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N.º Referencia: MTM2006-00478

MINISTERIO DE CIENCIA E INNOVACIÓN

DIRECCIÓN GENERAL DE INVESTIGACIÓN Y GESTIÓN DEL PLAN NACIONAL DE I+D+i

INFORME FINAL

Investigador Principal AMADEU DELSHAMS VALDÉS

Título de la actuación Dinámica Asociada a Conexiones entre Objetos Invariantes, Astrodinámica y otras aplicaciones

Organismo Universidad Politècnica de Catalunya

Centro Escuela Técnica Superior de Ingeniería Industrial de Barcelona

Departamento Matemática Aplicada I

Fecha de inicio 01/09/2006

Fecha de finalización 30/04/2010

PROYECTOS I+D+i, ACCIONES ESTRATÉGICAS Y ERANET

Sr. Subdirector General de Proyectos de Investigación Este documento no debe remitirse en papel/ por correo postal a la Subdirección

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A. MEMORIA. Resumen de las actividades realizadas y de los resultados del proyec-to en relacion con los objetivos propuestos (maximo 2.000 palabras)

Destaque su relevancia cientıfica y/o su interes tecnologico. En el caso de haber obtenidoresultados no previstos inicialmente, indique su relevancia para el proyecto. En caso deresultados fallidos, indıquense las causas.

Este proyecto DACOBIA (MTM2006-00478, Dinamica Asociada a Conexiones entreObjetos Invariantes, Astrodinamica, y otras aplicaciones) ha sido una continuacion delproyecto previo financiado OICEPA (BFM2003-09504-C02-02: Objetos Invariantes ensistemas dinamicos, sus Conexiones, Evolucion respecto a Parametros y Aplicaciones) y,a su vez, continua en el proyecto financiado DACOBIAN (MTM2009-06973, DinamicaAsociada a Conexiones entre Objetos Invariantes, Astrodinamica, Neurociencia y otrasaplicaciones). En lo que sigue, replicamos la estructura de la secion 3 (Objectivos) dela memoria DACOBIA.

1. Objetos invariantes en sistemas dinamicos y sus conexiones

a) Difusion de Arnold y escision de separatrices

Resultados: La existencia de difusion de Arnold en sistemas a priori inesta-bles se ha demostrado en [55, 56, 52] usando metodos geometricos. En [57]se han establecido propiedades de la “scattering map” de sistemas a prioriinestables. La validez del metodo de Melnikov en el estudio de la rotura deorbitas heteroclınicas por despliegues genericos de la singularidad Hopf-zerose ha demostrado en [22]. En [23] se resolvio la ecuacion “inner” en relacion alproblema de la distribucion de orbitas heteroclınicas para despliegues generi-cos de la singularidad Hopf-zero. Soluciones resurgentes de una EDP linealse obtuvieron en [136]. Usando tecnicas de “complex matching” se midieronescisiones singulares [136]. En [108, 109] se demostro la existencia de orbitashomoclınicas transversales a toros KAM hiperbolicos cerca de connexioneshomoclınicas asociadas a un equilibrio centro-centro-silla. La teorıa de Mel-nikov canonica se formalizo en [127]. La escision de separatrices en algunasaplicaciones que preservan volumen se estudio en [128].

b) Bifurcaciones, formas normales y calculo de objetos invariantes

Resultados: La existencia de atractores no-caoticos extranos en aplicacionesde tipo Harper se demostro en [104]. Nuevos aspectos de la bifurcacion Hopfhamiltoniana se descubrieron en [138]. La rotura de curvas invariantes res-onantes en aplicaciones de tipo billar se estudion en [144]. La aparicionde cascadas de puntos periodicos elıpticos cerca de tangencias homoclınicascuadraticas en aplicaciones que preservan area se establecion en [91]. El ran-go de frecuencias para el cual la forma normal de Kolmogorov converge seamplio en [152]. La existencia de pre-foliaciones invariantes no resonantes ensistemas no uniformemente hiperbolicos se demostro en [71]. La existenciade variedades invariantes unidimensionales asociadas a puntos parabolicos seestableceio en el caso diferenciable en [20], obteniendo resultados optimos so-bre la perdida de regularidad en el punto parabolico. Metodos numericos paracalcular con alta precision numeros de rotacion de aplicaciones del cırculo, yderivadas respecto de parametros, se han desarrollado en [150, 130], y se hanadaptado para estudiar numericamente el tamano asintotico del dominio derotacion para la familia de Arnold [151]. Metodos de paramatrizacion para elcalculo numerico de toros invariantes se han desarrollado en [101, 102, 103]para aplicaciones casi-periodicas y en [106] para sistemas hamiltonianos.

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c) Integrabilidad y aplicaciones de retorno

Resultados: El teorema de Morales-Ramis se ha ampliado para considerarvariacionales de orden superior y se ha aplicado para obtener la clasificacioncompleta de la integrabilidad de la familia de hamiltonianos de Henon-Heilesen [134]. La dependencia sensible respecto a condiciones iniciales en algu-nas ecuaciones diferenciales se explico mediante viages en las superfıcies deRiemann [65, 39].

2. Astrodinamica

Resultados: Algunos resultados sobre control distribuido de vuelos en formacionse obtuvieron en [149]. Una estrategia para la reconfiguracion de formaciones desatelites, usando metodos variacionales, se desarrollo e implemento en [80, 79].Usando tecnicas de variedades invariantes, se explico la formacion de brazos es-pirales en galaxias [147, 146, 19]. Transferencias de bajo empuje a puntos delibracion y rendezvous en orbitas de Lissajous fueron estudiadas en [62, 44].Conexiones homoclınicas y heteroclınicas se usaron para navegar en el Sistema So-lar [45, 43, 42]. La transferencia al punto Lagrangiano L1 del sistema Tierra-Lunase analizo en [15, 14]. (El grupo ha colaborado con la ESA.)

3. Neurociencia matematica y computacional

Resultados: Se descubrieron errores metodologicos en estimaciones experimen-tales de conductancias en el cerebro, dando una vision teorica de las causas detales errores [96]. Las “phase resetting curves” se relacionaron tant amb el metodode la parametritzacion de objectos invariantes como con simetrias de Lie, obte-niendo una extension de las PRC a un entorno del ciclo lımite y aplicacionesa osciladores biologicos [95]. Un modelo de red cortical lentamente oscilatoriocon depresion sinaptica, que replica correctamente resultados experimentales, seconstruyo en [29].

4. Aplicaciones fısicas

a) Mecanica cuantica, mecanica celeste i aceleradores

Resultados: La relacion entre el espectro de Cantor y la no existencia de on-das de Bloch debiles en operadores de Schrodinger unidimensionales con po-tencial ergodico se revelo en [139]. Parametrizaciones numericas de trayecto-rias de puntos de libracion y sus variedades invariantes se calcularon en [24].Las variedades invariantes del punto L3 y el movimiento de “horseshoe”en el RTBP se estudiaron en [28]. Los aspectos dinamicos de orbitas ho-moclınicas de tipo “horseshoe” con multi-ronda en el RTBP se analizaronen [26]. La “scattering map” se obtuvo numericamente en diversos problemasde mecanica celeste [41, 59, 58].

b) Teorıa de control

Resultados: El error de salida de un convertidor de potencia modelado poruna equacion diferencial de tipo “nonsmooth” es controlo usando metodosde promedios [16].

B. RESULTADOS MAS RELEVANTES ALCANZADOS EN EL PROYECTO(maximo 60 palabras)

Dentro de los logros del proyecto senalados en el apartado anterior, resene los mas relevanteshasta un maximo de tres

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1. Se ha construido una potente teorıa entorno al “scattering map”. Se han clarifi-cado sus propiedades geometricas [57], se ha usado para probar la existencia dedifusion de Arnold [54], se ha aplicado a problemas de mecanica celeste [58, 59],y esta siendo usado por otros investigadores.

2. En [94] han culminado los esfuerzos de casi dos decadas para cuantificar conexactitud la escision singular de separatrices en el pendulo rapidamente forzado,dando ası respuesta a varias cuestiones que permanecian abiertos.

3. En [147, 146, 19] se ha explicado, mediante tecnicas de variedades invariantes, laformacion de brazos espirales en galaxias. Estos resultados han sido bien recibidospor la comunidad, pues las explicaciones anteriores de este fenomeno no eransatisfactorias.

C. RESULTADOS DEL PROYECTO

C1. FORMACION DE PERSONAL EN EL PROYECTO, describir brevemente.

En el periodo 2006-2010 ocho personas del grupo han leıdo la tesis doctoral: P.B.Acosta-Humanez, D. Blazquez-Sanz, E. Canalias, L. Garcıa, M. Guardia, G. Huguet,A. Luque, C. Olive y P. Roldan todas dirigidas por miembros del grupo. Ademas, otrossiete estudiantes estan realizando la tesis bajo la direccion de miembros del grupo: I.Basak, J. Benita, M. Gonchenko, A. de la Rosa, O. Larreal, J. Soliz y A. Tamarit.El grupo ha estado vinculado al programa de doctorado de Matematica Aplicada de laUniversitat Politecnica de Catalunya y varios de sus investigadores son profesores deeste programa.Por un lado, dentro del marco del Programa de Doctorado de Matematica Aplicada dela UPC, se han contratado cada ano a diferentes profesores externos, expertos en temasde mucha actualidad que han participado en algunas asignaturas del programa. Estoscursos se han organizado en forma de Jornadas y a ellos han asistido un gran numerode estudiantes. Concretamente:Por un lado, dentro del marco del Programa de Doctorado de Matematica Aplicada dela UPC, se han contratado cada ano a diferentes profesores externos, expertos en temasde mucha actualidad que han participado en algunas asignaturas del programa. Estoscursos se han organizado en forma de Jornadas y a ellos han asistido un gran numerode estudiantes. A continuacion, detallamos la lista de profesores externos.

Curso 2005/2006 John Hogan (U. Bristol), Mirko Degli Esposti (U. Bologna), FrancoisHamel (U. Aix-Marseille III), Hiroshi Matano (U. Tokyo) y C. Eugene Wayne(Boston U.).

Curso 2006/2007 Dario Bambusi (Universita degli studi di Milano), Vadim Kaloshin(Pennsylvania State Univ. y Univ. of Maryland), Rafael de la Llave (Univ. ofTexas at Austin), Changfeng Gui (Univ. of Connecticut) y Regis Monneau (CERMICS-ENPC, France).

Curso 2007/2008 Juan Luis Vazquez (Universidad Autonoma de Madrid) y MarcoAntonio Teixeira (Universidade Estatual de Campinas).

Curso 2009/2010 Jean-Michel Roquejoffre (Univ. Paul Sabatier. Toulouse III), TimMyers (CRM, Bellaterra, Barcelona), Massimiliano Berti (Univ. Federico II) yAlfonso Sorrentino (Univ. of Cambridge).

Finalmente, pero con caracter muy destacado, el hecho de mantener un seminario de-nominado en la actualidad Seminari de Sistemes Dinamics UB-UPC y que existe de

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forma ininterrumpida desde 1978, con una o varias exposiciones cada miercoles en latarde, es una pieza clave en la formacion permanente de los investigadores del grupo.Ademas de este seminario, de caracter senior, en el que participan todos los miembrosdel grupo, tambien han surgido otras alternativas formadoras, mas enfocadas a losestudiantes de doctorado y a personas ya formadas que quieran iniciarse en temasnuevos.C2. TESIS DOCTORALES REALIZADAS TOTAL O PARCIALMENTE EN ELPROYECTO

Indicar: Tıtulo, nombre del doctorado, Universidad, Facultad o Escuela, fecha de comienzo,fecha de lectura, calificacion y director.

Tıtulo: Contributions to libration orbit mission design using hyperbolic invariantmanifolds. Doctorando: E. Canalias. Facultat de Matematiques i Estadıstica. Uni-versitat Politecnica de Catalunya. Lectura: 2007. Calificacion: Apto cum Laude.Director: J. Masdemont.

Tıtulo: Calcul de l’escissio de separatrius usant tecniques de matching complexi ressurgencia aplicades a l’equacio de Hamilton-Jacobi. Doctorando: C. Olive.Facultat de Matematiques i Estadıstica. Universitat Politecnica de Catalunya.Lectura: 2007. Calificacion: Apto cum Laude. Director: T. Seara.

Tıtulo: Differential Galois theory and Lie-Vessiot systems. Doctorando: D. Blazquez-Sanz. Facultat de Matematiques i Estadıstica. Universitat Politecnica de Catalun-ya. Lectura: 2008. Calificacion: Apto cum Laude. Director: J. J. Morales-Ruız.

Tıtulo: The role of hyperbolic invariant objects: From Arnold diffusion to biologi-cal clocks. Doctorando: G. Huguet. Facultat de Matematiques i Estadıstica. Uni-versitat Politecnica de Catalunya. Lectura: 2008. Calificacion: Apto cum Laude.Director: A. Delshams.

Tıtulo: Galoisian approach to supersymmetric quantum mechanics. Doctoran-do: P.B. Acosta-Humanez. Facultat de Matematiques i Estadıstica. UniversitatPolitecnica de Catalunya. Lectura: 2009. Calificacion: Apto cum Laude. Director:J. J. Morales-Ruız.

Tıtulo: Proximity maneuvering of libration point orbit formations using adaptedfinite element methods. Doctorando: L. Garcıa. Facultat de Matematiques i Es-tadıstica. Universitat Politecnica de Catalunya. Lectura: 2010. Calificacion: Aptocum Laude. Director: J. Masdemont.

Tıtulo: From non-smooth to analytic dynamical systems: low codimension bifurca-tions and exponentially small splitting of separatrices. Doctorando: M. Guardia.Facultat de Matematiques i Estadıstica. Universitat Politecnica de Catalunya.Lectura: 2010. Calificacion: Apto cum Laude. Director: T. Seara.

Tıtulo: Analytic and numerical tools for the study of quasi-periodic motions inHamiltonian systems. Doctorando: A. Luque. Facultat de Matematiques i Es-tadıstica. Universitat Politecnica de Catalunya. Lectura: 2010. Calificacion: Aptocum Laude. Director: J. Villanueva.

C3. ARTICULOS CIENTIFICOS

Indicar: Autor(es), tıtulo, referencia de la publicacion, (adjuntar primera pagina en formatodigital y aquella en la que se mencione a las entidades financiadoras del proyecto)

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La lista de artıculos cientıficos en revistas que sigue contiene la produccion cientıficadel grupo que ha aparecido publicada, esta pendiente de publicacion o esta pendientede aceptacion para su publicacion.

Referencias

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[4] P. Acosta-Humanez and F. Marcellan. Preface. In Differential Algebra, Complex Anal-ysis and Orthogonal Polynomials [3], pp. vii–viii. [URL]

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[8] P.B. Acosta-Humanez. Nonautonomous Hamiltonian systems and Morales-Ramis the-ory. I. The case x′′ = f(x, t). SIAM J. Appl. Dyn. Syst., 8(1):279–297, 2009. [URL];

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[12] P.B. Acosta-Humanez, M. Alvarez Ramırez and J. Delgado. Non-integrability of somefew body problems in two degrees of freedom. Qual. Theory Dyn. Syst., 8(2):209–239,2009.

[13] R.A. Adomaitis, I.G. Kevrekidis and R. de la Llave. A computer-assisted study ofglobal dynamic transitions for a noninvertible system. Internat. J. Bifur. Chaos Appl.Sci. Engrg., 17(4):1305–1321, 2007. Google Scholar; Math. Rev. MR 2008f:37193.

[14] E.M. Alessi, G. Gomez and J.J. Masdemont. Leaving and reaching the Moon from alibration point orbit. PAMM – Proc. Appl. Math. Mech., 7:1030911–1030912, 2007.

[15] E.M. Alessi, G. Gomez and J.J. Masdemont. Leaving the Moon by means of in-variant manifolds of libration point orbits. Commun. Nonlinear Sci. Numer. Simul.,14(12):4153–4167, 2009. [URL]; Math. Rev. MR2537575.

[16] F. Angulo, E. Fossas, T.M. Seara and G. Olivar. Bounding the output error in a buckpower converter using perturbation theory. Math. Probl. Eng., pp. Art. ID 732039, 20,2008. Math. Rev. MR 2009f:93030.

[17] A. Apte, R. de la Llave and E. Petrisor. Comment on: “Reconnection scenarios and thethreshold of reconnection in the dynamics of non-twist maps” [Chaos Solitons Fractals,14(1):117–127, 2002] by Petrisor. Chaos Solitons Fractals, 27(4):1115–1116, 2006. Math.

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[18] L. Arona and J.J. Masdemont. Computation of heteroclinic orbits between normallyhyperbolic invariant 3-spheres foliated by 2-dimensional invariant tori in Hill’s problem.Discrete Contin. Dyn. Syst., (Dynamical Systems and Differential Equations. Proceed-ings of the 6th AIMS International Conference, suppl.):64–74, 2007. Google Scholar; Math.

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[19] E. Athanassoula, M. Romero and J.J. Masdemont. Rings ans spirals in barred galaxies.I building blocks. Monthly Notices Roy. Astronom. Soc., 2008. Accepted.

[20] I. Baldoma, E. Fontich, R. de la Llave and P. Martın. The parameterization method forone-dimensional invariant manifolds of higher dimensional parabolic fixed points. Dis-crete Contin. Dyn. Syst., 17(4):835–865, 2007. Google Scholar; Math. Rev. MR 2008g:37024.

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[24] E. Barrabes, G. Gomez, J.M. Mondelo and M. Olle. Numerical parametrizations of li-bration point trajectories and their invariant manifolds. Adv. Astronaut. Sci., 129:1153–1168, 2008.

[25] E. Barrabes, J.M. Mondelo and M. Olle. Some particularities of the collinear point L3

in the RTBP. In Proceedings of the Ninth Conference on Celestial Mechanics (Span-ish), Monogr. Real Acad. Ci. Exact. Fıs.-Quım. Nat. Zaragoza, 30, pp. 35–41. Acad.Cienc. Exact. Fıs. Quım. Nat. Zaragoza, Zaragoza, 2007. Google Scholar; Math. Rev. MR

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[27] E. Barrabes, J.M. Mondelo and M. Olle. Numerical continuation of families of homo-clinic connections of periodic orbits in the RTBP. Nonlinearity, 22:2901–2918, 2009.

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[29] J.M. Benita, G. Deco, A. Guillamon and M.V. Sanchez-Vives. Reciprocal interactionof slow oscillatory activity and short-term synaptic depression in a biophysical networkmodel of the cerebral cortex, 2009. Preprint submitted to J. Neurosci.

[30] D. Blazquez-Sanz. Differential Galois theory and Lie-Vessiot systems. Ph.D. thesis,Univ. Politecnica de Catalunya. VDM Verlag, October 2008. [URL]

[31] D. Blazquez-Sanz. Affine structures on jet and Weil bundles. Colloq. Math., 114(2):291–305, 2009. [URL]; Math. Rev. MR 2009j:58007.

[32] D. Blazquez-Sanz and J.J. Morales-Ruiz. Differential Galois theory of algebraic Lie–Vessiot systems. In Acosta-Humanez and Marcellan [3], pp. 1–58. [URL]

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[36] F. Calogero and D. Gomez-Ullate. A new class of solvable many-body problems withconstraints, associated with an exceptional polynomial subspace of codimension 2. J.Phys. A–Math. Theor., 40(27):F573–F580, 2007. Google Scholar; Math. Rev. MR 2008i:37132.

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C4. CONFERENCIAS EN CONGRESOS, SIMPOSIOS Y REUNIONES (POR IN-VITACION)

La gran mayorıa de los trabajos publicados han sido expuestos en conferencias o semi-narios, tanto de caracter nacional como internacional. Ademas, se ha participado en laorganizacion de diversos congresos, workshops y cursos, tanto nacionales como interna-cionales:

Minisymposium: Instability in mechanical systems. SIAM Conference on Applica-tions of Dynamical Systems (DS09). Period: 19-21 May 2009. Utah. Co-organiser:Tere M. Seara.

CRM Research Programme for the academic year 2008-2009. Mathematical Bi-ology: Modelling and Differential Equations Period: January to June 2009. Co-ordinators: Angel Calsina (Universitat AutAnoma de Barcelona), Jose AntonioCarrillo (ICREA and Universitat AutAnoma de Barcelona), Antoni Guillamon(Universitat PolitAcnica de Catalunya).

UPC INTERNATIONAL INTEGRABILITY SEMINAR 2010 - UIIS. Jun 7-82010. Organisers: Primitivo Acosta-Humanez, David Blazquez Sanz, J. TomasLazaro.

Mini-simposia on Dynamical Systems and Space Exploration at SIAMmeeting on Dynamical Systems and Partial Differential Equations, Barcelona(Spain) May 31st - June 4th, 2010 (G. Gomez, M.W. Lo, J.J. Masdemont, organ-isers)

Mini-simposia on Applications of Dynamical Systems to Dynamical As-tronomy at SIAM meeting on Dynamical Systems and Partial Differential Equa-tions, Barcelona (Spain) May 31st - June 4th, 2010 (G. Gomez, M.W. Lo, J.J.Masdemont, organisers)

Referencia: MTM2006-00478 14

Joint SIAM/RSME-SCM-SEMA Meeting: Emerging Topics in Dynamical Sys-tems and Partial Differential Equations DSPDEs’10. May 31st, a June 4th, 2010.Barcelona, Spain Tere Martınez-Seara, co-organiser.

3rd Conference on Nonlinear Science and Complexity NSC’10. Ankara(Turkey) 28-31 July, 2010 (M. Gidea, J.J. Masdemont, organisers)

D. CARACTER DE LOS RESULTADOS DEL PROYECTO (senalar hasta dos op-ciones)

(X) Teoricos (X) Teorico-practicos( ) Practicos ( ) De inmediata aplicacion industrial

E. COLABORACIONES

E1. SI EL PROYECTO HA DADO LUGAR A COLABORACIONES CON OTROSGRUPOS DE INVESTIGACION, comentelas brevemente.

Ha habido diversas colaboraciones con otros grupos de investigacion, como se desprendede las publicaciones que se han incluido en este informe. Concretamente, se ha publicadocon los siguientes (grupos de) investigadores:

X. Li (Hunan Normal University, China),

M. Gidea (Northeastern Univ. of Chicago, IL, USA),

J. D. Meiss (Univ. of Colorado at Boulder, CO, USA),

H. Lomelı (Univ. of Texas at Austin, TX, USA),

A. Windsor (Univ. of Memphis, TN, USA),

M. Shub (Univ. of Toronto, Canada),

A. Olvera (Univ. Nacional Autonoma de Mexico, Mexico),

S. V. Gonchenko, O. Koltsova y L. Lerman (Univ. de Nizhny Novgorod, Rusia),

A.V. Borisov y I.S. Mamaev (Udmurt State Univ., Rusia),

A. J. Maciejewski (Univ. of Zielona Gora, Polonia),

M. Przybylska (Nicolaus Copernicus Univ., Polonia),

B. Jovanovic (Mathematical Institute SANU, Serbia),

S. J. Hogan (Univ. of Bristol, UK),

H.W. Braden (Univ. of Edinburgh, UK),

J.-P. Ramis y L. Gavrilov (Laboratoire Emile Picard, Univ. P. Sabatier, Francia),

E. Athanassoula (Observatorio de Marsella, Francia),

F. Calogero y P. M. Santini (Univ. di Rome“La Sapienza”, Italia),

E. Valdinoci (Univ. di Roma “Tor Vergata”, Italia),

S. Abenda (C.I.R.A.M., Univ. di Bologna, Italia),

M. Sommacal (Istituto Nazionale di Fisica Nucleare, Italia),

Referencia: MTM2006-00478 15

F. Marcellan (Univ. Carlos III, Espana),

E. Barrabes (Univ. de Girona, Espana),

J. Llibre (Univ. Autonoma de Barcelona, Espana),

E. Fontich, A. Haro, G. Gomez, C. Simo y S. Simon (Univ. de Barcelona, Espana),

E. Fossas, R. M. Massa (Univ. Politecnica de Catalunya, Espana).

Ademas, de acuerdo con las previsiones de colaboraciones previstas en la memoria delproyecto, han tenido lugar diversas visitas por parte de los siguientes investigadores:

Ano 2006

[V1] Chong-Qing Cheng (Nanjing University, China), 8–12 Jan 2006.

[V2] Matteo Sommacal (Universite Paris VI, France), 25–30 Jan 2006.

[V3] John Hogan (University of Bristol, UK), 4–10 Feb 2006.

[V4] Vladimir Gonchenko (Research Institute for Applied Mathematics and Cyber-netics, Nizhny Novgorod, Russia), 11–16 Feb 2006.

[V5] Lubomir Gavrilov (Universite Paul Sabatier, Toulouse, France), 17–19 Feb 2006.

[V6] Andrew Hone (University of Kent at Canterbury, UK), 15–22 Feb 2006.

[V7] Paolo Santini (Universita di Roma “La Sapienza”, Italy), 25–28 Feb 2006.

[V8] Alex Bombrun (Ecole des Mines de Paris, France), 26–28 Apr 2006.

[V9] Niky Kamran (McGill University, Canada), 29 May – 1 Jun 2006.

[V10] Sonia Carvalho (Universidade Federal de Minas Gerais de Brasil), 14 Mar – 20Jun 2006.

[V11] John Hogan (University of Bristol, UK), 17–24 Jun 2006.

[V12] Ma. Jose Romero Valles (Universidad de Granada), 18–24 Jun 2006.

[V13] Mirko Degli Esposti (Universita degli Studi di Bologna, Italy), 16–26 Jun 2006.

[V14] Simonetta Abenda (Universita di Bologna, Italy), 9–15 Jul 2006.

[V15] Armando Treibish (Universite d’Artois, France), 9–15 Jul 2006.

[V16] Clareence Eugene Wayne (Boston University, USA), 9–16 Jul 2006.

[V17] Francois Hamel (Universite Aix-Marseille III, France), 8–18 Jul 2006.

[V18] Hiroshi Matano (University of Tokyo, Japan), 7–19 Jul 2006.

[V19] Pedro Didonato (Instituto Tecnologico de Aeronautica, Brasil), 9 Jan – 21 Jul2006.

[V20] Bozidar Jovanovic (University of Belgrade, Serbia), 20–24 Jul 2006.

[V21] Vadim Kaloshin (California Institute of Technology, USA), 6–13 Sep 2006.

[V22] Mark Alber (Stanford University, California, USA), 18–19 & 23–24 Sep 2006.

Referencia: MTM2006-00478 16

[V23] Pedro Veerman (Portland State University, USA), 22 Nov – 3 Dic 2006.

[V24] Alexey Borisov (Universidad de Izevsk, Russia), 2–6 Dic 2006.

[V25] Ivan Mamaev (Universidad de Izevsk, Russia), 2–6 Dic 2006.

Ano 2007

[V26] Mercelena Hernandez (Universidad Sergio Arboleda, Bogota, Colombia), 10–17Feb 2007.

[V27] Carlos Pena Rincon (Universidad Sergio Arboleda, Bogota, Colombia), 10–17 Feb2007.

[V28] Johan Manuel Redondo (Universidad Sergio Arboleda, Bogota, Colombia), 10–17Feb 2007.

[V29] Gonzalo Contreras (CIMAT, Guanajuato, Mexico), 12–18 Feb 2007.

[V30] Dario Bambusi (Universita degli studi di Milano, Italy), 10–13 Jun 2007.

[V31] Vadim Kaloshin (California Institute of Technology, USA), 5–22 Jun 2007.

[V32] Robert Milson (Dalhousie University, Canada), 24–26 Jun 2007.

[V33] Sebastian Benzekry (Universite Aix-Marseille III, France), 5 Mar – 30 Jun 2007.

[V34] Oksana Koltsova (Nizhny Novgorod State University, Russia), 17 Jun – 2 Jul2007.

[V35] David Sauzin (CNRS – IMCCE, Observatoire de Paris, France), 1–7 Jul 2007.

[V36] Armando Treibish (Universite d’Artois, France), 7–11 Jul 2007.

[V37] Simonetta Abenda (Universita di Bologna, Italy), 9–15 Jul 2007.

[V38] Jean Pierre Marco (Universite Pierre et Marie Curie, Paris 6, France), 14–20 Jul2007.

[V39] John Hogan (University of Bristol, UK), 3–7 Sep 2007.

[V40] Eric Lombardi (CNRS de France, France), 3–7 Sep 2007.

[V41] Luca Zampogni (Universita di Perugia, Italy), 12–19 Nov 2007.

[V42] Marco Antonio Teixeira (UNICAMP - IMECC, Brasil), 19–22 Dec 2007.

Ano 2008

[V43] Florentino Borondo (Universidad Autonoma de Madrid), 23–25 Jan 2008.

[V44] David Farrellly (Universidad Autonoma de Madrid), 23–25 Jan 2008.

[V45] Luis Garcia Naranjo (Ecole Politechnique, Lausanne, Switzerland), 28 Jan – 2Feb 2008.

[V46] Sonia Pinto de Carvalho (Universidade Federal de Minas Gerais de Brasil), 17Jan – 8 Feb 2008.

[V47] Alexander Seyranian (Lomonosov Institute of Mechanics, University of Moscow,Russia), 12–17 Mar 2008.

Referencia: MTM2006-00478 17

[V48] Marco Antonio Teixeira (UNICAMP - IMECC, Brasil), 1 Jan – 15 Apr 2008.

[V49] Mike Jeffrey (University of Bristol, UK), 7–12 Apr 2008.

[V50] Alan Champneys (University of Bristol, UK), 6–14 Apr 2008.

[V51] Vladimir Dragovic (Mathematics Institute of Belgrade, Serbia), 9–14 Apr 2008.

[V52] Sergey V. Gonchenko (Research Institute for Applied Mathematics and Cyber-netics, Russia), 24 Mar – 20 Apr 2008.

[V53] Andrew Pickering (Universidad Rey Juan Carlos, Madrid), 20–25 Apr 2008.

[V54] Miguel Carriegos (Universidad de Leon), 10 Apr – 23 May 2008.

[V55] Shouchuan Hu (Missouri State University, USA), 5–8 Jun 2008.

[V56] John Hogan (University of Bristol, UK), 16–20 Jun 2008.

[V57] Turgay Uzer (Georgia Institute of Technology, USA), 14–26 Sep 2008.

[V58] Massimiliano Berti (Universita di Napoli, Italy), 15–26 Sep 2008.

[V59] Alain Chenciner (Universite de Paris VII, France), 21–26 Sep 2008.

[V60] George Haller (Massachusetts Institute of Technology, USA), 22–26 Sep 2008.

[V61] Robert MacKay (University of Warwick, UK), 22–26 Sep 2008.

[V62] Yingfei Yi (Georgia Institute of Technology, USA), 22–26 Sep 2008.

[V63] Vassili Gelfreich (University of Warwick, UK), 1–27 Sep 2008.

[V64] Chris Jones (University of North Carolina at Chapel-Hill, USA – University ofWarwick, UK), 14–27 Sep 2008.

[V65] Anatoli Neishtadt (Moscow Space Research Institute, Russia – LoughboroughUniversity, UK), 14–27 Sep 2008.

[V66] Dmitry Treschev (Moscow State University, Russia), 14–27 Sep 2008.

[V67] Luigi Chierchia (Universia Roma Tre, Italy), 15–27 Sep 2008.

[V68] Hakan Eliasson (Universite de Paris VII, France), 21–28 Sep 2008.

[V69] Ernesto Perez-Chavela (Universidad Autonoma Metropolitana, Mexico), 30 Aug– 29 Sep 2008.

[V70] David Sauzin (CNRS – IMCCE, Observatoire de Paris, France), 21 Sep – 3 Oct2008.

[V71] Alessandra Celletti (Universia di Roma “Tor Vergata”, Italy), 21 Sep – 4 Oct2008.

[V72] Enrico Valdinoci (Universita degli Studi di Roma “Tor Vergata”, Italy), 7–10 &19–24 Oct 2008.

[V73] Hector Lomelı (Instituto Tecnologico Autonomo de Mexico), 16–26 Oct 2008.

[V74] Jun Yan (Fudan University, China), 1–29 Oct 2008.

[V75] Yannick Sire (Universite Aix-Marseille III, France), 1–31 Oct 2008.

Referencia: MTM2006-00478 18

[V76] Albert Fathi (Ecole Normale Superieure de Lyon, France), 28 Oct – 2 Nov 2008.

[V77] Jacques Fejoz (Institut de Mathematiques de Jussieu, France), 3 – 13 Nov 2008.

[V78] Vadim Kaloshin (University of Maryland, USA), 7–13 Nov 2008.

[V79] Wang Wei-Min (Universite de Paris-Sud, Paris XI, France), 8–14 Nov 2008.

[V80] Daniel Clyde Offin (Queens University, USA), 26 Oct – 15 Nov 2008.

[V81] Laurent Niederman (Universite de Paris-Sud, Orsay, France), 26 Oct – 21 Nov2008.

[V82] Michael Benedicks (Matematik Institutionen KTH, Germany), 23–27 Nov 2008.

[V83] Vered Rom-Kedar (Weizmann Institute of Science, Israel), 20–28 Nov 2008.

[V84] Federico Conglitore (Albert-Ludwigs-Universitat, Germany), 16–30 Nov 2008.

[V85] Vadim Zharnitsky (University of Illinois at Urbana, USA), 23–30 Nov 2008.

[V86] Christina Roeckerath (RWTH Aachen University, Germany), 13 Nov – 1 Dec2008.

[V87] Eli Shlizerman (Tel Aviv University, Israel), 24 Nov – 4 Dec 2008.

[V88] Roberto Barrio (Universidad de Zaragoza), 1–5 Dec 2008.

[V89] Luis Benet (Universidad Nacional Autonoma de Mexico), 1–5 Dec 2008.

[V90] Luca Biasco (Universita di Roma III, Italy), 1–5 Dec 2008.

[V91] Florentino Borondo (Universidad Autonoma de Madrid), 1–5 Dec 2008.

[V92] Fernando Casas (Universitat Jaume I), 1–5 Dec 2008.

[V93] Cristel Chandre (Universite de Marseille, France), 1–5 Dec 2008.

[V94] Charles Jaffe (University of West Virginia, USA), 1–5 Dec 2008.

[V95] Ugo Locatelli (Universita di Roma “Tor Vergata”), 1–5 Dec 2008.

[V96] Paul Milewski (University of Wisconsin-Madison, USA), 1–5 Dec 2008.

[V97] Rytis Paskauskas (Univesita di Trieste, Italy), 1–5 Dec 2008.

[V98] Jesus Pelaez (Universidad Politecnica de Madrid), 1–5 Dec 2008.

[V99] Ettore Perozzi (SpaceOPSAccademy & Telespazio, Italy), 1–5 Dec 2008.

[V100] Dolors Puigjaner (Universitat de Rovira i Virgili), 1–5 Dec 2008.

[V101] Clark Robinson (Northwestern University, USA), 1–5 Dec 2008.

[V102] Philippe Robutel (Observatoire de Paris, France), 1–5 Dec 2008.

[V103] Shane Ross (Virginia Tech, USA), 1–5 Dec 2008.

[V104] Piotr Zgliczynski (Jagiellonian University, Krakov, Poland), 1–5 Dec 2008.

[V105] Vıctor Munoz Villarragut (Universidad de Valladolid), 1 Nov – 6 Dec 2008.

Referencia: MTM2006-00478 19

[V106] Sergey Bolotin (Moscow State University, Russia – University of Wisconsin,USA), 6 Nov – 6 Dec 2008.

[V107] Rodrigo Trevino (University of Maryland, USA), 30 Nov – 6 Dec 2008.

[V108] Shane D. Ross (Virginia Tech, USA), 22 Nov – 7 Dec 2008.

[V109] Jean-Pierre Marco (Universite Pierre et Marie Curie, Paris 6, France), 23 Nov –7 Dec 2008.

[V110] Maciej Capinski (AGH University of Science and Technology, Krakov, Poland),30 Nov – 7 Dec 2008.

[V111] Philip Holmes (Princeton University, USA), 3–7 Dec 2008.

[V112] Hinke Osinga (Cornell University, USA), 30 Nov – 10 Dec 2008.

[V113] Arturo Olvera Chavez (Universidad Nacional Autonoma de Mexico), 16 Nov – 12Dec 2008.

[V114] Sergey Gonchenko (Research Institute for Applied Mathematics and Cybernetics,Russia), 17 Nov – 17 Dec 2008.

[V115] Pierre Berger (Centre de Recerca Matematica), 1 Sep – 20 Dec 2008.

[V116] Renato Calleja (University of Texas at Austin, USA), 1 Sep – 20 Dec 2008.

[V117] Chong-Qing Cheng (Nanjing University, China), 1 Sep – 20 Dec 2008.

[V118] Alfonso Sorrentino (Princeton University, USA), 1 Sep – 20 Dec 2008.

[V119] Xifeng Su (Nanjing University, China), 1 Sep – 20 Dec 2008.

[V120] Xuemei Li (University of Texas at Austin, USA), 2 Oct – 20 Dec 2008.

[V121] Marian Gidea (Northestern Illinois University, USA), 1 Sep – 31 Dec 2008.

Ano 2009

[V122] Matthieu Desroches (U. Bristol), 3–15 May 2009.

[V123] Vincent Hakim (Ecole Normal Superieure, Paris), 10–23 May 2009.

[V124] David Terman (The Ohio State U.), 4–28 May 2009.

[V125] Louis Tao (Beijing Univ.), 3–31 May 2009.

[V126] Nicolas Brunel (Univ. Paris Descartes), 22 May – 6 Jun 2009.

[V127] Joaquın J. Torres (Univ. de Granada), 11 May – 9 Jun 2009.

[V128] Luis Garcıa Naranjo (Ecole Polytechnique de Lausanne), 19–21 & 27–30 Jun 2009.

[V129] Jacques Fejoz (Universite Pierre et Marie Curie, Paris 6), 12–23 Jul 2009.

[V130] Olesya Grishchenko (Smeal College of Business, Pennsylvania), 12–24 Jul 2009.

[V131] Vadim Kaloshin (California Institute of Technology), 12–24 Jul 2009.

[V132] Andrzej Maciejewski (Institute of Astronomy, University of Zielona Gora, Poland),20–30 Sep 2009.

Referencia: MTM2006-00478 20

[V133] Maria Przbulska (Nikolaus Copernicus University, Torun, Poland), 20–30 Sep2009.

[V134] Christina Roeckerath (Lehrstuhl A fur Mathematik), 11 Sep – 3 Oct 2009.

[V135] Mike Jeffrey (University of Bristol), 25 Sep – 9 Oct 2009.

[V136] Alan R. Champneys (University of Bristol), 13–15 Oct 2009.

[V137] Marco Antonio Teixeira (IMECC, Unicamp, Brasil), 7–23 Oct 2009.

[V138] Christina Roeckerath (Lehrstuhl A fur Mathematik), 25–30 Oct 2009.

[V139] Taras Skrypnyk (International School for Advanced Studies, Trieste), 27–31 Oct2009.

[V140] John Hogan (University of Bristol), 2–5 Dec 2009.

E2. SI HA PARTICIPADO EN PROYECTOS DEL PROGRAMA MARCO DE I+DDE LA UE Y/O EN OTROS PROGRAMAS INTERNACIONALES EN TEMA-TICAS RELACIONADAS CON LAS DE ESTE PROYECTO, indique programa,tipo de participacion y beneficios para el proyecto.

Mencione las solicitudes presentadas al Programa Marco de la UE durante la ejecucion delproyecto, aunque no hayan sido aprobadas.

Yu. Fedorov ha participado en el proyecto ALISA (Algoritmically Integrable Sys-tems), perteneciente al EU Framework Program on Research and TechnologicalDevelopment. (Periodo: 2005–2007. Objetivo: Desarrollo de algoritmos de com-putacion simbolica en el estudio de sistemas integrables.)

J. Masdemont ha participado en dos proyectos del Centre Nacionale d’Etudes Spa-tiales (CNES) con referencias R-S06/VF-0001-052 y R-S07/TG-0004-005. (Peri-odo: 2007–2008. Objetivo primer proyecto: Desarrollo de una librerıa de softwarepara el analisis de misiones alrededor de los puntos de libracion. Objetivo segundoproyecto: Estudio de transferencias altenativas a orbitas geoestacionarias.)

J. J. Morales ha participado en el proyecto AMDS (Algebraic Methods in Dynam-ical Systems), perteneciente a la Agence Nationale de la Recherche (ANR) con lareferencia JC05-41465. (Periodo: 2008. Objetivo: Financiacion de un congreso.)

R. de la Llave ha participado en el proyecto Analytical and numerical studies oflong range order in Dynamical Systems and PDEs, perteneciente a la NationalScience Foundation (NSF), con la referencia DMS-0354567. (Periodo: 2004-2009.Objetivo: Estudiar el comportamiento global a largo plazo de ciertos sistemasdinamicos y ecuaciones en derivadas parciales.)

T. M. Seara ha participado en el proyecto MicroMed, perteneciente a las AccionesComplementarias Internacionales, con la referencia PCI2005-A7-0284. (Periodo:2006–2009. Objetivo: Diseno y estudio de viabilidad de un microtron compactopara aplicaciones medicas.)

T. M. Seara y J. Villanueva han participado en el proyecto CODY, ConformalStructures and Dynamics, perteneciente al EU Research Training Network den-tro de las Marie Curie Mobility Actions RTN call FP6-MRTN-CT-2006-035651.(Periodo: 2006–2009. Objetivo: Utilizacion de tecnicas cuentitativas de cirugıaquasiconforme en el estudio de anillos de Hermann.)

Referencia: MTM2006-00478 21

J. Masdemont esta participando en el proyecto AstroNet, The AstrodynamicsNetwork, perteneciente al EU Research Training Network dentro de las MarieCurie Grant, call MCRTN-CT-2006-035151. (Periodo: 2007–2010. Objetivo: For-macion de estudiantes de doctorado en temas de Astrodinamica para trabajar enel sector aeroespacial.)

Los beneficios, ademas de las publicaciones conjuntas, estan basados en una estrechay continuada interaccion con estos grupos, todos ellos de primera lınea, que, gracias ala complementariedad de tecnicas y herramientas, ha dado lugar a una mayor profun-dizacion conjunta en los temas estudiados.

F. PROYECTOS COORDINADOS1

Describa las actuaciones de coordinacion entre subproyectos, y los resultados de dichacoordinacion en relacion a los objetivos globales del proyecto.

El proyecto no es coordinado.G. RELACIONES O COLABORACIONES CON DIVERSOS SECTORES

G1. SI EN EL PROYECTO HA HABIDO COLABORACION CON ENTES PRO-MOTORES OBSERVADORES (EPO) PARTICIPANTES:

1. Describa en detalle la relacion mantenida con los EPO’s, y la participacion concretade estos en el proyecto, especificando, si procede, su aportacion al mismo en todossus aspectos. (Si se ha modificado la relacion y/o el apoyo del EPO, en relacion conlo previsto a la aprobacion del proyecto, descrıbalo brevemente).

Desde 2003 se han venido realizando labores de consultorıa y soporte para NASA-JPL en relacion con la ya terminada mision espacial Genesis y las futuras TPF(Terrestrial Planet Finder) y JIMO (Jupiter Icy Moons Orbiter).

Durante el perıodo 2003-2005 en colaboracion con la empresa Deimos SpaceS.L. (Madrid) se ha llevado a termino el contrato de investigacion HerramientaAnalıtica y Numerica para el Control Distribuido de Satelites en Formacion den-tro del programa PROFIT de transferencia de tecnologıa Universidad-Empresa.

Durante el perıodo 2004-2006, en colaboracion con la empresa Deimos Space S.L.(Madrid) y Alcatel Space (Toulouse) se ha participado en el contrato de la AgenciaEspacial Europea 18426/04/D/HK Development of a Libration Orbit Design Tooldel departamento de Mission Analisys ESA-ESOC.

2. Describa, si procede, las transferencias realizadas al (los) EPO (s) de los resultadosobtenidos, indicando el caracter de la transferencia y el alcance de su aplicacion.

Dentro del proyecto PROFIT se desarrollaron algoritmos de control y transfer-encia de formaciones de satelites artificiales que fueron implementados en lossimuladores de la empresa Deimos Space.

En el proyecto 18426/04/D/HK se desarrollo software operacional que se im-plemento en ESA-ESOC. El software desarrollado contiene los ultimos avancesobtenidos por el grupo en la parte de investigacion sobre dinamica ligada a lospuntos de libracion. Tambien se implementaron algunos algoritmos relativos avuelo en formacion.

3. Indique si esta colaboracion ha dado lugar a la presentacion de nuevos proyectos o sise tiene intencion de continuarla en el futuro. En caso afirmativo, describa brevementecomo va a concretarse.

1A rellenar solo por el coordinador del proyecto.

Referencia: MTM2006-00478 22

Se ha iniciado una consultorıa con Deimos Space y Alcatel Space sobre estudiosreferentes a la mision Darwin de la Agencia Espacial Europea de la cual Alcateles el contratista principal.

Se estan manteniendo negociaciones con el Centre National d’Estudes Spatiales(CNES-Toulouse) para la implementacion de librerias sobre dinamica de libracionen su sistema de calculo.

G2. SI EL PROYECTO HA DADO LUGAR A OTRAS COLABORACIONES CONEL ENTORNO SOCIOECONOMICO (INDUSTRIAL, ADMINISTRATIVO, DESERVICIOS, ETC.), NO PREVISTAS INICIALMENTE EN EL PROYECTO, des-crıbalas brevemente.No ha habido otras colaboraciones no previstas incialmente en el proyecto.

H. GASTOS REALIZADOS H1. GASTOS REALIZADOS EN LA ÚLTIMA ANUALIDAD Nota: Debe cumplimentarse este apartado independientemente de la justificación económica enviada por el organismo. 1.- Indique el total de gasto realizado en el proyecto:

Concepto Total gasto de la anualidad

(€)

Personal

Costes de ejecución 59.188,15

TOTAL GASTO REALIZADO 59.188,15

2.- Comente brevemente si ha h abido algún tipo de incidencia en este apartado que desee

reseñar. H2. GASTOS REALIZADOS DURANTE TODO EL PROYECTO Nota: Debe cumplimentarse este apartado independientemente de la justificación económica enviada por el organismo. Euros 1. Gastos de personal (indicar número de personas, situación laboral y función desempeñada)

Total 64.764,85 € 2. Material inventariable (describir brevemente el material adquirido) Material bibliográfico y equipamiento informático (ordenadores de sobremesa y portátiles, impresoras y pantallas). Total 71.639,25 € ____________________________________________________________________________ 3. Material fungible (describir brevemente el tipo de material) Software no inventariable (bobinas de CD's, DVD's, tarjetas de memoria), toner, separatas, papel de impresora… Total 2.502,61 € ____________________________________________________________________________ 4. Viajes y dietas (describir brevemente) Asistencia a c ongresos, eventos científicos, cursos, escuelas especializadas, etc. Visitas realizadas por miembros del grupo a otros grupos de investigación, nacionales

e internacionales. Contribución económica para sufragar gastos de visitantes nacionales y extranjeros. Total 169.290,59 € ____________________________________________________________________________ 5. Otros gastos (describir brevemente) Inscripciones a congresos, gastos de visitantes nacionales y extranjeros… Total 56.361,10 € ____________________________________________________________________________ 6. Costes indirectos Total 94.355,10 € ____________________________________________________________________________ 7. Dotación adicional o complementos salariales, si procede Total 86.100,00 € ____________________________________________________________________________ TOTAL GASTOS EJECUTADOS DEL PROYECTO 545.013,50 € ____________________________________________________________________________

I. INFORMACIÓN CORRESPONDIENTE A LA ÚLTIMA JUSTIFICACIÓN DE GASTO. I1. PERSONAL ACTIVO EN EL PROYECTO DURANTE EL ÚLTIMO PERÍODO DE JUSTIFICACIÓN. En el cuadro siguiente debe recogerse la situación de todo el personal del o de los Organismos participantes que haya prestado servicio en el proyecto en la anualidad que se justifica, o que no haya sido declarado anteriormente, y cuyos costes (salariales, dietas, desplazamientos, etc.), se imputen al mismo. Si la persona estaba incluida en la solicitud original, marque “S” en la casilla correspondiente y no rellene el resto de las casillas a la derecha. Indique en la casilla “Categoría Profesional” el puesto de trabajo ocupado, el tipo de contratación: indefinida, temporal, becarios (con indicación del tipo de beca: FPI, FPU, etc.), etc. En el campo “Función en el proyecto” indique el tipo de función/actividad realizada en el proyecto, (p. e., investigador, técnico de apoyo,…). Recuerde que: - En este capítulo solo debe incluir al personal vinculado a los Organismos participantes en el proyecto. Los gastos de personal externo (colaboradores científicos, autónomos…) que haya realizado tareas para el proyecto debe ser incluido en el capítulo de “Varios”. - Las “Altas” y “Bajas” deben tramitarse de acuerdo con las “Instrucciones para el desarrollo de los proyectos de I+D” expuestas en la página web del MICINN.

Si no incluido en solicitud original:

Apellido 1 Apellido 2 Nombre NIF/NIE Catg.ª Profesional Incluido en solicitud original

Función en el proyecto

Fecha de alta Observaciones

Delshams Valdés Amadeu 38490659K Prof. CU S Martinez-Seara Alonso M. Teresa 46526613M Prof. CU S

Ramirez Ros Rafael 40933296N Prof. TU S Villanueva Castelltort Jordi 46545752P Prof. TU S

Ollé Torner Mercedes 33869546E Prof. TU S Puig Sadurní Joaquin 46762416N Prof. TU S

Pantazi Chara X2882936R Prof. Ayudante S Gutierrez Serrés Pere 35070712J Prof. TU S

Martin De la Torre Pablo 43046286S Prof. TU S Masdemont Soler Josep Joaquim 40602280N Prof. TU S Guillamon Grabolosa Antoni 46670968N Prof. TU S

Lázaro Ochoa José Tomás 43673294C Prof. Colaborador S Pacha Andujar Juan Ramón 46590657V Prof. Colaborador S

Fedorov Yuri X2158441Y Prof. Contratado doctor S Benita Bordes José Manuel X6115287R Becario Predoctoral S Larreal Barreto Oswaldo José X6346842S Becario Predoctoral S Acosta Humánez Primitivo X06157638D Profesor Investigador S

Belén

Luque Jiménez Alejandro 53089946N Titulado Superior Contratado S

Guardia Munarriz Marcel 47724038G Investigador Asociado S 15/08/10

Roldan González Pablo 38142528H Titulado Superior Contratado S

Gonchenko Marina X7591774A Becario Predoctoral S De la Rosa Ibarra Abraham X9276704E Becario Predoctoral S

Huguet Casades Gemma 78090864E Becario Posdoctoral S Olivé Farré M. Carme 39682550Y Prof TEU S

Morales Ruiz Juan José 33816674G Prof. CU S Blázquez Sanz David 70933642D Prof. de Investigación S

Canalias Vila Elisabet 46671640V Investigador Contratado S

Gómez-Ullate Oteiza David 52368263T Prof. TU S De la Llave Canosa Rafael 00786544J Prof. CU S

García Taberner Laura 40317220Z Prof. Colaborador S

Tamarit Sariol Anna 46872163A Becaria FPI Objetivo 1b):

Escisión separatrices

15/02/10

I2. GASTOS DE EJECUCIÓN: MODIFICACIONES DE CONCEPTOS DE GASTO CON RESPECTO A LA SOLICITUD ORIGINAL PARA EL ÚLTIMO PERÍODO DE JUSTIFICACIÓN. Recuerde que los trasvases entre gastos de personal y gastos de ejecución deben tramitarse de acuerdo con las “Instrucciones para el desarrollo de los proyectos de I+D+i” expuestas en la página web del MICINN.

a) Equipamiento: En el cuadro adjunto, rellene una línea por cada equipo adquirido incluido en la justificación de gastos y no previsto en la solicitud inicial que dio lugar a la concesión de la ayuda para el proyecto, y justifique brevemente su adquisición. Si se ha adquirido un equipo en sustitución de otro que figuraba en la solicitud de ayuda inicial (por mejorar sus prestaciones, por obsolescencia del anterior…), indíquelo también en la casilla correspondiente. Identificación del equipo Importe Justificación adquisición Sustituye a ...(en su caso).

b) Viajes/Dietas: En el cuadro adjunto se justificará la imputación de gasto en viajes y dietas solo en el caso de que este tipo de gasto no estuviera previsto en la solicitud inicial

c) Material fungible:

Se describirá y razonará en el siguiente cuadro la adquisición del material fungible incluido en la justificación de gastos, sólo cuando este tipo de gasto no estuviera previsto en la solicitud original.

d) Varios: Se describirán en el siguiente cuadro los gastos varios más relevantes incluidos en la justificación de gastos y no previstos en la solicitud original, justificando brevemente su inclusión. En este apartado se incluirá, entre otros, al personal externo y, en el caso de que el gasto justificado se refiera a colaboraciones científicas, se identificará al colaborador.

FIN DEL INFORME FINAL

Advances in Mathematics 217 (2008) 1096–1153www.elsevier.com/locate/aim

Geometric properties of the scattering mapof a normally hyperbolic invariant manifold

Amadeu Delshams a, Rafael de la Llave b,∗, Tere M. Seara a

a Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spainb Department of Mathematics, University of Texas, Austin, TX 78712-1802, USA

Received 5 November 2006; accepted 23 August 2007

Available online 24 October 2007

Communicated by Michael J. Hopkins

Abstract

Given a normally hyperbolic invariant manifold Λ for a map f , whose stable and unstable invariantmanifolds intersect transversally, we consider its associated scattering map. That is, the map that, given anasymptotic orbit in the past, gives the asymptotic orbit in the future.

We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map issymplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, thereare extremely easy formulas for the primitive function, which have a variational interpretation as differenceof actions.

We use this geometric information to obtain efficient perturbative calculations of the scattering mapusing deformation theory. This perturbation theory generalizes and extends several results already obtainedusing the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained bygeometrically natural methods and do not involve the use of particular coordinate systems, hence the resultscan be used to obtain intersection properties of objects of any type.

We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodicpotential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafaelde la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows,Adv. Math. 202 (1) (2006) 64–188] using methods of fast–slow systems. Moreover, the geometric theoryallows to compute perturbatively the dependence on the slow variables, which does not seem to be accessibleto the previous methods.© 2007 Elsevier Inc. All rights reserved.

* Corresponding author.E-mail addresses: Amadeu.Delshams@upc.es (A. Delshams), llave@math.utexas.edu (R. de la Llave),

Tere.M-Seara@upc.es (T.M. Seara).

0001-8708/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2007.08.014

A. Delshams et al. / Advances in Mathematics 217 (2008) 1096–1153 1147

and, deformation theory gives that

s∗1 (J,ϕ,B, θ) = S0 ◦ s∗

0 (J,ϕ,B, θ) = S0(J,ϕ + �,B, θ) = J∇S0(J,ϕ + �,B, θ)

where J is the symplectic matrix.We denote the coordinates of x∗−, by (J,ϕ + a−,B−, θ) as in (83) and we obtain that

s∗1 (J,ϕ + a−,B, θ) = J∇S0(J,ϕ + a+,B, θ) = −J∇L(J,ϕ, θ), and then:

s∗ε (J,ϕ + a−,B−, θ) = (J,ϕ + a+,B−, θ) − ε2J∇L(J,ϕ, θ) + O

(ε4).

In particular, for the coordinates B we have

B+ = B− − ε2 ∂S0

∂θ(E,ϕ + a−,B−, θ) + O

(ε4) = B− + ε2 ∂L

∂θ(J,ϕ, θ) + O

(ε4).

Finally, using that A± = B± with the normalized parameterization, we obtain that this formulaagrees with (84) provided in [26].

The calculations in those papers were done by very different methods using averaging theorywhich relies on the fact that the energy is a slow variable. The method in [22,26], also used in[25,27], allowed only to compute the energy component of the scattering map but it does notallow us to compute the ϕ component since ϕ is not a slow variable. The method of this paper,also gives the ϕ component of the first order expansion of the scattering map.

The applications in [22,26] also involve comparing the scattering map with the effect on theKAM tori. This calculation will not be undertaken here, since it involves asymptotic computationof KAM tori and it involves extra cancellations (the leading term in the intersection is O(ε3)).We nevertheless note that the calculation of intersections simplifies when one uses the geometricproperties indicated above. We hope to come back to this application.

We finish the presentation of this example by noting that the primitive function Pε = P s∗ε of

the scattering map takes the form:

Pε = P0 + ε2P1 + O(ε4)

where the leading term P1 can be controlled from Eq. (74). It is worth noting that d

d(ε2)(αHε)|ε=0

= 0, so that P1(J,ϕ, θ) = −L(J,ϕ − a+, θ) = S0(J,ϕ,B, θ) as in the computations that lead toformula (87).

The expression L had played an important role in the variational calculation in [48]. This isrelated to the variational interpretation of the scattering map discussed in Section 3.4.3.

Acknowledgments

This work has been supported by the MCyT–FEDER Grants BFM2003-9504 and MTM2006-00478. Visits of R.L. to Barcelona which were crucial for this work have been supported byInvestigador Visitant of I.C.R.E.A. The work of R.L. has been also supported by NSF grants. Wethank M. Gidea, B. Fayad, P. Roldán for discussions and suggestions. In particular, M. Gideabrought [33] to our attention and P. Roldán made valuable observations that removed impreci-sions.

J Nonlinear Sci (2010) 20: 595–685DOI 10.1007/s00332-010-9068-8

Exponentially Small Splitting for the Pendulum:A Classical Problem Revisited

Marcel Guardia · Carme Olivé · Tere M. Seara

Received: 10 May 2009 / Accepted: 1 April 2010 / Published online: 7 May 2010© Springer Science+Business Media, LLC 2010

Abstract In this paper, we study the classical problem of the exponentially smallsplitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptoticformula for the distance between the perturbed invariant manifolds in the so-calledsingular case and we compare it with the prediction of Melnikov theory. Secondly,we give exponentially small upper bounds in some cases in which the perturbation isbigger than in the singular case and we give some heuristic ideas how to obtain anasymptotic formula for these cases. Finally, we study how the splitting of separatricesbehaves when the parameters are close to a codimension-2 bifurcation point.

Keywords Exponentially small splitting of separatrices · Melnikov method ·Resurgence theory · Averaging · Complex matching

Mathematics Subject Classification (2000) 34C29 · 34C37 · 37C29 · 34E10

Communicated by J. Marsden.

M. Guardia · T.M. Seara (�)Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028Barcelona, Spaine-mail: tere.m-seara@upc.edu

M. Guardiae-mail: marcel.guardia@upc.edu

C. OlivéDepartament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avda PaïsosCatalans 26, 43007 Tarragona, Spaine-mail: carme.olive@urv.cat

J Nonlinear Sci (2010) 20: 595–685 683

system (which is integrable). In fact, μ(ε) = μi ± c±i ε2 + o(ε2) for certain c±

i > 0,are codimension-1 curves γ ±

i on which occurs a pitchfork bifurcations in which aparabolic critical point appears and then bifurcates to two elliptic and a hyperbolicpoints, giving birth to a figure eight made of two homoclinic connections. The studyof the splitting in the domains in the parameter space delimited by the pairs of curvesγ ±i in which both hyperbolic structures coexist remains open.

In Fig. 12, we illustrate the set of values of the parameters (μ, ε) for which we canprovide results about the exponentially small splitting.

The authors think that this paper shows that the problem of the exponentially smallsplitting of separatrices is still a problem that is far from being completely understoodand bring some new ideas that open new points of view to deal with it.

Acknowledgements The authors want to thank I. Baldomá and E. Fontich for valuable remarks andsuggestions, and also the referees, and especially the editor P. Holmes, for their help to improve the finalversion of the manuscript.

The authors have been partially supported by the Spanish MCyT/FEDER grant MTM2006-00478.Moreover, the research of M.G. has been supported by the Spanish Ph.D. grant FPU AP2005-1314.

References

Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables. Dover, New York (1992). Reprint of the 1972 edition

Angenent, S.: A variational interpretation of Mel’nikov’s function and exponentially small separatrix split-ting. In: Symplectic Geometry. London Math. Soc. Lecture Note Ser., vol. 192, pp. 5–35. CambridgeUniv. Press, Cambridge (1993)

Baldomá, I.: The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems.Nonlinearity 19(6), 1415–1445 (2006)

Baldomá, I., Fontich, E.: Exponentially small splitting of invariant manifolds of parabolic points. Mem.Am. Math. Soc. 167(792), x+83 (2004)

Baldomá, I., Fontich, E.: Exponentially small splitting of separatrices in a weakly hyperbolic case. J. Differ.Equ. 210(1), 106–134 (2005)

Baldomá, I., Seara, T.M.: Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zerosingularity. J. Nonlinear Sci. 16(6), 543–582 (2006)

Baldomá, I., Seara, T.M.: The inner equation for generic analytic unfoldings of the Hopf-zero singularity.Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 323–347 (2008)

Balser, W.: From Divergent Power Series to Analytic Functions. Lecture Notes in Mathematics, vol. 1582.Springer, Berlin (1994). Theory and application of multisummable power series

Benseny, A., Olivé, C.: High precision angles between invariant manifolds for rapidly forced Hamiltoniansystems. In: Proceedings Equadiff91, pp. 315–319 (1993)

Bonet, C., Sauzin, D., Seara, T., València, M.: Adiabatic invariant of the harmonic oscillator, complexmatching and resurgence. SIAM J. Math. Anal. 29(6), 1335–1360 (1998) (electronic)

Candelpergher, B., Nosmas, J.-C., Pham, F.: Approche de la résurgence. Actualités Mathématiques [Cur-rent Mathematical Topics]. Hermann, Paris (1993)

Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor. 60(1),144 (1994)

Delshams, A., Gutiérrez, P.: Splitting potential and the Poincaré–Melnikov method for whiskered tori inHamiltonian systems. J. Nonlinear Sci. 10(4), 433–476 (2000)

Delshams, A., Ramírez-Ros, R.: Melnikov potential for exact symplectic maps. Commun. Math. Phys.190(1), 213–245 (1997)

Delshams, A., Ramírez-Ros, R.: Exponentially small splitting of separatrices for perturbed integrablestandard-like maps. J. Nonlinear Sci. 8(3), 317–352 (1998)

Delshams, A., Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forcedpendulum. Commun. Math. Phys. 150(3), 433–463 (1992)

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPLIED DYNAMICAL SYSTEMS c© 2008 Society for Industrial and Applied MathematicsVol. 7, No. 4, pp. 1527–1557

Separatrix Splitting in 3D Volume-Preserving Maps∗

Hector E. Lomelı† and Rafael Ramırez-Ros‡

Abstract. We construct a family of integrable volume-preserving maps in R3 with a two-dimensional hetero-clinic connection of spherical shape between two fixed points of saddle-focus type. In other contexts,such structures are called Hill’s spherical vortices or spheromaks. We study the splitting of the sepa-ratrix under volume-preserving perturbations using a discrete version of the Melnikov method. First,we establish several properties under general perturbations. For instance, we bound the topologicalcomplexity of the primary heteroclinic set in terms of the degree of some polynomial perturbations.We also give a sufficient condition for the splitting of the separatrix under some entire perturbations.A broad range of polynomial perturbations verify this sufficient condition. Finally, we describe theshape and bifurcations of the primary heteroclinic set for a specific perturbation.

Key words. separatrix splitting, volume-preserving maps, primary heteroclinic set, Melnikov method, bifurca-tions

AMS subject classifications. 34C37, 34C23, 37C29, 33E20

DOI. 10.1137/080713173

1. Introduction. A fundamental question in dynamical systems is the effect that smallperturbations of a dynamical system cause on its unperturbed invariant sets. The most studiedunperturbed invariant sets are tori and stable/unstable invariant manifolds of hyperbolic sets.Usually, the unperturbed dynamical system is integrable and has separatrices; that is, itsstable and unstable invariant manifolds overlap. After a generic perturbation, the perturbedstable and unstable invariant manifolds intersect transversely, which gives rise to the onset ofchaos, through the creation of Smale horseshoes. This phenomenon is known as the problemof splitting of separatrices. A widely used technique for detecting such intersections is theMelnikov method.

Our goal is to apply the Melnikov method to the splitting of separatrices in the discretevolume-preserving framework. Similar questions have been considered before. However, webelieve this is the first time that detailed analytical results about the structure of the primaryheteroclinic set and its bifurcations are established for specific maps. This represents a stepforward with respect to previous works [23, 24], in which once a formula for the Melnikov func-tion in terms of an infinite series is written down, the approach becomes mainly numerical,because of the technical difficulties that obstruct the analytical one. Here, we have overcome

∗Received by the editors January 11, 2008; accepted for publication (in revised form) by J. Meiss July 22, 2008;published electronically December 10, 2008.

http://www.siam.org/journals/siads/7-4/71317.html†Department of Mathematics, Instituto Tecnologico Autonomo de Mexico, Mexico, DF 01000 (lomeli@itam.mx).

Current address: Department of Mathematics, The University of Texas, Austin, TX 78712. This paper was finishedwhile this author held a Research Scholar position at the University of Texas at Austin and was supported in partby Asociacion Mexicana de Cultura.

‡Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona,Spain (Rafael.Ramirez@upc.edu). This author was supported in part by MCyT-FEDER grant MTM2006-00478.

1527

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPLIED DYNAMICAL SYSTEMS c© 2009 Society for Industrial and Applied MathematicsVol. 8, No. 3, pp. 1005–1042

A Computational and Geometric Approach to Phase Resetting Curves andSurfaces∗

Antoni Guillamon† and Gemma Huguet‡

Abstract. This work arises from the purpose of applying new tools in dynamical systems to time problems inbiological systems. The main aim of this paper is to develop a numerical method to perform theeffective computation of the phase advancement when we stimulate an oscillator which has not yetreached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phaseresetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood ofthe limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal wefirst perform a careful study of the theoretical grounds (the parameterization method for invariantmanifolds and another approach using Lie symmetries), which allows us to describe the isochronoussections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoreticalframework applicable, we use the numerical algorithms of the parameterization method and othersemianalytical tools to extend invariant manifolds; as a result, we design a numerical scheme tocompute both the isochrons and the PRSs of a given oscillator. Finally, to illustrate this algorithm,we apply it to some well-known biological models and we include a discussion on different biologicaland numerical aspects suggested by these examples.

Key words. parameterization method, Lie symmetries, isochrons, phase resetting curves, numerical computa-tion of invariant objects, biological oscillators

AMS subject classifications. 34C14, 34C20, 92-08, 37N25, 92B05

DOI. 10.1137/080737666

1. Introduction. The behavior of coupled oscillators in biology and, more intensively,in neuroscience has been the subject of a great deal of recent interest, and there is a wideliterature on this topic (see [17] for a survey), mainly because many oscillators can be describedby their phase variables. Moreover, under generic conditions, the phase of the oscillation canalso be defined outside the hyperbolic limit cycle via asymptotic phase. Thus, the stablemanifold of a point x0 on a limit cycle is the union of points having equal phases, and it isoften referred to as the isochron of x0.

To study synchronization, a useful measurable property of a neural oscillator is its phaseresetting curve (PRC). The PRC is found by perturbing the oscillation with a brief stimulusat different times on its cycle and measuring the resulting phase-shift from the unperturbed

∗Received by the editors October 9, 2008; accepted for publication (in revised form) by D. Terman May 26,2009; published electronically August 19, 2009. This work was partially supported by the MCyT/FEDER grantMTM2006-00478 (DACOBIA) and Generalitat de Catalunya grant 2005SGR-986.

http://www.siam.org/journals/siads/8-3/73766.html†Dept. de Matematica Aplicada I, Universitat Politecnica de Catalunya, Dr. Maranon 44-50, E-08028, Barcelona,

Catalonia (antoni.guillamon@upc.edu).‡Centre de Recerca Matematica, Apartat 50, E-08193, Bellaterra (Barcelona), Catalonia (gemma.huguet@upc.

edu). The work of this author was supported by the Spanish fellowship AP2003-3411 and the NSF grant DMS0354567.

1005

Physica D 227 (2007) 120–134www.elsevier.com/locate/physd

Dynamical systems on infinitely sheeted Riemann surfaces

Yuri N. Fedorov, David Gomez-Ullate∗

Department de Matematica Aplicada I, Universitat Politecnica de Catalunya, Barcelona, E-08028, Spain

Received 25 July 2006; accepted 30 January 2007Available online 11 February 2007

Communicated by A.C. Newell

Abstract

This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems inrelation with the multi-valuedness of the solutions as functions of complex time τ . In this work we consider a family of systems whose solutionscan be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R → C = {τ } is known to be an infinitelysheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the τ -plane is dense. The main novelty of thispaper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study globalproperties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions.The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real timetrajectories of the system are given by paths on R that are projected to a circle on the complex plane τ . Due to the branching of R, the solutionsmay have different periods or may be aperiodic.c© 2007 Elsevier B.V. All rights reserved.

Keywords: Complex dynamics; Riemann surface; Inversion of hyperelliptic integrals; Isochronicity; Sensitive dependence

1. Introduction

It has now become a classical subject to study theconnections between the integrability of a dynamical system(or the absence of it) and the singularity structure and multi-valuedness of its solutions. The discovery of this relation datesback to the work of Painleve and his collaborators [22,30],who classified all second order ODEs whose solutions havemovable poles as their only singularities (now known as thePainleve property) and are therefore single-valued functions.Then Kowalewskaya found a new integrable case of theheavy top by requiring that the solutions have the Painleveproperty [24,25]. A number of techniques collectively knownas Painleve–Kowalewskaya analysis have been developed overthe last thirty years (for a review see, for instance, [32]) totest for this property of the solutions, essentially by seekingfor a formal solution near a singularity in terms of a Laurentseries, introducing it in the equations and determining the

∗ Corresponding author.E-mail addresses: yuri.fedorov@upc.edu (Y.N. Fedorov),

david.gomez-ullate@upc.edu (D. Gomez-Ullate).

leading orders and resonances (terms in the expansion atwhich arbitrary constants appear). Painleve analysis has beenextended to test for the presence of algebraic branching (weakPainleve property, [31,32]) by considering a Puiseaux seriesinstead of a Laurent series. These analytic techniques (whichhave been algorithmized and are now available in computerpackages) constitute a useful tool in the investigation ofintegrability: in many non-linear systems where no solution inclosed form is known, Painleve analysis provides informationon the type of branching the general solution or special familiesmight have. It has also proved to be useful to identify specialvalues of the parameters for which generically chaotic systemssuch as Henon-Heiles or Lorenz become integrable.

It is also natural to investigate the singularity structureof solutions of chaotic dynamical systems. Tabor and hiscollaborators initiated this study in the early eighties for theLorenz system [33] and the Henon-Heiles Hamiltonian [18].They realized that the singularities of the solutions incomplex-time are important for the real-time evolution of thesystem. The complex time analytic structure was studied byextensions of the Painleve analysis involving the introductionof logarithmic terms in the expansion – the so-called

0167-2789/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2007.02.001

Y.N. Fedorov, D. Gomez-Ullate / Physica D 227 (2007) 120–134 133

6. Conclusions

In this paper we introduced a class of dynamical systemson R4 whose solutions can be interpreted as lifting a circularpath on the complex plane to a generally infinitely sheetedRiemann surface R describing the inversion of a hyperellipticintegral. For the integral of the holomorphic differential dη/µ

on a generic genus 2 surface Γ , we provided an explicitdescription of the global structure of R and showed that thesurface can be identified with the universal covering of thetheta divisor of Jac(Γ ). (It should be stressed that in caseof another holomorphic differential, in particular λ dλ/µ, thestructure ofR and type of branchingR 7→ C are different.) Wealso noted that for some regular curves with automorphisms,like the sextic curve Γ6, the four periods of the differentialare commensurable, and the corresponding surface R becomesfinitely sheeted. This implies that almost all the trajectoriesof the corresponding systems are periodic with period T or2T , although the trajectories increase in complexity as |E |increases.

Using the global description of R, we provided a geometricalgorithm for the circular travelling on R, that replaces thenumeric integration of the corresponding ODEs and associatesto the path γ (t) onR a sequence on a Z2-lattice. We conjecturethat for a regular value of the energy, all solutions are periodic,although the periods can be an arbitrarily high integer multipleof the fundamental period.

For didactic reasons we have restricted to genus 2 curvesΓ , although it is also possible to describe the global geometricstructure of Riemann surfaces R for holomorphic integrals ona generic hyperelliptic curve of genus g > 2. Namely, dividingits 2g periods into two allowed and 2g − 2 forbidden ones,one constructs an analog of a D-sheet by removing from C theunion of identical (4g − 4)-gonal windows {Wi j | i, j ∈ Z2

}

which have pairs of parallel edges formed by the forbiddenperiods. Different windows are obtained from each other byshifts by the allowed periods. Then one can show that thesurface R is itself a union of Z2g−2 identical copies of Dglued to each other along the opposite edges of the windows.A trajectory γ (t) on R can be encoded as a path on a Z2g−2-lattice.

Note added in proof

We would like to point out a very interesting developmentclosely related to this work due to Grinevich and Santini [23]. Intheir recently submitted paper, they explore Riemann surfacesof higher genus finding strong evidence of chaotic behaviourbeyond a certain critical genus. They provide an explanationbased on random walks on some effective dimension related tothe genus of the surface.

Acknowledgements

We thank L. Gavrilov, P. Santini, and V. Enolskifor discussions and valuable remarks. Our research waspartially supported by the Spanish Ministry of Science andTechnology under grants MTM2006-00478, MTM2006-14603and FIS2005-00752.

References

[1] S. Abenda, Yu. Fedorov, On the weak Kowalevski–Painleve propertyfor hyperelliptically separable systems, Acta Appl. Math. 60 (2) (2000)137–178.

[2] S. Abenda, V. Marinakis, T. Bountis, On the connection betweenhyperelliptic separability and Painleve integrability, J. Phys. A 34 (17)(2001) 3521–3539.

[3] M.S. Alber, Y.N. Fedorov, Algebraic geometrical solutions for certainevolution equations and Hamiltonian flows on nonlinear subvarietiesof generalized Jacobians, Inverse Problems 17 (4) (2001) 1017–1042.Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier,2000).

[4] L. Bates, R. Cushman, Complete integrability beyond liouville-arnold,Rep. Math. Phys. 56 (1) (2005) 77–91.

[5] E.D. Belokolos, V.Z. Enolskii, Reduction of abelian functions andalgebraically integrable systems. I, J. Math. Sci. (NY) 106 (6) (2001)3395–3486. Complex analysis and representation theory, 2.

[6] E.D. Belokolos, A.I. Bobenko, V.Z. Enolskii, A.R. Its, V.B. Matveev,Algebro-geometric approach to nonlinear integrable equations,in: Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin,1994.

[7] C.M. Bender, J.-H. Chen, D.W. Darg, K.A. Milton, Classical trajectoriesfor complex hamiltonians, J. Phys. A 39 (2006) 4219–4238.

[8] C. Birkenhake, H. Lange, Complex abelian varieties, second ed.,in: Grundlehren der Mathematischen Wissenschaften (FundamentalPrinciples of Mathematical Sciences), vol. 302, Springer-Verlag, Berlin,2004.

[9] O. Bolza, On binary sextics with linear transformations into themselves,Amer. J. Math. 10 (1) (1887) 47–70.

[10] T. Bountis, L. Drossos, I.C. Percival, Nonintegrable systems withalgebraic singularities in complex time, J. Phys. A 24 (14) (1991)3217–3236.

[11] F. Calogero, J.-P. Francoise, Periodic motions galore: How to modifynonlinear evolution equations so that they feature a lot of periodicsolutions, J. Nonlinear Math. Phys. 9 (1) (2002) 99–125.

[12] F. Calogero, J.-P. Francoise, Periodic motions galore: How to modifynonlinear evolution equations so that they feature a lot of periodicsolutions, J. Nonlinear Math. Phys. 9 (1) (2002) 99–125.

[13] F. Calogero, J.-P. Francoise, M. Sommacal, Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions, J. NonlinearMath. Phys. 10 (2) (2003) 157–214.

[14] F. Calogero, D. Gomez-Ullate, P.M. Santini, M. Sommacal, The transitionfrom regular to irregular motions, explained as travel on Riemannsurfaces, J. Phys. A 38 (41) (2005) 8873–8896.

[15] F. Calogero, M. Sommacal, Periodic solutions of a system of complexODEs. II. Higher periods, J. Nonlinear Math. Phys. 9 (4) (2002) 483–516.

[16] Y.F. Chang, G. Corliss, Ratio-like and recurrence relation tests forconvergence of series, J. Inst. Math. Appl. 25 (4) (1980) 349–359.

[17] Y.F. Chang, J.M. Greene, M. Tabor, J. Weiss, The analytic structure ofdynamical systems and self-similar natural boundaries, Physica D 8 (1–2)(1983) 183–207.

[18] Y.F. Chang, M. Tabor, J. Weiss, Analytic structure of the Henon-HeilesHamiltonian in integrable and nonintegrable regimes, J. Math. Phys. 23(4) (1982) 531–538.

[19] V.Z. Enolskii, M. Pronine, P.H. Richter, Double pendulum and θ -divisor,J. Nonlinear Sci. 13 (2) (2003) 157–174.

[20] H. Flaschka, A remark on integrable Hamiltonian systems, Phys. Lett. A131 (9) (1988) 505–508.

[21] A.S. Fokas, T. Bountis, Order and the ubiquitous occurrence of chaos,Physica A 228 (1–4) (1996) 236–244.

[22] B. Gambier, Sur les equations differentielles du second ordre et dupremier degre dont lintegrale generale est a points critiques fixes, ActaMath. 33 (1910) 1–55.

[23] P. Grinevich, P.M. Santini, Newtonian dynamics in the planecorresponding to straight and cyclic motions on the hyperelliptic curveµ2= νn

− 1, n ∈ Z: ergodicity, isochrony, periodicity and fractals,nlin.CD/0607031.

Physica D 237 (2008) 2599–2615www.elsevier.com/locate/physd

Computation of derivatives of the rotation number for parametric families ofcircle diffeomorphisms

Alejandro Luque, Jordi Villanueva∗

Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received 25 July 2007; received in revised form 22 February 2008; accepted 28 March 2008Available online 4 April 2008

Communicated by J. Stark

Abstract

In this paper we present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphismswith high accuracy. Our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averagesof iterates of the map and Richardson extrapolation. We focus on analytic circle diffeomorphisms, but the method also works if the maps aredifferentiable enough. In order to justify the method, we also require the family of maps to be differentiable with respect to the parameters and therotation number to be Diophantine. In particular, the method turns out to be very efficient for computing Taylor expansions of Arnold Tongues offamilies of circle maps. Finally, we adapt these ideas to study invariant curves for parametric families of planar twist maps.c© 2008 Elsevier B.V. All rights reserved.

PACS: 02.30.Mv; 02.60.-x; 02.70.-c

Keywords: Families of circle maps; Derivatives of the rotation number; Numerical approximation

1. Introduction

The rotation number, introduced by Poincare, is an importanttopological invariant in the study of the dynamics of circlemaps and, by extension, invariant curves for maps or twodimensional invariant tori for vector fields. For this reason,several numerical methods for approximating rotation numbershave been developed during the last years. We refer tothe works [3,4,8,13,14,21,24,31] as examples of methods ofdifferent nature and complexity. This last ranges from puredefinition of the rotation number to sophisticated and involvedmethods like frequency analysis. The efficiency of thesemethods varies depending if the approximated rotation numberis rational or irrational. Moreover, even though some of themcan be very accurate in many cases, they are not adequatefor every kind of application, for example due to violation oftheir assumptions or due to practical reasons, like the requiredamount of memory.

∗ Corresponding author. Tel.: +34 934015887; fax: +34 934011713.E-mail addresses: alejandro.luque@upc.edu (A. Luque),

jordi.villanueva@upc.es (J. Villanueva).

Recently, a new method for computing Diophantine rotationnumbers of circle diffeomorphisms with high precision atlow computational cost has been introduced in [27]. Thismethod is built assuming that the circle map is conjugate toa rigid rotation in a sufficiently smooth way and, basically,it consists in averaging the iterates of the map together withRichardson extrapolation. This construction takes advantage ofthe geometry and dynamics of the problem, so it turns out to bevery efficient in multiple applications. The method is speciallysuited if we are able to compute iterates of the map with a highprecision, for example if we can work with computer arithmetichaving a large number of decimal digits.

The goal of this paper is to extend the method of [27] inorder to compute derivatives of the rotation number with respectto parameters in families of circle diffeomorphisms. We followthe same averaging-extrapolation process applied to derivativesof iterates of the map. To this end, we require the family tobe differentiable with respect to parameters. Hence, we areable to obtain accurate variational information at the same timethat we approximate the rotation number. Consequently, themethod allows us to study parametric families of circle maps

0167-2789/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2008.03.047

2614 A. Luque, J. Villanueva / Physica D 237 (2008) 2599–2615

function

g(x, y) = 1 − 2y(cos(x))2 sin(x) + y2(cos(x))4,

so we can write Y ◦ Fα(x, y) = y√

g(x, y). First, we observethat for any s ∈ Q we have ∂

k,l,mα,x,y(ygs)|y=0 = 0 provided k 6= 0

or m = 0. Otherwise, the required derivatives can be computedby means of the following recurrent expressions

∂ l,mx,y (ygs) = ∂ l,m−1

x,y (gs)

+ sl∑

i=0

m−1∑j=0

(l

i

)(m − 1

j

)∂

i, jx,y(ygs−1)∂

l−i,m− jx,y (g)

and

∂ l,mx,y (gs) = s

l∑i=0

m−1∑j=0

(l

i

)(m − 1

j

)∂

i, jx,y(ygs−1)∂

l−i,m− jx,y (g).

Finally, we observe that the derivatives ∂l−i,m− jx,y (g) can be

computed easily by expanding the function as a trigonometricpolynomial

g(x, y) = 1 −y

2(sin(3x) + sin(x))

+y2

2

(34

+ cos(2x) +14

cos(4x)

).

Computations are performed by using double–double datatype, p = 7 and 221 iterates, at most. We stop the computationsif the estimated error is less than 10−25. Derivatives of theexpansion (32) and their estimated error, are given in Table 3.Finally, in order to verify the results, we compare truncatedexpansions of the curve with the numerical approximationcomputed in Section 5.2. The deviation is plotted in log10 scalein Fig. 5(right).

Acknowledgements

We wish to thank Rafael de la Llave, Tere M. Searaand Joaquim Puig for interesting discussions and suggestions.We are also very grateful to Rafael Ramırez-Ros forintroducing and helping us in the PARI-GP software.Finally, we acknowledge the use of EIXAM, the UPCApplied Math cluster system for research computing (seehttp://www.ma1.upc.edu/eixam/), and in particular Pau Roldanfor his support in the use of the cluster. The authors havebeen partially supported by the Spanish MCyT/FEDER grantMTM2006-00478. Moreover, the research of A.L. has beensupported by the Spanish phD grant FPU AP2005-2950.

References

[1] PARI/GP Development Headquarter. http://pari.math.u-bordeaux.fr/.[2] V.I. Arnold, Small denominators. I. Mapping the circle onto itself, Izv.

Akad. Nauk SSSR Ser. Mat. 25 (1961) 21–86.[3] H. Broer, C. Simo, Resonance tongues in Hill’s equations: A geometric

approach, J. Differential Equations 166 (2) (2000) 290–327.[4] H. Bruin, Numerical determination of the continued fraction expansion of

the rotation number, Physica D 59 (1–3) (1992) 158–168.

[5] E. Castella, A Jorba, On the vertical families of two-dimensional tori nearthe triangular points of the bicircular problem, Celestial Mech. Dynam.Astronom. 76 (1) (2000) 35–54.

[6] R. de la Llave, A tutorial on KAM theory, in: Smooth Ergodic Theory andits Applications (Seattle, WA, 1999), in: Proc. Sympos. Pure Math., vol.69, Amer. Math. Soc., 2001, pp. 175–292.

[7] R. de la Llave, A. Gonzalez, A Jorba, J. Villanueva, KAM theory withoutaction-angle variables, Nonlinearity 18 (2) (2005) 855–895.

[8] R. de la Llave, N.P. Petrov, Regularity of conjugacies between criticalcircle maps: An experimental study, Experiment. Math. 11 (2) (2002)219–241.

[9] W. de Melo, S. van Strien, One-dimensional dynamics, in: Ergebnisseder Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics andRelated Areas (3)), vol. 25, Springer-Verlag, Berlin, 1993.

[10] H.R. Dullin, J.D. Meiss, D. Sterling, Generic twistless bifurcations,Nonlinearity 13 (1) (2000) 203–224.

[11] C. Gole, Symplectic Twist Maps: Global Variational Techniques, WorldScientific Publishing, 2001.

[12] G. Gomez, A Jorba, J. Masdemont, C. Simo, Study of Poincare maps fororbits near Lagrangian points, ESA-ESOC contract 8711/91/D/IM/(SC),Darmstadt, Germany, 1993.

[13] G. Gomez, A Jorba, C. Simo, J. Masdemont, Advanced methods fortriangular points, in: Dynamics and Mission Design near Libration Points.vol. IV, in: World Scientific Monograph Series in Mathematics, vol. 5,World Scientific Publishing Co. Inc., River Edge, NJ, 2001.

[14] G. Gomez, J.M. Mondelo, C. Simo, Refined Fourier analysis: Procedures,error estimates and applications. Preprint 2001. Available electronicallyat: http://www.maia.ub.es/dsg/2001/.

[15] M.R. Herman, Mesure de Lebesgue et nombre de rotation, in: Geometryand Topology (Proc. III Latin Amer. School of Math., Inst. Mat. PuraAplicada CNPq, Rio de Janeiro, 1976), in: Lecture Notes in Math., vol.597, Springer, Berlin, 1977, pp. 271–293.

[16] M.R. Herman, Sur la conjugaison differentiable des diffeomorphismes ducercle a des rotations, Inst. Hautes Etudes Sci. Publ. Math. (49) (1979)5–233.

[17] Y. Hida, X. Li, D.H. Bailey, (QD, double–double and quad doublepackage), http://crd.lbl.gov/˜dhbailey/mpdist/.

[18] A. Katok, B. Hasselblatt, Introduction to the Modern Theoryof Dynamical Systems, in: Encyclopedia of Mathematics and itsApplications, vol. 54, Cambridge University Press, 1995.

[19] Y. Katznelson, D. Ornstein, The differentiability of the conjugation ofcertain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems9 (4) (1989) 643–680.

[20] S.G. Krantz, H.R. Parks, A primer of real analytic functions,in: Birkhauser Advanced Texts: Basler Lehrbucher. (Birkhauser AdvancedTexts: Basel Textbooks), second edn, Birkhauser Boston Inc., Boston,MA, 2002.

[21] J. Laskar, C. Froeschle, A. Celletti, The measure of chaos by the numericalanalysis of the fundamental frequencies. Application to the standardmapping, Physica D 56 (2–3) (1992) 253–269.

[22] A. Luque, J. Villanueva, Numerical computation of rotation numbers forquasi-periodic planar curves (in preparation).

[23] J. Moser, On invariant curves of area-preserving mappings of an annulus,Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II 1962 (1962) 1–20.

[24] R. Pavani, A numerical approximation of the rotation number, Appl.Math. Comput. 73 (2–3) (1995) 191–201.

[25] E. Risler, Linearisation des perturbations holomorphes des rotations etapplications, Mem. Soc. Math. Fr. (N.S.) 77 (1999) viii+102.

[26] T.M. Seara, J. Villanueva, Numerical computation of the asymptotic sizeof the rotation domain for the Arnold family. Preprint 2007. Availableelectronically at: http://www.ma1.upc.edu/recerca/2006-2007.html.

[27] T.M. Seara, J. Villanueva, On the numerical computation of Diophantinerotation numbers of analytic circle maps, Physica D 217 (2) (2006)107–120.

[28] C. Simo, Effective computations in celestial mechanics and astrodynam-ics, in: Modern Methods of Analytical Mechanics and their Applications(Udine, 1997), Springer, Vienna, 1998, pp. 55–102.

IOP PUBLISHING NONLINEARITY

Nonlinearity 21 (2008) 1759–1811 doi:10.1088/0951-7715/21/8/005

Kolmogorov–Arnold–Moser aspects of the periodicHamiltonian Hopf bifurcation

Merce Olle, Juan R Pacha and Jordi Villanueva

Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647,08028 Barcelona, Spain

E-mail: merce.olle@upc.edu, juan.ramon.pacha@upc.edu and jordi.villanueva@upc.edu

Received 25 October 2007, in final form 22 May 2008Published 26 June 2008Online at stacks.iop.org/Non/21/1759

Recommended by A Chenciner

AbstractIn this work we consider a 1 : −1 non-semi-simple resonant periodic orbit of athree degrees of freedom real analytic Hamiltonian system. From the formalanalysis of the normal form, we prove the branching off of a two-parameterfamily of two-dimensional invariant tori of the normalized system, whosenormal behaviour depends intrinsically on the coefficients of its low-orderterms. Thus, only elliptic or elliptic together with parabolic and hyperbolic torimay detach from the resonant periodic orbit. Both patterns are mentioned in theliterature as the direct and inverse, respectively, periodic Hopf bifurcation. Inthis paper we focus on the direct case, which has many applications in severalfields of science. Our target is to prove, in the framework of Kolmogorov–Arnold–Moser (KAM) theory, the persistence of most of the (normally) elliptictori of the normal form, when the whole Hamiltonian is taken into account, andto give a very precise characterization of the parameters labelling them, whichcan be selected with a very clear dynamical meaning. Furthermore, we givesharp quantitative estimates on the ‘density’ of surviving tori, when the distanceto the resonant periodic orbit goes to zero, and show that the four-dimensionalinvariant Cantor manifold holding them admits a Whitney-C∞ extension. Dueto the strong degeneracy of the problem, some standard KAM methods forelliptic low-dimensional tori of Hamiltonian systems do not apply directly, soone needs to properly suit these techniques to the context.

Mathematics Subject Classification: 37J20, 37J40

1. Introduction

This paper is related to the existence of quasiperiodic solutions linked to a Hopf bifurcationscenario in the Hamiltonian context. In its simpler formulation, we shall consider a real

0951-7715/08/081759+53$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1759

Kolmogorov–Arnold–Moser aspects of the periodic Hamiltonian Hopf bifurcation 1803

Due to the super-exponential term (κ(0))2n−1—compared with the geometric growth in r(n−1)—it

is easy to realize the existence of such S. Hence, lemma A.9 ensures that the limit vector-function U(∞) = �(∞) − �(−1) is of class Whitney-Cβ with respect to Λ ∈ E (∞), for anyβ > 0, and so is �(∞) (observe that �(−1) is analytic in Λ). Consequently, using the Whitneyextension theorem A.10, the function�(∞) can be extended to aC∞-function ofΛ in the wholeR2. Abusing notation, we keep the name �(∞) for this extension. As pointed out before, itkeeps the analytic and periodic dependence with respect to θ ∈ �2(ρ

(∞)).

Acknowledgments

The authors thank the anonymous referees and the Board Member for their valuable commentsand remarks which have contributed to improving this paper. This work has been partiallysupported by the Spanish MCyT/FEDER grant MTM2006-00478.

Appendix A

In this final section we have compiled those contents that, in our opinion, are necessary for theself-containment of this paper, but which we have preferred not to include in the body of thepaper in order to facilitate its readability. Concretely, in appendix A.1 we present the technicalresults on weighted norms we use to prove theorem 3.1. In appendix A.2 we prove a technicalbound concerning the statement of theorem 4.1. Finally, in appendix A.3 we present a briefintroduction to Whitney-smoothness.

A.1. Basic properties of the weighted norm

The following lemmas review some properties of the weighted norm | · |ρ,R introduced in (3).These properties are completely analogous to those for the usual supremum norm.

In lemmas from A.1 to A.4 we discuss the bounds in terms of this weighted norm forthe product of functions, partial derivatives (Cauchy estimates), composition of functions, themean value theorem, estimates on Hamiltonian flows and on small divisors. In lemma A.5 wediscuss the convergence of an infinite composition of canonical transformations. In lemma A.6we give a technical result on the norm of the square root and, finally, in lemma A.7 we giveanother technical result referring to the norm | · |R introduced at the end of section 2.

For most of these results we omit the proof, because it can be done simply by expandingthe functions in Taylor–Fourier series (2) and then bounding the resulting expressions. For fulldetails we refer to [43]. Throughout this section we use the notation introduced in section 2,sometimes without explicit mention.

Lemma A.1. Let f = f (θ, x, I, y) and g = g(θ, x, I, y) be analytic functions defined inDr,s(ρ, R) with 2π -periodic dependence in θ . Then we have the following.

(i) |f · g|ρ,R � |f |ρ,R · |g|ρ,R .(ii) For any 0 < δ � R, 0 � χ < 1, i = 1, . . . , r and j = 1, . . . , 2r we have

|∂θi f |ρ−δ,R � |f |ρ,Rδ exp(1)

, |∂Ii f |ρ,Rχ � |f |ρ,R(1 − χ2)R2

, |∂zj f |ρ,Rχ � |f |ρ,R(1 − χ)R

,

with z = (x, y). All these bounds can be extended to the case in which f and g take values inCn or Mn1,n2(C) (assuming that the matrix product of (i) is defined).

IOP PUBLISHING NONLINEARITY

Nonlinearity 22 (2009) 1997–2077 doi:10.1088/0951-7715/22/8/013

Geography of resonances and Arnold diffusion ina priori unstable Hamiltonian systems

Amadeu Delshams1 and Gemma Huguet2

1 Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647,08028 Barcelona, Spain2 Centre de Recerca Matematica, Apartat 50, 08193 Bellaterra (Barcelona), Spain

E-mail: Amadeu.Delshams@upc.edu and Gemma.Huguet@upc.edu

Received 4 December 2008, in final form 3 June 2009Published 20 July 2009Online at stacks.iop.org/Non/22/1997

Recommended by C-Q Cheng

AbstractIn this paper we consider the case of a general Cr+2 perturbation, for r largeenough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees offreedom, and we provide explicit conditions on it, which turn out to be C2

generic and are verifiable in concrete examples, which guarantee the existenceof Arnold diffusion.

This is a generalization of the result in Delshams et al (2006 Mem. Am.Math. Soc.) where the case of a perturbation with a finite number of harmonicsin the angular variables was considered.

The method of proof is based on a careful analysis of the geography ofresonances created by a generic perturbation and it contains a deep quantitativedescription of the invariant objects generated by the resonances therein. Thescattering map is used as an essential tool to construct transition chains ofobjects of different topology. The combination of quantitative expressions forboth the geography of resonances and the scattering map provides, in a naturalway, explicit computable conditions for instability.

Mathematics Subject Classification: 37J40, 37C29, 35B34, 34C29, 37C50

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The goal of this paper is to present a generalization of the geometric mechanism for globalinstability (popularly known as Arnold diffusion) in a priori unstable Hamiltonian systemsintroduced in [DLS06a]. That paper developed an argument to prove the existence of

0951-7715/09/081997+81$30.00 © 2009 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1997

Geography of resonances and Arnold diffusion 2071

Conditions H3′, H3′′ and H3′′′ can also be checked in the example (172) for any resonanceI = −l0/k0 and any M � 1, independently of ε.

Thus, if we consider I = −l0/k0 for any (k0, l0) ∈ N2, k0 �= 0 and gcd(k0, l0) = 1, thefunction Uk0,l0 in hypothesis H3 on (k0, l0) has the following expression:

Uk0,l0(θ) =M∑t=1

atk0,t l0 cos(tθ) = ak0,l0 cos(θ) + O2(ρk0 , rl0), (177)

where θ = k0ϕ + l0s and M � 1 can be chosen arbitrarily.Therefore, θ1 = 0 and θ2 = π are the unique critical points for the function Uk0,l0(θ).

Hence hypothesis H3′ on (k0, l0) is clearly verified.Next, for I = −l0/k0 we want to check hypothesis H3′′ on (k0, l0). This condition requires

to show that the function f in (15) is not constant. To that end, we will consider two values ofθ and we will show that their images for this function are different. For instance, note that thefunction f in (15) takes the same values as Uk0,l0 evaluated on its critical points θ1 and θ2 aslong as ∂2L∗

∂ϕ2 (I, θi/k0) �= 0, for i = 1, 2. Hence, hypothesis H3′′ on (k0, l0) is clearly satisfied if

the function Uk0,l0 has two extrema θi taking different values which satisfy ∂2L∗∂ϕ2 (I, θi/k0) �= 0,

which is the case as can be checked just looking at non-degenerate extrema of the function L.They give rise to non-degenerate extrema of the function L∗, which coincide with the ones ofthe function Uk0,l0 .

Similarly, we can check hypothesis H3′′′ on (k0, l0). In this case we need to show that thedeterminant (157) given in remark 4.8 does not vanish. It is clearly non-zero if we choose,for the two first columns, the two critical points θ1 and θ2 discussed above, and for the thirdcolumn θ3 �= 0, π , such that ∂2L∗

∂ϕ2 (−l0/k0, θ3/k0) = 0, but otherwise U′k0,l0(θ3) �= 0 and

∂L∗∂ϕ

(−l0/k0, θ3/k0) �= 0. The existence of this point θ3 is guaranteed by the fact that if one

considers the first order trigonometric polynomial of the reduced Poincare function L∗[�1],one can see that its critical points are always non-degenerate.

Hence, we apply theorem 2.1 and we conclude that

Proposition 5.1. Given the Hamiltonian (171) with g as in (172), 0 < ρ < r 1 and[I−, I+] ⊂ R−, for |ε| � ε∗(ρ, r) there exist orbits following the mechanism described in thispaper and such that I (0) � I−, I (T ) � I+, for some T > 0.

Acknowledgments

The authors are very grateful to M Gidea, R de la Llave, T M Seara and C Simo for valuablecomments and suggestions.

This work has been partially supported by the Spanish Grant MTM2006-00478. GHhas also been supported by the Spanish Fellowship AP2003-3411 and the NSF Grant DMS0354567 during her visit to UT Austin. The final version was written while the authors werevisiting CRM during the Research Programme Stability and Instability in Mechanical Systems(SIMS08), for whose hospitality they are very grateful.

Appendix A. Double Fourier series

Proposition A.1. Letf be aCr function with respect to (J, ϕ, s, ε), r � 1 and 2π -periodic withrespect to (ϕ, s). Then its Fourier coefficients fk,l(J, ε), (k, l) ∈ Z2, satisfy, for � = 0, . . . , r∣∣fk,l

∣∣C� � C

|f |Cr

|(k, l)|r−�, (A.1)

where C is a constant that depends only on r and � and |(k, l)| = max(|k|, |l|).

J Dyn Diff EquatDOI 10.1007/s10884-010-9199-5

Resonance Tongues and Spectral Gaps in Quasi-PeriodicSchrödinger Operators with One or More Frequencies.A Numerical Exploration

Joaquim Puig · Carles Simó

Received: 4 November 2009© Springer Science+Business Media, LLC 2010

Abstract In this article we investigate numerically the spectrum of some representativeexamples of discrete one-dimensional Schrödinger operators with quasi-periodic poten-tial in terms of a perturbative constant b and the spectral parameter a. Our examplesinclude the well-known Almost Mathieu model, other trigonometric potentials with a singlequasi-periodic frequency and generalisations with two and three frequencies. We computednumerically the rotation number and the Lyapunov exponent to detect open and collapsedgaps, resonance tongues and the measure of the spectrum. We found that the case with onefrequency was significantly different from the case of several frequencies because the latterhas all gaps collapsed for a sufficiently large value of the perturbative constant and thus thespectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the caseswith one frequency considered, gaps are always dense in the spectrum, although some gapsmay collapse either for a single value of the perturbative constant or for a range of values. Inall cases we found that there is a curve in the (a, b)-plane which separates the regions wherethe Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve,which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.

Keywords Quasi-periodic Schrödinger operators ·Quasi-periodic cocycles and skew-products · Spectral gaps · Resonance tongues ·Rotation number · Lyapunov exponent · Numerical explorations

To Prof. Russell Johnson in his 60th anniversary, with a deep appreciation for his outstanding works.

J. Puig (B)Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647,08028 Barcelona, Spaine-mail: joaquim.puig@upc.edu

C. SimóDepartament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes,585, 08007 Barcelona, Spaine-mail: carles@maia.ub.es

123

J Dyn Diff Equat

details on the maximal deviation d(N ) of the current Lyapunov and rotation sums, L andR, and the values given by a linear fit, are shown as a function of the number of iterates N .Some hints on a relation between d(N ) and the simultaneous Diophantine properties of thefrequencies are also given.

Finally we want to stress that, beyond the spectral problems in Schrödinger operators,equations of the type (4.1) appear in the study of the stability of invariant tori, as normalvariational equations along a quasi-periodic solution. They are a natural generalisation ofthe classical Hill equations which appear in the periodic case. Furthermore, more generalequations arise in higher dimensional normal variational equations. Most of the problems inthis case, away from perturbative ones, are open.

4.4 A Summary of Detected Phenomena

We summarize here some of the phenomena observed in this numerical study.

(1) The separation between the reducible and non-reducible domains in (a, b) has a com-plicated structure. A wild line seems to separate both domains. That line was simplyb = 2 in the Almost Mathieu case. Of course, the line can be drawn in different waysinside the uniformly hyperbolic zones.

(2) In contrast with the case of one frequency, the collapse of resonances is produced in asharp way when there are two or more frequencies. Before reaching the collapse, thewidth of resonance tongues behaves almost linearly. Some of the collapsed tongues canreopen and have a definitive collapse for larger values of b.

(3) In the nonuniformly hyperbolic zones, the boundaries of the resonant tongues seem tobe nondifferentiable, with lateral derivatives, at some points.

(4) In the case of several frequencies and even for moderate values of b, the spectrum seemsto have no gaps inside.

All these phenomena should be checked on a variety of examples and analysed theoreti-cally.

Acknowledgements The research of J.P. has been supported by grant MTM2006-00478. The research ofC.S. has been supported by grant MTM2006-05849/Consolider (Spain). The massive computations (a totalnumber of iterates close to 1016) have been carried out at the EIXAM and MAYA clusters of the UPC teamon Dynamical Systems and the HIDRA cluster of the UB team on Dynamical Systems. We also want to thankHåkan Eliasson for fruitful discussions.

References

1. Avila, A.: Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunovexponent and the boundary of nonuniform hyperbolicity. Arxiv preprint arXiv:0905.3902 (2009)

2. Avila, A., Bochi, J., Damanik, D.: Cantor spectrum for Schrödinger operators with potentials arising fromgeneralized skew-shifts. Duke Math. J. 146(2), 253–280 (2009)

3. Avila, A., Jitomirskaya, S.: The ten martini problem. Ann. Math. 170(1), 303–343 (2009)4. Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12(1),

93–131 (2010)5. Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocy-

cles. Ann. Math. 164(3), 249–294 (2006)6. Bjerklöv, K.: Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic

Schrödinger equations. Ergod. Theory Dyn. Syst. 25(4), 1015–1045 (2005)7. Bjerklöv, K.: Positive Lyapunov exponents for continuous quasiperiodic Schrödinger equations. J. Math.

Phys. 47, 022702 (2006)

123

A&A 472, 63–75 (2007)DOI: 10.1051/0004-6361:20077504c© ESO 2007

Astronomy&

Astrophysics

The formation of spiral arms and rings in barred galaxies

M. Romero-Gómez1,2, E. Athanassoula1, J. J. Masdemont3, and C. García-Gómez2

1 LAM, Observatoire Astronomique de Marseille Provence, CNRS, 2 place Le Verrier, 13248 Marseille Cedex 04, Francee-mail: [merce.romerogomez;lia]@oamp.fr

2 D.E.I.M., Universitat Rovira i Virgili, Campus Sescelades, Avd. dels Països Catalans 26, 43007 Tarragona, Spain3 I.E.E.C & Dep. Mat. Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received 19 March 2007 / Accepted 11 May 2007

ABSTRACT

In this and in a previous paper of 2006, we propose a theory to explain the formation of both spirals and rings in barred galaxiesusing a common dynamical framework. It is based on the orbital motion driven by the unstable equilibrium points of the rotatingbar potential. Thus, spirals, rings, and pseudo-rings are related to the invariant manifolds associated to the periodic orbits aroundthese equilibrium points. We examine the parameter space of three barred galaxy models and discuss the formation of the differentmorphological structures according to the properties of the bar model. We also study the influence of the shape of the rotation curvein the outer parts, by making families of models with rising, flat, or falling rotation curves in the outer parts. The differences betweenspiral and ringed structures arise from differences in the dynamical parameters of the host galaxies. The results presented here will bediscussed and compared with observations in a forthcoming paper.

Key words. galaxies: structure – galaxies: kinematics and dynamics – galaxies: spiral

1. Introduction

Bars are a very common feature of disc galaxies. In a sample of186 spirals drawn from the Ohio State University Bright SpiralGalaxy Survey, Eskridge et al. (2000) find that 56% of the galax-ies in the near infrared are strongly barred, while an additional6% are weakly barred. Only 27% can be classified as non-barred.A large fraction of barred galaxies show two clearly defined spi-ral arms (e.g. Elmegreen & Elmegreen 1982), often departingfrom the end of the bar at nearly right angles. This is the casefor instance in NGC 1300, NGC 1365 and NGC 7552. Deep ex-posures, moreover, show that these arms wind around the barstructure and extend to large distances from the centre (see forinstance Sandage & Bedke 1994). Almost all researchers agreethat spiral arms and rings are driven by the gravitational fieldof the galaxy (see Toomre 1977; and Athanassoula 1984, for re-views). In particular, spirals are believed to be density waves in adisc galaxy (Lindblad 1963). Toomre (1969) found that the spi-ral waves propagate towards the principal Lindblad resonancesof the galaxy, where they damp down, and thus concludes thatlong-lived spirals need some replenishment.

There are essentially three different possibilities for a spiralwave to be replenished. First, it can be driven by a companionor satellite galaxy. A direct, relatively slow, and close passageof another galaxy can form trailing shapes (e.g. Toomre 1969;Toomre & Toomre 1972; Goldreich & Tremaine 1978, 1979;Toomre 1981, and references therein). They can also be excitedby the presence of a bar. Several studies have shown that a ro-tating bar or oval can drive spirals (e.g. Lindblad 1960; Toomre1969; Sanders & Huntley 1976; Schwarz 1979, 1981; Huntley1980). Athanassoula (1980) studied the self-consistent quasi-stationary response of a galaxy composed of both gas and starsto a growing and rotating bar. She showed that the response ofboth components is bar-like up to corotation, where it turns intoa trailing two-armed spiral, ending approximately at the outer

Lindblad resonance (OLR). The third alternative, proposed byToomre (1981), is the swing amplification feedback cycle. Thisstarts with a leading wave propagating from the centre towardscorotation. In doing so, it unwinds and then winds in the trailingsense, while being very strongly amplified. This trailing wavewill propagate towards the centre, while a further trailing waveis emitted at CR and propagates outwards, where it is dissipatedat the OLR. The inwards propagating trailing wave, when reach-ing the centre will reflect into a leading spiral, which will prop-agate outwards towards the CR, thus closing the feedback cycle.Note that, if there is an inner Lindblad resonance (ILR), the wavepropagating inwards is damped at that resonance and the cycleis cut.

Strongly barred galaxies can also show prominent and spec-tacular rings or partial rings. The origin of such morphologieshas been studied by Schwarz (1981, 1984, 1985), who followedthe response of a gaseous disc galaxy to a bar perturbation.He proposed that ring-like patterns are associated to the prin-cipal orbital resonances, namely ILR, CR, and OLR. The ILRwould be responsible for the nuclear rings, CR would be asso-ciated with the inner rings, which are indicated by an r in thede Vaucouleurs classification, and the OLR would be the ori-gin of the outer rings, which are indicated by an R precedingthe Hubble type. Nuclear rings are small rings of star formationoften found near the centres or nuclei of early-type barred galax-ies, but we will not discuss these structures here. Inner rings arethe well-defined rings that encircle the bars of barred galaxies.They are sometimes active sites of star formation. Outer ringsare the larger, more diffuse rings, about twice the size of the bar.When these structures are incomplete, we designate them withthe term pseudo-rings. There are different types of outer rings.Buta (1995) classified them according to the relative orientationof the ring and bar major axes. If these two axes are perpendic-ular, the shape of the ring is similar to an “8” and the outer ringis classified as R1. If the two axes are parallel, the outer ring is

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20077504

M. Romero-Gómez et al.: The formation of rings and spirals in barred galaxies 75

Acknowledgements. We thank Albert Bosma for stimulating discussions ofthe properties of observed rings. This work is supported by the SpanishMCyT-FEDER Grant MTM2006-00478. M.R.G. acknowledges her “BecarioMAE-AECI”.

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