Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Giovanni B. Valsecchi, IASF-Roma, INAF, Roma, Italy
Títol: A numerical exploration of periodic orbits close to that of the Moon
Resum: The Saros cycle, known since antiquity, is an eclipse cycle in which 223 synodic months are nearly equal to 242 nodical months, as well as to 239 anomalistic months. The study of this cycle has led to the discovery that the lunar orbit is very close to a set of 8 periodic orbits of the restricted circular 3-dimensional Sun-Earth-Moon problem, differing from each other for the initial phases, whose duration is 223 synodic months.
These periodic orbits, even if of long duration when compared to those usually found in literature, are by no means the longest ones that can be found close to the orbit of the Moon; actually, according to a conjecture put forward by Poincare', there should be infinitely many periodic orbits, of longer and longer period, that get closer and closer to the actual lunar motion.
It is possible to show, with the help of Delaunay's expressions for the motion of the lunar perigee and node, that the longer periodic orbits are arranged in the eccentricity-inclination plane in a characteristic pattern, that is simply a deformation of the arrangement, in frequency space, of the set of points corresponding to the frequencies of the periodic orbits themselves. This allows to set up a numerical procedure to find periodic orbits, whose durations are multiples of the current synodic month, in an efficient way. This procedure is used to make a systematic exploration of the periodic orbits in the Sun-Earth-Moon problem over the entire eccentricity-inclination plane.
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Amadeu Delshams (UPC)
Títol: A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
Resum: We consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.
Joint work with Gemma Huguet (CRM i CNS), preprint available at http://arxiv.org/abs/1007.2739
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Piotr Zgliczynski, Institute of Computer Science, Jagiellonian University
Títol: Existence of homoclinic tangency for the Hénon map and for the forced-damped pendulum - a computer assisted proof
Resum: We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Hénon map and the forced damped pendulum ODE. This is a joint work with Daniel Wilczak
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Arturo Vieiro (UPC)
Títol: Planar radial weakly-dissipative diffeomorphisms
Resum: Many physical problems are usually described as a Hamiltonian system plus a (relatively) small perturbation. In this work we study the effect of a small dissipative radial perturbation acting on a one parameter family of area preserving diffeomorphisms. By "radial" dissipation we refer to a specific (but general enough) type of dissipative perturbation. The interest of this work is on the global effect of the dissipation on a fixed domain around an elliptic fixed/periodic point of the family of maps, rather than on the effects around a single resonance. We describe the local/global bifurcations observed in the transition from the conservative to a weakly dissipative case: the location of the resonant islands, the changes in the domains of attraction of the foci inside these islands, how the resonances disappear, etc. The possible ω-limits are determined in each case. For a fixed dissipative perturbation size, we distinguish three main regions in the phase space according to the topological properties of the "resonant" structures which survive the dissipation. In each region the dynamics around a surviving resonant chain can be described using suitable models which can be obtained, according to the presence or not of homoclinic points in the resonant "islands", as an interpolating flow of the Birkhoff normal form or as a return map (in this case the return model obtained can be seen as a weakly dissipative version of the well-known separatrix map). Some considerations and results on the probability of capture by the attractors will be given. This is a joint work with C. Simó.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Pablo Gutiérrez Barrientos, Universidad de Oviedo i UAB
Títol: Blenders
Resum: In 1996, L. Díaz and C. Bonatti constructed a local model that resembles a high dimensional skew horseshoe and that they called Blender. The Blenders are a powerful tool to show robust properties. For instance, in the presence of a Blender, a quase-transverse intersection between stable and unstable manifolds of hyperbolic sets of different indices turns out to be C1-persistence.
In this lecture we introduce a symbolic definition of Blender to provide a generalization of the geometric model of Díaz and Bonatti with grea central dimension. This is a joint work with L. Díz.
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Gabor Kiss
Títol: Dynamics of delay differential equations with distributed delays
Resum: We compare the stability properties of some families of delay differential equations with one delay to associated families of equations with distributed delays. With the aid of some examples, we indicate some differences between the non-linear oscillations of equations with one and distributed delays
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Sergei Gonchenko, Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, Russia
Títol: On appearance of visibility and invisibility effects for stable periodic orbits at homoclinic bifurcations
Resum: We observe various types of bifurcations of homoclinic tangencies leading to birth of stable periodic orbits (and, sometimes, invariant tori and strange attractors). We consider parameter families (general unfoldings) and discuss peculiarities of the corresponding bifurcations. In particular, we outline both cases where, in the families, stable periodic orbits are detected with "zero probability" and when stability domains are abnormally extended in contrast with certain situations
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Josep Maria Mondelo, UAB.
Títol: Dynamical analysis of 1:1 resonance around Vesta
Resum: The Dawn mission is scheduled to arrive at Vesta on July 2011. In order to reach its lowest science orbit, the Dawn spacecraft needs to go across the 1:1 resonant region ("Vesta-stationary orbits"). Motivated by this, this talk will show preliminary results of a study of this region, done by analogy with the collinear libration points of the spatial, circular RTBP. Some transfers across it using invariant manifolds of periodic orbits will be shown.
Joint work with: S. Broschart (NASA-JPL), B.F. Villac (UC Irvine)
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Núria Fagella, UB.
Títol: A separation Theorem for entire transcendental functions.
Resum: Let f be an entire transcendental map for which the set of singularities of the inverse function is bounded, and of finite order (or a finite composition of finite order functions). For this class of maps we prove a separation theorem analogous to the Goldberg-Milnor separation theorem for polynomials. Such results have several applications, as for example that there cannot be any nonlinearizable irrationally indifferent periodic points in the boundary of a periodic Siegel disk.
(This is joint work with Anna Benini.)
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Yakov Pesin, Pennsylvania State University
Títol: Pugh-Shub Stable Ergodicity, Partial Hyperbolicity and Lyapunov Exponents.
Resum: This talk is about recent advances in the Pugh-Shub stable ergodicity theory for partially hyperbolic diffeomorphisms. I describe two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Inna Basak (Universitat Politècnica de Catalunya)
Títol: Separation of variables and explicit theta-function solution of the classical Steklov-Lyapunov systems: A geometric and algebraic geometric background
Resum: We revise the separation of variables and explicit integration of the classical Steklov-Lyapunov systems, which was first made by F. Kötter in 1900. Namely, we give a geometric interpretation of the separating variables and, then, applying the Weierstrass root functions, obtain an explicit theta-function solution to the problem.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Jordi Villadelprat, UB.
Títol: A Chebyshev criterion for Abelian integrals
Resum: Let E be an n-dimensional linear space of analytic functions on a real interval I. E is said to be an extended Chebyshev space if any nonzero element of E has at most n-1 zeros on I counted with multiplicities. Abelian integrals appear naturally when studying planar differential systems. For instance, the first approximation of the displacement function of the Poincar? map associated to a small deformation of a Hamiltonian system is an Abelian integral. Thus, under generic conditions, its zeros will determine the number and location of limit cycles born in the perturbation.
Bounding the number of zeros of an Abelian integral is usually a very long and highly non-trivial problem. In some papers the authors study the geometrical properties of the so-called centroid curve using the fact that it verifies a Riccati equation (which is itself deduced from a Picard-Fuchs system). In other papers the authors use complex analysis and algebraic topology (analytic continuation, argument principle, monodromy, Picard-Lefschetz formula, etc.).
In this talk I will explain a very simple condition that guarantees that a collection of Abelian integrals is Chebyshev. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.
This is a joint work with Maite Grau(UdL) and Francesc Mañosas(UAB).
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Radu Saghin (Centre de Recerca Matemàtica)
Títol: Integrability of invariant bundles
Resum: I will discuss the integrability (in the Frobenius sense) of distributions (sub-bundles of the tangent bundle) which are invariant under a diffeomorphism of a compact manifold. This problem arises naturally in the study of partially hyperbolic diffeomorphisms. I will present several results, examples of non-integrability and non-unique integrability due to Smale and Hertz-Hertz-Ures, and a conjecture which relates the integrability of an invariant bundle with the expansion rates of the diffeomorphism.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Jordi-Lluís Figueras, UB
Títol: Breakdown mechanisms of Fiberwise Hyperbolic Invariant Tori in skew product systems
Resum: In this talk we will expose several mechanisms of breakdown of Fiberwise Hyperbolic Invariant Tori (FHIT) in volume preserving skew product systems.
First we will revise the concepf of FHIT, its associated cocycle and its associated transfer operator.
Second we will expose 4 mechanisms in volume preserving skew product systems:
Lloc: Aula 103(1r pis), FME, UPC (podria canviar a la 102).
A càrrec de: Alejandro Luque (UPC)
Títol: Nudos y lazos en la ecuación de Euler estacionaria
Resum: Me gustaría discutir de forma informal y divulgativa (quien me conozca puede esperar más de lo primero, aunque espero daros más de lo segundo) sobre un trabajo reciente de Alberto Enciso y Daniel Peralta-Salas.
En concreto, toda distribución localmente finita de lazos en el espacio puede deformarse un poquito (a especificar) de forma que sea una trayectoria de un campo vectorial de Beltrami. Estos campos son solución de la ecuación de Euler estacionaria (que todos recordaréis de cursos de fluidos).
La demostración de este resultado és un "matxembrat" de diferentes técnicas. Por ejemplo, en algún momento se usa que las cosas hiperbólicas son robustas...
El artículo en cuestión podéis encontrarlo en arxiv.org/pdf/1003.3122
Nota: me acabo de enterar que la parabra "matxembrat" es un barbarismo atroz, pues proviene de la palabra castellana machihembrado...
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Gonchenko S.V. (Nizhny Novgorod, Russia)
Títol: On dynamical systems with mixed dynamics
Resum: We say that a dynamical system possesses "mixed dynamics" if:
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: M.A. Teixeira, IMECC - UNICAMP, Campinas (Brasil)
Títol: Non-Smooth Systems and Singular Perturbation Problem
Resum: Main goal is to present a mechanism for studying perturbations of certain non-smooth dynamical system via Singular Perturbation Theory
A càrrec de: Maciej Capinski, Institute of Mathematics, AGH University of Science and Technology, Krakow
Títol: Computer Assisted Proof for Fibers of Invariant Manifolds in the Planar Restricted Circular Three Body Problem
Resum: We first present a computer assisted proof for detection of the family of Lyapounov orbits around the equilibrium point L1 in the Planar Restricted Circular Three Body Problem. The method provides very tight estimates and can be applied over a broad range of energies of the orbits. We then present a method for detection of fibers of stable/unstable manifolds associated with the family of orbits. The method is based on a combination of a parameterization method together with cone conditions and topological arguments. We also discuss propagation of the fibers along the flow to prove transversal intersections of the manifolds
Lloc: Aula IMUB (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marco Abate, Dipartimento di Matematica, Universita' di Pisa
Títol: Stable manifolds for holomorphic automorphisms
Resum: It is well known that the stable manifolds of a compact hyperbolic invariant (with respect to a holomorphic automorphism) subset of a complex manifold are smoothly diffeomorphic to a complex Euclidean space; in 2000, Bedford asked whether they also are biholomorphic to a complex Euclidean space. In this talk I shall describe a new technique for dealing with this problem, providing in particular a positive answer in all points where Lyapunov exponents exist, generalizing previous results by Jonsson, Varolin and Peters (Joint work with A. Abbondandolo and P. Majer)
A càrrec de: Jasmin Raissy, Dipartimento di Matematica e Applicazioni, Universita' di Milano Bicocca.
Títol: Holomorphic linearization of commuting germs of holomorphic maps
Resum: Given
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Viktor Enolskii Institute of Magnetism (Kiev, Ukrania) Institute Herriot-Watt (Edinburg, UK)
Títol: Algebraic geometric background of the charge 3 monopoles the Yang-Mills-Higgs theory
Resum: We consider so called Bogomolny equations in the Yang-Mills-Higgs theory and construct their solutions that called monopoles, because they generalize Dirac monople to the case of higher gauge group. Namely, we integrate the Bogomolny equation for the case of charge 3 in terms of the Riemann theta-functions of the associated algebraic curve by using the method of finite gap integration.
Our work involve various remarkable results of mathematics: Ramanujan hyperegeometric relations, Schottky-Jung proportionalities for unramified covers, results on modili of genus two algebraic curve. Some movies visualizing our results will be presented.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Warwick Tucker, Uppsala University, Suècia
Títol: Challenges for validated numerics in dynamical systems
Resum: We will present recent work on two intensely studied problems in dynamical systems. The first deals with the discrete quadratic map; here the long-standing question concerns the regions of stability when varying the parameter of the map. The second topic concerns Hilbert's 16th problem, which asks for the possible number (and configuration) of limit cycles for planar polynomial vector fields. Both problems have been thoroughly studied for more than a century, and with a formidable variation of techniques. We will show how validated numerics can contribute to gaining further insight into these problems. This work is joint with Daniel Wilczak and Tomas Johnson, respectively.
Lloc: Aula 103(1r pis), FME, UPC.
A càrrec de: Lorenzo Díaz, Catholic University of Rio de Janeiro, Brasil
Títol: Porcupine-like horsehoes: transitivity, Lyapunov spectrum, and phase transitions
Resum: We discuss simple, but representative, examples of local diffeomorphisms defined as one-step skew products modeled over a horsehose map. These systems are naturally asociated to a heterodimensional cylce. This cycle gives rise to a homoclinic class on which the diffeomorphism is topologically transitive and partially hyperbolic. It can be conveniently studied in terms of an iterated function system of interval maps that are genuinely non-contracting. These examples have topologically a rich fiber structure (justifying the porcupine terminology). Moreover, they exhibit a rich phase transition in the pressure function (coexistence of equilibrium states with positive entropies) that is associated to a gap in the spectrum of Lyapunov exponents in the central direction.
This is a joint work with K. Gelfert (UFRJ) and M. Rams (IM PAN Warsaw).
A càrrec de: Vadim Kaloshin, University of Maryland
Títol: On central configurations for 4 and 5 bodies
Resum: The 6th Smale problem for the XXI century is about finiteness of planar central configurations for any combinations of positive masses. The problem was first posed by Chazy in 1918, then by Wintner in 1941. Hampton-Moeckel in 2004 gave a computer assisted proof of finiteness for 4 bodies. We show that the number of central configurations of the 5 body problem is finite, except perhaps if masses certain algebraic relations. We also give a purely analytic proof of Hampton-Moeckel result for 4 bodies.
This is a joint work with Alain Albouy.
Lloc: Aula S01, FME, UPC.
A càrrec de: Richard Montgomery, University of California at Santa Cruz
Títol: The Brake-to-Syzygy Map
Resum: In the three-body problem, we study solutions with zero initial velocity (brake orbits). Following such a solution until the three masses become collinear (syzygy) defines a kind of Poincar\'e map which we call the `syzygy map'. Viewed after reduction, the syzygy map is a map from one punctured open two-dimensional disc to another, the punctures corresponding to the Lagrange triple collision orbit. Among the results we discuss are the following. For most masses, the map extends continuously to the punctures. The image of the map contains the binary collision locus. These binary collision brake orbits can be realized Jacobi-Maupertuis minimizers. There exists a new periodic brake orbit whose second `hit' with the syzygy locus is orthogonal. The syzygy map grew out of a desire to better understand which syzygy sequences can occur within the zero-angular momentum three-body problem. We will describe this motivation, and what the map tells us about stutters within syzygy sequences. We present several open questions, very simple to state, regarding the nature of the orbits realizing the map. This is joint work with Rick Moeckel and Andrea Venturelli.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rodrigo Treviño. University of Maryland
Títol: El cociclo de Kontsevich-Zorich, desviaciones de promedios ergódicos y la homología de foliaciones
Resum: Using a recently-developed criterion of Forni applied to measures on the moduli space of Abelian differentials supported on points coming from quadratic differentials by a standard, double cover construction, we can prove that the Kontsevich-Zorich cocycle is non-uniformly hyperbolic for this measure. I will discuss applications to the study of deviations in homology of typical leaves of the corresponding non-orientable foliations as well as applications to the study of deviations of ergodic averages, which exhibit phenomena different from the Abelian case