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  • Exponentially small splitting of separatrices for 2D and 3D invariant tori

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Gonchenko, Marina
    International Conference Dynamics, Bifurcations and Strange Attractors
    Presentation's date: 2013-07-01
    Presentation of work at congresses

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  • Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
    Physica. D, Nonlinear phenomena
    Date of publication: 2013-01-15
    Journal article

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  • Homoclinic phenomena in conservative systems  Open access

     Gonchenko, Marina
    Defense's date: 2013-04-29
    Universitat Politècnica de Catalunya
    Theses

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    The goal of this thesis is the study of homoclinic orbits in conservative systems (area-preserving maps and Hamiltonian systems). We consider homoclinic (bi-asymptotic) orbits either to saddle periodic orbits or to whiskered tori. Such type orbits, called homoclinic by Poincaré, are of great interest in the theory of dynamical systems since their presence implies complicated dynamics. The thesis is divided in two parts according to two quite different topics considered. In the first part, we study area-preserving maps (APMs) with a nontransversal homoclinic orbit (homoclinic tangency) to a saddle fixed point in order to know the behavior of orbits near the given homoclinic trajectory. To this end, we construct first return maps, for which we use finitely-smooth normal forms of the saddle maps and introduce cross-coordinates. The fixed points of the first return maps correspond to single-round periodic orbits of the maps under consideration. Applying rescaling methods we derive the first return maps to the Hénon-like maps whose bifurcations are well known. Thus, translating the results obtained for the fixed points of the return maps to the periodic orbits, we prove the existence of cascades of elliptic periodic points. We also study the phenomenon of the coexistence of infinitely many single-round periodic orbits of different large periods (called global resonance). We consider the related problems in different types of APMs (symplectic maps and non-orientable APMs) with quadratic or cubic tangencies. We also establish the structure of 1:4 resonance for some conservative Hénon-like maps. The second part of the thesis is dedicated to the study of exponentially small splitting of separatrices arising from a perturbation of a Hamiltonian system with a homoclinic connection (separatrix). We consider a perturbation of an integrable Hamiltonian system having whiskered tori with coincident stable and unstable whiskers. Generally, in the perturbed system, the whiskers do not coincide anymore and our goal is to detect the transverse homoclinic orbits associated to the persistent whiskered tori. The perturbed system turns out to be not integrable due to the presence of these homoclinic trajectories and, consequently, there is chaotic dynamics near them. We give a suitable parametrization to the whiskers to determine the distance between them. This distance is given by the splitting function, and the simple zeros of this function give rise to transverse homoclinic orbits. We use the classical Poincaré-Melnikov approach to measure the splitting, although in the case of exponential smallness we have to ensure that the first order approximation overcome the error term. We consider Hamiltonian systems possessing two-dimensional whiskered tori with quadratic frequencies and three-dimensional whiskered tori with cubic golden frequency. In the two-dimensional case, we find 23 new quadratic numbers for which the Poincaré-Melnikov method can be applied and establish the existence of 4 transverse homoclinic orbits. We also study the continuation of the homoclinic orbits for all values of the parameter of perturbation in the case of the silver number sqrt(2)-1. For the three-dimensional whiskered torus with frequency vector given by the so-called "cubic golden number", we establish the existence of exponentially small splitting of separatrices and detect the transversality of 8 homoclinic orbits.

  • DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES. APLICACIONES A

     Delshams i Valdes, Amadeu
    Participation in a competitive project

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    Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps  Open access

     Delshams i Valdes, Amadeu; Gonchenko, Sergey; GONCHENKO, VLADIMIR; Lazaro Ochoa, Jose Tomas; Stenkin, Oleg
    Nonlinearity
    Date of publication: 2013-01-01
    Journal article

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    We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle fixed points. We consider one-parameter families of reversible maps unfolding the initial heteroclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations and birth of asymptotically stable, unstable and elliptic periodic orbits

  • Transition map and shadowing lemma for normally hyperbolic invariant manifolds

     Delshams i Valdes, Amadeu; Gidea, Marian; Roldan Gonzalez, Pablo
    Discrete and continuous dynamical systems. Series A
    Date of publication: 2012
    Journal article

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  • Explicit Integration of some Integrable Systems of Classical Mechanics  Open access

     Basak Gancheva, Inna
    Defense's date: 2012-03-28
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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    The main objective of the thesis is the analytical and geometrical study of several integrable finite-dimentional dynamical systems of classical mechanics, which are closely related, namely: - the classical generalization of the Euler top: the Zhukovski-Volterra (ZV) system describing the free motion of a gyrostat, i.e., a rigid body carrying a symmetric rotator whose axis is fixed in the body; - the Steklov-Lyapunov integrable case of the Kirchhoff equations describing the motion of a rigid body in an ideal incompressible liquid; - a nontrivial integrable generalization of the Steklov-Lyapunov system found by V.Rubanovskii: it describes the motion of a gyrostat in an ideal fluid in presence of a non-zero circulation. In our study we obtained explicit solution of the Zhukovski-Volterra ([2] and the Steklov-Lyapunov systems in terms of sigma- or theta-functions, and performed a bifurcation analysis of these systems, as well as of the Rubanovskii generalization. One should note that the solution of the ZV system was first given by V. Volterra, who, however, presented only its structure, but not the explicit formulas. The thesis gives a new alternative solution of this system by using an algebraic parametrization of the angular momentum. This allowed us to find poles and zeros of angular momentum in an algebraic way. The parametrization was also used to find an explicit solution for the Euler precession angle, and, as a consequence, to solve the Poisson equations describing the motion of the gyrostat in space. Similarly, by giving a geometric interpretation of the separating variables, and using the Weierstrass root functions, we reconstructed the thetafunctional solution of the Steklov-Lyapunov systems, which was first given by F. Kötter in 1899 without a derivation ([3]). In the study of bifurcations and singularities of the ZV system we used its bi-Hamiltonian structure ([1]. According the new method, the solution is critical, if there exist a parameter of corresponding family of Poisson brackets, for wich the rang of the brackets with this parameter drops. Applying new technics, based on the property of the system of being bi-Hamiltonian, we construct the bifurcation diagram of the ZV system. We also find the equilibrium points of the system, check the non-degeneracy condition for such points in the sense of the singularity theory of Hamiltonian systems, determine the types of equilibria points, and verify whether they are stable or not. We also describe the topological type of common levels of the first integrals of the ZV system. Similar problems have been discussed in many papers, but the goal of our work is to study the system and demonstrate the above techniques. It is a remarkable fact that using the bi-Hamiltonian property makes it possible to answer all the above questions practically without any difficult computations. The same method is applied to construct the bifurcation diagram for the Steklov-Lyapunov system, describe the zones of real motion, and analyze stability of critical periodic solutions. Then the bifurcation analysis is extended to the Rubanovskii generalizaton. Here the main difficulty is that the number of different types of the bifurcation diagram is quite high, so we only describe general properties of the bifurcation curves, do stability analysis for closed trajectories, and equilibria.

    El objetivo principal de la tesis es el estudio analítico y geométrico de varios sistemas integrables dinámicos y finito-dimensionales de la mecánica clásica que están estrechamente vinculados, a saber: -La generalización clásica de Euler top: el sistema Zhukovski-Volterra (ZV) que describe el movimiento libre de un giróstato, es decir, un cuerpo rígido que lleva un rotor simétrico cuyo eje es fijo al cuerpo. - El caso del sistema integrable de Steklov-Lyapunov de las ecuaciones de Kirchhoff que describen el movimiento de un cuerpo rígido en un líquido incompresible ideal; - Una generalización no trivial del sistema integrable de Steklov-Lyapunov encontrado por V. Rubanovskii que describe el movimiento de un giróstato en un fluido ideal en presencia de una circulación distinta de cero. En nuestro estudio hemos obtenido una solución explícita de los sistemas de Zhukovski-Volterra [2] y de Steklov-Lyapunov en términos de funciones sigma- o theta y hemos realizado un análisis de la bifurcación de estos sistemas, así como de la generalización de Rubanovskii. Hay que señalar que la solución del sistema de ZV fue dado por primera vez por V. Volterra, que, sin embargo, presenta sólo su estructura, pero no las fórmulas explícitas. La tesis ofrece una nueva solución alternativa de este sistema mediante el uso de una parametrización algebraica del momento angular. Esto nos ha permitido encontrar polos y ceros del momento angular en forma algebraica. La parametrización también se utilizó para encontrar una solución explícita para el ángulo de precesión de Euler, y, en consecuencia, para resolver las ecuaciones de Poisson que describen el movimiento de un giróstato en el espacio. Del mismo modo, al dar una interpretación geométrica de las variables de separación, y utilizando las funciones de las raíces Weierstrass, hemos reconstruido la solución thetafunctional de los sistemas de Steklov-Lyapunov, que fue dado por primera vez por F. Kotter en 1899 sin una derivación ([3]). En el estudio de las bifurcaciones y las singularidades del sistema ZV hemos utilizado su estructura bi-Hamiltoniana ([1]). Según el nuevo método, la solución es crítica, si existe un parámetro de la familia correspondiente del paréntesis de Poisson, para que el rango de las paréntesis con este parámetro se disminuye. Aplicando las nuevas técnicas, basadas en la propiedad del sistema de ser bi-Hamiltoniana, construimos el diagrama de bifurcación del sistema ZV. También hemos encontrado los puntos de equilibrio del sistema, verificando la condición de no-degeneración de estos puntos, en el sentido de la teoría de singularidad de los sistemas hamiltonianos, determinando los tipos de puntos de equilibrio, y comprobando si son estables o no. También hemos descrito el tipo topológico de los niveles comunes de los primeros integrales del sistema de ZV. Problemas similares se han discutido en muchas obras, pero el objetivo de nuestro trabajo es estudiar el sistema y demostrar las técnicas anteriormente mencionadas. Es un hecho notable que el uso de la propiedad bi-Hamilton permite responder a todas las preguntas anteriores, prácticamente sin ningún cálculo difícil. El mismo método se aplica para construir el diagrama de bifurcación para el sistema de Steklov-Lyapunov, describir las zonas de movimiento real, y analizar la estabilidad de soluciones periódicas críticas. A continuación, el análisis de bifurcación se extiende a la generalización Rubanovskii. Aquí la principal dificultad consiste en que el número de diferentes tipos del diagrama de bifurcación es bastante alto, por lo que sólo se describen las propiedades generales de las curvas de bifurcación, y el análisis de estabilidad se hace para trayectorias cerradas, y equilibrios .

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    Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation  Open access

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
    Date: 2012-07
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    The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton-Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel0nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example.

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    Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps  Open access

     Delshams i Valdes, Amadeu; Lazaro Ochoa, Jose Tomas; Gonchenko, Vladimir; Gonchenko ., S.V.; Sten'kin, Oleg
    Date: 2012-01-12
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    Abstract. We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

  • A geometric mechanism of diffusion: rigorous verification in a priori unstable Hamiltonian systems

     Delshams i Valdes, Amadeu; Huguet, G.
    Journal of differential equations
    Date of publication: 2011-03-01
    Journal article

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    In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom and we apply the geometric mechanism for diffusion introduced in [A. Delshams, R. de la Llave, T.M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (844) (2006), viii + 141 pp.], and generalized in [A. Delshams, G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity 22 (8) (2009) 1997– 2077]. We provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform the straightforward computations along the proof and present the geometric mechanism of diffusion in an easily understandable way. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

  • Jornades d'interacció entre sistemes dinàmics i equacions entre sistemes dinàmics i equacions en derivades parcials (JISD2011)

     Martinez-seara Alonso, Maria Teresa; Delshams i Valdes, Amadeu; Cabre Vilagut, Xavier; Gonzalez Nogueras, Maria Del Mar
    Participation in a competitive project

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    Transversality of homoclinic orbits to hyberbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation  Open access

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
    Date: 2011
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    We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and suffcient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop.

  • Proximity maneuvering of libration point orbit formations using adapted finite element methods  Open access

     Garcia Taberner, Laura
    Defense's date: 2010-02-04
    Universitat Politècnica de Catalunya
    Theses

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    This doctoral dissertation is structured in four chapters as follows. The first chapter contains a summary of formation flying projects that have been taken into consideration since few years ago. We specially focus on the missions that have been planned to be located in a libration point regime. For completeness, this chapter also contains a general state of the art about the main reconfiguration techniques for satellite formations. The main new contributions of the thesis are contained in chapters 2, 3 and 4. Chapter 2 introduces the general methodology that will be considered in all the dissertation. It is based on a discretization in time by means of a finite element approximation, and at the same time, is suitable to incorporate optimal control problems. In this chapter we study the reconfigurations using linearized equations about a nominal Halo orbit minimizing the functional given by the sum of the square of the magnitude of the maneuvers. This functional is not directly related to the fuel consumption, but has good properties concerning minimization and regularity. In chapter 3 we are still working with the linearized model about the nonlinear orbit, but the functional that we optimize, given by the sum of the modulus of the maneuvers, is directly related to fuel consumption. As a consequence, the methodology can be tuned in such a way that, if possible, the user can choose to converge to bang-bang optimal controls (when possible) or to low thrust trajectories in general situations. In this chapter, our objective is not only to study how the reconfigurations can be accomplished. We also consider the problem of obtaining good meshes for our finite element discretization, and up to a certain extent, to decide which is the best mesh for each kind of problem. Finally, in chapter 4, we deal with non-linear and perturbed problems. In a first step we consider reconfigurations in the Restricted Three Body Problem and in a second one with JPL ephemeris. This fact slightly changes the trajectories of the spacecraft with respect to the ones obtained in the previous chapters. To correct for such deviations we design and implement a methodology based on adding small corrective maneuvers on top of the nominal ones. We also study the magnitude of corrective maneuvers that will need to be applied in case of errors in the execution of the nominal ones. Finally, this chapter ends with some other applications that can be performed using the methodology we have developed.

    Aquesta tesi doctoral està estructurada en quatre capítols. El primer capítol comprèn un resum dels projectes de vol en formació que s'han tingut en consideració els últims anys, especialment els que estan planejats de situar-se al voltant dels punts de libració. En aquest capítol també fem un estat de l'art de les principals tècniques de reconfiguració de formacions de satèl•lits. Les principals contribucions noves d'aquesta tesi es troben als capítols 2, 3 i 4. En el capítol 2 introduïm la metodologia general que s'utilitzarà en tota la dissertació. Aquesta metodologia està basada en una discretització del temps usant una aproximació en elements finits, que al mateix temps la fa factible d'incorporar en problemes d'optimització. En aquest capítol es consideren les equacions linealitzades al voltant d'una òrbita Halo. El problema d'optimització minimitza el funcional obtingut per la suma dels quadrats de les maniobres. Encara que aquest funcional no estigui directament relacionat amb el consum de combustible, es comporta bé a l'hora de minimitzar. En el capítol 3 es segueixen utilitzant les equacions linealitzades al voltant de l'òrbita Halo, però ara el funcional que es minimitza és la suma dels mòduls de les maniobres, que està directament relacionat amb el consum de combustible. Com a conseqüència, la metodologia permet que es pugui convergir a controls bang-bang en el cas que sigui possible, o a avanç continu en les altres situacions. En aquest capítol, el nostre objectiu no consisteix només en estudiar com fer les reconfiguracions, sinó que també considerem el problema d'obtenir una bona discretització per al nostre problema d'elements finits, i decidir quina és la millor malla per cada tipus de problema. Finalment, al capítol 4 considerem problemes no lineals i incloem perturbacions. Comencem considerant les reconfiguracions en el problema restringit de tres cossos, per després veure com es comporta usant les efemèrides JPL. Aquests nous models canvien una mica les trajectòries dels satèl•lits respecte les que havíem obtingut en els capítols anteriors. Per corregir aquestes desviacions implementem una metodologia basada en afegir petites correccions a les maniobres que estan donades. També estudiem la magnitud de les maniobres que cal aplicar quan es produeixen errors d'execució en les maniobres nominals. Finalment, aquest capítol acaba amb altres aplicacions que es poden dur a terme usant la metodologia que hem desenvolupat.

     Esta tesis doctoral está estructurada en cuatro capítulos. El primer capítulo contiene un resumen de los proyectos de vuelo en formación que se han tenido en consideración en los últimos años, especialmente aquellos que están planeados de situarse alrededor de los puntos de libración. En este capítulo también se hace un estado del arte de las principales técnicas de reconfiguración de formaciones de satélites.Las principales contribuciones nuevas de esta tesis se encuentran en los capítulos 2, 3 y 4. En el capítulo 2 introducimos la metodología general que se usará en toda la disertación. Esta metodología está basada en una discretización del tiempo usando una aproximación en elementos finitos, que al mismo tiempo la hace factible de incorporar en problemas de optimización. En este capítulo, se consideran las ecuaciones linealizadas alrededor de una órbita Halo. El problema de optimización minimiza el funcional obtenido como la suma de los cuadrados de las maniobras. Aunque este funcional no está directamente relacionado con el consumo de combustible, tiene un buen comportamiento en la minimización.En el capítulo 3 se siguen usando las ecuaciones linealizadas alrededor de la órbita Halo, pero ahora el funcional al minimizar es la suma de los módulos de las maniobras, que está directamente relacionado con el consumo de combustible. Como consecuencia, la metodología permite que se pueda convergir a controles bang-bang en el caso de que sea posible, o a avance continuo en las otras situaciones.En este capítulo, nuestro objetivo no consiste únicamente en estudiar cómo hacer las reconfiguraciones, sino que consideramos el problema de obtener una buena malla para el problema de los elementos finitos, y decidir cuál es la mejor malla para cada tipo de problema.Finalmente, en el capítulo 4 se consideran problemas no lineales y se incluyen perturbaciones. Empezamos considerando las reconfiguraciones en el problema restringido de 3 cuerpos, y luego observamos qué pasa cuando usamos las efemérides JPL. Estos nuevos modelos cambian un poco las trayectorias de los satélites respecto las obtenidas en los capítulos anteriores. Para corregir estas desviaciones, se implementa una metodología basada en añadir pequeñas correcciones a las maniobras dadas. También estudiamos la magnitud de las maniobras que hace falta aplicar cuando se producen errores de ejecución en las maniobras nominales. Para finalizar, este capítulo acaba con otras aplicaciones que se pueden llevar a término con la metodología desarrollada.

  • Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Koltsova, O.; Pacha Andujar, Juan Ramon
    Regular and chaotic dynamics
    Date of publication: 2010-10
    Journal article

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    We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).

    hyperbolic KAM tori - transverse homoclinic orbits - Melnikov method

  • An accounting device for biasymptotic solutions: the scattering map in the restricted three body problem

     Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
    Date of publication: 2010
    Book chapter

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  • 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
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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.

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    A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems  Open access

     Delshams i Valdes, Amadeu; Huguet Casades, Gemma
    Date: 2010-07
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    In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

  • 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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  • Euler's beta integral in Pietro Mengoli's works

     Massa Esteve, Maria Rosa; Delshams i Valdes, Amadeu
    Archive for history of exact sciences
    Date of publication: 2009-03
    Journal article

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    Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo (1672) and displayed the results in triangular tables. In particular, his new arithmetic–algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli’s view of his algorithmic computation of quadratures.

  • Galoisian Approach to Supersymmetric Quantum Mechanics.

     Acosta Humanez, Primitivo Belen
    Defense's date: 2009-07-02
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
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  • Arnold¿s mechanism of diffusion in the spatial circular Restricted Three Body Problem: a semi- numerical argument

     Delshams i Valdes, Amadeu; Gidea, Marian; Roldan Gonzalez, Pablo
    Dynamics, Topology and Computations
    Presentation's date: 2009-06-03
    Presentation of work at congresses

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  • SISTEMES DINAMICS DE LA UPC

     Lazaro Ochoa, Jose Tomas; Roldan Gonzalez, Pablo; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Masdemont Soler, Josep Joaquim; Gomez-Ullate Oteiza, David; Luque Jiménez, Alejandro; Larreal Barreto, Oswaldo José; Huguet Casades, Gemma; Pacha Andujar, Juan Ramon; Ramirez Ros, Rafael; Martinez-seara Alonso, Maria Teresa; Guillamon Grabolosa, Antoni; Martin De La Torre, Pablo; Canalias Vila, Elisabet; Puig Sadurni, Joaquim; Benita Bordes, Jose Manuel; Morales Ruiz, Juan Jose; Guardia Munarriz, Marcel; Olive Farre, Maria del Carme; de la Rosa Ibarra, Abraham; Baldoma Barraca, Inmaculada Concepcion; Blazquez Sanz, David; Fedorov, Yury; Acosta Humanez, Primitivo Belen; Gonchenko, Marina; Pantazi, Chara; de la Llave Canosa, Rafael; Villanueva Castelltort, Jordi; Delshams i Valdes, Amadeu
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    Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method  Open access

     Gutiérrez Serrés, Pere; Delshams i Valdes, Amadeu; Pacha Andujar, Juan Ramon
    Date: 2009-12
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    We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel'nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound# and can be approximated by a trigonometric polynomial #which gives an upper bound)

  • Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

     Delshams i Valdes, Amadeu; Huguet Casades, Gemma
    Nonlinearity
    Date of publication: 2009
    Journal article

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  • Geometric properties of the scattering map of a normally hyperbolic invariant manifold

     Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
    Advances in mathematics
    Date of publication: 2008-02
    Journal article

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  • Consideracions al voltant de la funció beta a la obra de Leonhard Euler (1707-1783)

     Massa Esteve, Maria Rosa; Delshams i Valdes, Amadeu
    Quaderns d'història de l'enginyeria
    Date of publication: 2008-01
    Journal article

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  • Computing the scattering map in the spatial hill's problem

     Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
    Discrete and continuous dynamical systems. Series B
    Date of publication: 2008-09
    Journal article

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  • The role of hyperbolic invariant objects: From Arnold diffusion to biological clocks  Open access

     Huguet Casades, Gemma
    Defense's date: 2008-10-16
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
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    El marc d'aquesta tesi són els objectes invariants hiperbòlics (tors amb bigotis, cicles límit, NHIM,. . .), que constitueixen, per aquesta tesi, els objectes essencials per a l'estudi de diversos problemes des de la difusió d'Arnold fins als rellotges biològics. Treballem en tres temes diferents des d'un enfocament tant teòric com numèric, amb una especial atenció per a les aplicacions, especialment en neurobiologia:· Existència de difusió d'Arnold per a sistemes Hamiltonians a priori inestables· Algorismes numèrics ràpids per al càlcul de tors invariants i els "bigotis" associats, per a sistemes Hamiltonians utilitzant el mètode de la parametrització.· Càlcul d'isòcrones i corbes de resposta de fase (PRC) en sistemes neurobiològics usant el mètode de la parametrització.En la primera part de la tesi, hem considerat el cas d'un sistema Hamiltonià a priori inestable amb 2+1/2 graus de llibertat sotmès a una pertorbació de tipus general. "A priori inestable" significa que el sistema no pertorbat presenta un punt d'equilibri hiperbòlic amb una òrbita homoclínica associada. El resultat principal d'aquesta part de la tesi és que per a un conjunt genèric de pertorbacions prou regulars, el sistema presenta el fenòmen de la difusió d'Arnold, és a dir, existeixen trajectòries la variable acció de les quals experimenta un canvi d'ordre 1. La demostració es basa en un estudi detallat de les zones ressonants i els objectes invariants generats en elles, i ofereix una descripció completa de la geografia de les ressonàncies generades per una pertorbació genèrica.En la segona part d'aquest memòria, desenvolupem mètodes numèrics eficients que requereixen poca memòria i operacions per al càlcul de tors invariants i els "bigotis" associats en sistemes Hamiltonians (aplicacions simplèctiques i camps vectorials Hamiltonians).En particular, això inclou els objectes invariants involucrats en el mecanisme de la difusió d'Arnold, estudiat en el capítol anterior. Els algorismes es basen en el mètode de la parametrització i segueixen de prop demostracions recents del teorema KAM que no usen variables acció-angle. Donem detalls de la implementació numèrica que hem dut a terme i mostrem alguns exemples.En la darrera part de la tesi relacionem problemes de temps en sistemes biològics amb algunes eines conegudes de sistemes dinàmics. En particular, usem el mètode de la parametrització i les simetries de Lie per a calcular numèricament les isòcrones i les corbes de resposta de fase (PRC) associades a oscil·ladors i ho apliquem a diversos models biològics ben coneguts. A més a més, aconseguim estendre el càlcul de PRCs en un entorn de l'oscil·lador. Les PRCs són útils per a l'estudi de la sincronització d'oscil·ladors acoblats i una eina bàsica en biologia experimental (ritmes circadians, acoblament sinàptic i elèctric de neurones,. . . ).

  • Differential Galois Theory and Lie-Vessiot Systems  awarded activity

     Blazquez Sanz, David
    Defense's date: 2008-07-07
    Department of Applied Mathematics II, Universitat Politècnica de Catalunya
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  • Diophantine conditions for cubic frequency vectors

     Gonchenko, Marina; Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere
    Conference on Stability and Instability in Mechanical Sysyems: Recent Progress and Mathematical Theory
    Presentation of work at congresses

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  • Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

     Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
    Date of publication: 2008-02
    Book chapter

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  • Contributiones a la sexta edicion del congreso Nolineal 2008

     Marques Truyol, Francisco; Delshams i Valdes, Amadeu
    Date of publication: 2008-06
    Book

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  • Diophantine conditions for cubic frequency vectors

     Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Gonchenko, Marina
    Conference on Stability and Instability in Mechanical Sysyems: Recent Progress and Mathematical Theory
    Presentation's date: 2008-09-25
    Presentation of work at congresses

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  • Diophantine conditions for cubic frequency vectors

     Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
    Date: 2008-06
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  • Study of resonances for Diophantine cubic frequency vectors

     Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
    Date: 2008-09
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  • An accounting device for biasymtotic solutions: the scattering map in the restricted three body problem

     Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
    2nd Conference on Nonlinear Science and Complexity
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  • Diophantine conditions for cubic frequency vectors

     Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
    NoLineal 2008
    Presentation of work at congresses

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  • Propuesta de acciones de la red temática DANCE a partir del año 2007

     Delshams i Valdes, Amadeu; Alseda Soler, Lluis; Jorba Monte, Angel
    Participation in a competitive project

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  • Jornades d'Introducció als Sistemes Dinàmics i a les EDP'S 2007 (JISD2007)

     Delshams i Valdes, Amadeu; Martinez-seara Alonso, Maria Teresa
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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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  • Computing the scattering map in the spatial Hill's problem

     Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
    Date: 2007-08
    Report

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  • Euler's Beta Function in Pietro Mengoli's Works

     Massa Esteve, Maria Rosa; Delshams i Valdes, Amadeu
    Date: 2007-08
    Report

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  • The scattering map in the planar circular restricted Tree Body Problem

     Canalias Vila, Elisabet; Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
    Celestial mechanics and dynamical astronomy
    Date of publication: 2006-05
    Journal article

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  • Homoclinic Orbits for Invariant Tori of Nearly Integrable Hamiltonian Systems

     Koltsova, O; Lerman, L; Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere
    Doklady mathematics
    Date of publication: 2006-01
    Journal article

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  • Homoclinic trajectories towards invariant tori of a nearly-integrable Hamiltonian system

     Koltsova, O; Lerman, L M; Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere
    Rossiiskaya Akademiya nauk. Doklady, svodnyi vypusk
    Date of publication: 2006-05
    Journal article

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  • Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

     Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
    Advances in mathematics
    Date of publication: 2006-04
    Journal article

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  • A geometric mechanism for diffusion in Hamiltonia systems overcoming the large gap problem: heuristics and rigorous varification on a model

     Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
    Memoirs of the American Mahematical Society
    Date of publication: 2006-01
    Journal article

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  • Preface: Special issue on nonautonomous, stochastic and Hamiltonian dynamical systems

     Caraballo, Tomás; Delshams i Valdes, Amadeu; Jorba, Àngel; Kloeden, Peter; Rafael, Obaya
    Discrete and continuous dynamical systems. Series A
    Date of publication: 2006-10
    Journal article

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