is presented as an illustration.

  • Numerical computation of the asymptotic size of the rotation domain for the Arnold family

     Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi
    Physica. D, Nonlinear phenomena
    Date of publication: 2009-01
    Journal article

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  • Numerical computation of rotation numbers of quasi-periodic planar curves

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Physica. D, Nonlinear phenomena
    Date of publication: 2009-10
    Journal article

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  • SISTEMES DINAMICS DE LA UPC

     Lazaro Ochoa, Jose Tomas; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Masdemont Soler, Josep Joaquim; Pantazi, Chara; Huguet Casades, Gemma; Morales Ruiz, Juan Jose; Pacha Andujar, Juan Ramon; Ramirez Ros, Rafael; Martinez-seara Alonso, Maria Teresa; Gomez-Ullate Oteiza, David; Luque Jiménez, Alejandro; Larreal Barreto, Oswaldo José; Guillamon Grabolosa, Antoni; Martin De La Torre, Pablo; Canalias Vila, Elisabet; Puig Sadurni, Joaquim; Benita Bordes, Jose Manuel; Baldoma Barraca, Inmaculada Concepcion; de la Rosa Ibarra, Abraham; Guardia Munarriz, Marcel; Olive Farre, Maria del Carme; Fedorov, Yury; Acosta Humanez, Primitivo Belen; Blazquez Sanz, David; Gonchenko, Marina; Roldan Gonzalez, Pablo; de la Llave Canosa, Rafael; Villanueva Castelltort, Jordi; Delshams i Valdes, Amadeu
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  • A KAM Theorem without action-angle variables for elliptic reducible lower dimensional tori

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    SIAM Conference on Applications of Dynamical Systems
    Presentation's date: 2009-05-17
    Presentation of work at congresses

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  • Existence of elliptic lower dimensional tori using the parameterization method

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Date: 2009-10
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  • Numerical computation of rotation numbers of quasi-periodic planar curves

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Date: 2009-03
    Report

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  • Kolmogorov Theorem Revisited

     Villanueva Castelltort, Jordi
    Journal of differential equations
    Date of publication: 2008-01
    Journal article

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  • Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Physica. D, Nonlinear phenomena
    Date of publication: 2008-10
    Journal article

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  • Normal-Internal Resonances in Quasi-Periodically Forced Oscillators

     Villanueva Castelltort, Jordi
    2nd Workshop on Quantum Chaos. Theory and Applications
    Presentation's date: 2008-02-12
    Presentation of work at congresses

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  • Vortices of Bohmian Trajectories

     Borondo, F; Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Wisniacki, D A
    Conference on Bifurcations: Mathematical & Quantum Aspects & Applications, BIFUR 08
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  • A Dynamical System Approach to the Study of Bohmian Trajectories

     Villanueva Castelltort, Jordi
    2nd Workshop on Quantum Chaos. Theory and Applications
    Presentation's date: 2008-02-12
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  • Cuánticas para el Oscilador Armónico desde el Punto de Vista de los Sistemas Dinámicos

     Villanueva Castelltort, Jordi
    NoLineal 2008
    Presentation's date: 2008-06-16
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  • A dynamical system approach to the study of Bohmian trajectories

     Borondo, F; Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Wisniacki, D A
    2nd Workshop on Quantum Chaos. Theory and Applications
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  • Computation of rotation numbers and derivatives for invariant curves of general planar maps

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    The 7th AIMS Conference on Dynamical Systems and Diferential Equations, Special Session on Hamiltonian Systems and Applications
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  • Computation of Rotation Numbers and Derivatives for Invariant Curves of General Planar Maps

     Villanueva Castelltort, Jordi
    The 7th AIMS Conference on Dynamical Systems and Diferential Equations, Special Session on Hamiltonian Systems and Applications
    Presentation's date: 2008-05-18
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  • Trayectorias Cuánticas para el Oscilador Armónico desde el Punto de Vista de los Sistemas Dinámicos

     Borondo, F; Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Wisniacki, D A
    NoLineal 2008
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  • Estudio de curvas invariantes de aplicaciones planas generales mediante el cálculo de números de rotación

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Tercer encuentro Internacional de Matemáticas EI-MUSA
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  • A dynamical system approach to the study of Bohmian trajectories

     Borondo, F; Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Wisniacki, D A
    Date: 2008-02
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  • Kolmogorov-Arnold-Moser Aspects of the Periodic Hamiltonian Hopf Bifurcation

     Villanueva Castelltort, Jordi; Pacha Andujar, Juan Ramon; Olle Torner, Maria Mercedes
    Nonlinearity
    Date of publication: 2008-01
    Journal article

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  • KAM aspects of the quasi-periodic Hamiltonian Hopf bifurcation. Summary of results

     Villanueva Castelltort, Jordi
    6th International Symposium on Classical and Celestial Mechanics
    Presentation's date: 2007-08-01
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  • Numerical continuation of invariant curves using variational information of the rotation number

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    International Conference on Dynamical Methods and Mathematical Modelling
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  • Numerical study of rotation numbers and its variationals for parametric families of circle diffeomorphisms

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Congreso de Ecuaciones Diferenciales y Aplicaciones / Congreso de Matemática Aplicada
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  • Cálculo numérico de números de rotación y sus variacionales

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi; Martinez-seara Alonso, Maria Teresa
    Nolineal 2007 : Nuevos retos y perspectivas de la dinámica no lineal y sus aplicaciones
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  • Preservation of low-dimensional tori in Hamiltonian 1:-1 resonance

     Pacha Andujar, Juan Ramon; Villanueva Castelltort, Jordi; Olle Torner, Maria Mercedes
    International Conference Analytical Methods of Celestial Mechanics
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  • Preservation of low dimensional tori in Hamiltonian Hopf bifurcacion

     Villanueva Castelltort, Jordi
    X Workshop on Celestial Mechanics
    Presentation's date: 2007-09-05
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  • Numerical continuation of invariant curves using variational information of the rotation number

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Date: 2007-08
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  • Dinámica, Atractores y Nolinealidad: Caos y Estabilidad

     Delshams i Valdes, Amadeu; Gonchenko, Marina; Olle Torner, Maria Mercedes; Baldoma Barraca, Inmaculada Concepcion; Villanueva Castelltort, Jordi; Larreal Barreto, Oswaldo José
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  • On the Numerical Computation of Diophantine rotation numbers of analytic circle maps

     Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi
    Physica. D, Nonlinear phenomena
    Date of publication: 2006-05
    Journal article

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  • Numerical computation of rotation numbers for analytic maps

     Villanueva Castelltort, Jordi
    Carles Simó Fest (60th birthday). Conference on Dynamical Systems
    Presentation's date: 2006-05-29
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  • Numerical computation of rotation numbers for quasi-periodic planar curves

     Luque Jiménez, Alejandro; Villanueva Castelltort, Jordi
    Carles Simó Fest (60th birthday). Conference on Dynamical Systems
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  • Efficient management of multiple agent tracking through observation handling

     González, J; Rowe, D; Andrade Cetto, Juan; Villanueva Castelltort, Jordi
    6th IASTED International conference on visualization, imaging, and image processing
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  • Dinámica Asociada a Conexiones entre Objetos Invariantes, Astrodinámica y otras aplicaciones (DACOBIA)

     Delshams i Valdes, Amadeu; Fedorov, Yury; Gonchenko, Marina; Pantazi, Chara; Sánchez Casas, José Pablo; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Roldan Gonzalez, Pablo; Masdemont Soler, Josep Joaquim; Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi; Pacha Andujar, Juan Ramon; Guillamon Grabolosa, Antoni; Guardia Munarriz, Marcel; Larreal Barreto, Oswaldo José
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  • Dinámica, Atractores y Nolinealidad: Caos y Estabilidad

     Delshams i Valdes, Amadeu; Fedorov, Yury; Pantazi, Chara; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Baldoma Barraca, Inmaculada Concepcion; Lazaro Ochoa, Jose Tomas; Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi; Pacha Andujar, Juan Ramon; Larreal Barreto, Oswaldo José
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  • Dinámica, Atractores y Nolinealidad: Caos y Estabilidad

     Delshams i Valdes, Amadeu; Fedorov, Yury; Pantazi, Chara; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Baldoma Barraca, Inmaculada Concepcion; Lazaro Ochoa, Jose Tomas; Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi; Pacha Andujar, Juan Ramon
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  • KAM Theory without Action-Angle Variables

     de la Llave Canosa, Rafael; Gonzalez Colas, Antonio Maria; Jorba, A; Villanueva Castelltort, Jordi
    Nonlinearity
    Date of publication: 2005-03
    Journal article

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  • Dinàmica no lineal en dimensió baixa i atractors estranys

     Pantazi, Chara; Alseda Soler, Lluis; Delshams i Valdes, Amadeu; Mañosa Fernández, Víctor; Gutiérrez Serrés, Pere; Baldoma Barraca, Inmaculada Concepcion; Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi; Puig Sadurni, Joaquim
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  • Grup de recerca consolidat 2005. Sistemas Dinámicos de la UPC

     Delshams i Valdes, Amadeu; Pantazi, Chara; Gutiérrez Serrés, Pere; Roldan Gonzalez, Pablo; Baldoma Barraca, Inmaculada Concepcion; Lazaro Ochoa, Jose Tomas; Martinez-seara Alonso, Maria Teresa; Villanueva Castelltort, Jordi; Pacha Andujar, Juan Ramon; Guillamon Grabolosa, Antoni; Larreal Barreto, Oswaldo José
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  • Introducció al Teorema KAM

     Villanueva Castelltort, Jordi
    XV Congreso Nacional de Matemáticas
    Presentation's date: 2005-08-08
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  • Kolmogorov Theorem Revisited

     Villanueva Castelltort, Jordi
    Date: 2005-07
    Report

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  • Dinámica, Atractores y Nolinealidad: Caos y Estabilidad

     Olle Torner, Maria Mercedes; Alseda Soler, Lluis; Baldoma Barraca, Inmaculada Concepcion; Villanueva Castelltort, Jordi
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