Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Andre Vanderbauwhede, University of Ghent, Belgica .
Títol: Bananas and banana-splits - Degenerate subharmonic branching in reversible systems .
Resum: In autonomous time-reversible systems symmetric periodic orbits typically appear in one-parameter families; along such families simple multipliers can be locked on the unit circle, and when they pass a root of unity one sees under generic conditions two bifurcating branches of subharmonic periodic orbits, one stable, one unstable. These generic conditions are: (i) the simplicity of the multiplier (together with the non-existence of other resonant multipliers), and (ii) a transversality condition which requires that the root of unity is passed with non-zero speed (using an appropriate parametrization of the primary family). In this talk we will describe what happens when either one of these generic conditions is not satisfied, as can happen in one- or more parameter families of reversible systems. The main emphasis will be on case (ii) where so-called "bananas" and "banana-splits" appear. Such bananas can be found for example along the short period family of periodic orbits emanating from L4 (and L5) for a certain range of the mass ratio in the restricted 3-body problem; they play an important role in the bifurcations of the long period family as L4 passes through a sequence of resonances.
Most of the talk will be on an introductory level, using simple examples and lots of graphics to clarify the ideas. The work presented in this talk was done in collaboration with Maria-Cristina Ciocci (Imperial College), Francisco Javier Muñoz Almaraz (Barcelona), Emilio Freire and Jorge Galan (University of Sevilla).
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Turgay Uzer (Georgia Tech) .
Títol: Applying nonlinear dynamics to physics and chemistry: Some open problems .
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Ana M. Sanz, Depto. de Análisis Matemático y Didáctica de la Matemática, Universidad de Valladolid .
Títol: Atractores globales en algunos modelos de dinámica de poblaciones y de redes neuronales dados por ecuaciones diferenciales funcionales con retardo .
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Joaquim Puig, Dept. Matemàtica Aplicada I, UPC .
Títol: Lyapunov exponent and weak L2-reducibility of skew-products .
Resum: In this talk we will consider bounded linear skew-products on SL(2,R)×Ω where the dynamics on the measure space (Ω,μ) is given by an ergodic transformation. Focusing in the case of Schrödinger skew-products, we will introduce the averaged Lyapunov exponent and the problem of reducibility to an SO(2,R)-skew product by an L2-transformation. The quasi-periodic and almost-periodic cases are included in this formulation.
Kotani, in the early eighties, showed that almost everywhere in the set where the Lyapunov exponent vanishes there is such L2-reducibility. We will present Kotani's result and prove that whenever the set with zero Lyapunov exponents is a Cantor set, it has a residual subset where L2-reducibility is not possible. This can be used to characterize which quasi-periodic Schrödinger skew-products are reducible to constant coefficients for all values of the energy .
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Tomasz Kapela, Institute of Computer Science Jagiellonian University, Krakow, Polonia .
Títol: Computer assisted proofs and applications to Celestial Mechanics .
Resum: I will give a basic review of tools which we use to produce rigorous computer assisted proofs such as: an interval arithmetics, set representation methods and C1-Lohner algorithm for rigorous integration of ODE.
I will also present two very useful theorems, which are the base of many computer assisted proofs: Interval Newton Method and its modification Interval Krawczyk Method.
As an example I will show how to use these tools to provide rigorous proofs of the existence of many choreograpies in the N-body problem .
A càrrec de: Martín Lara, Real Instituto y Observatorio de la Armada, San Fernando .
Títol: Europa science orbit design. Frozen orbits, periodic orbits, ephemeris .
Resum: The strong evidence of liquid water close to the surface of Europa motivates high scientific interest in a mission to Europa. The preliminary design of a spacecraft mission requires a deep insight in the dynamics. It can be obtained using simplified models that retain the underlying dynamics of the problem. For dealing with the variety of orbits related to a spacecraft mission (transfer, parking, science, quarantine) we use two different models: the circular, restricted three-body problem, and a Hill problem perturbed by the non-sphericity of the central body. Then, the phase space around planetary satellites can be described using different techniques such as averaging, periodic orbits computation, or stability maps based on chaoticity indicators. Since the candidates for a science orbit about Europa fall among the set of unstable orbits, we resort to dynamical systems theory to maximize the lifetime of the orbiter by determining initial conditions on the stable manifold of a frozen orbit. Ephemeris computation in a realistic model are essential to validate the results .
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jacques Féjoz, Institut de Mathématiques de Jussieu and Observatoire de Paris (Astronomie et Systèmes dynamiques) .
Títol: On the proof of Arnold's theorem on the stability of the planetary problem .
Resum: In 1963, Vladimir Arnold¹ stated and partly proved the following theorem: in the Newtonian model of the Solar system with n≥2 planets in space, if the masses of the planets are small enough compared to the mass of the Sun, there is a subset of the phase space of positive measure, in the neighborhood of circular and coplanar Keplerian motions, leading to quasiperiodic motions with 3n-1 frequencies. In 1998, in a series of lectures Michael Herman sketched a complete and more conceptual proof of this theorem². In reviewing this proof, I will focus on a couple of ideas which make it so powerful and, I believe, elegant. These ideas mainly pertain to some normal forms of Hamiltonians, which epitomize the structure of KAM theory as understood by M. Herman.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Henk Broer, Dept of Mathematics, Univ of Groningen .
Títol: A predator prey model with nonmonotonic response function .
Resum: An adaptation of the Lotka-Volterra equations that covers many other cases will be discussed. It is a 5-parameter family of planar vector fields that allows for a systematic mathematical exploration mainly in terms of codimension 3 bifurcations as developed by Dumortier-Roussarie-Sotomayor and others. These act as organising centers for planar sections in the parameter space, that order the various bifurcation diagrams. The background mathematics uses Poincaré-Hopf index and Poincaré Bendixson theory. A start is made with the exploration of cases where the system is subject to seasonal or other time periodic excitations, in which case the planar vector field acts as principal part.
(this is a joint work with V. Naudot, R. Roussarie and K. Saleh)
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Chong-Qing Cheng, Dept of Mathematics, Nanjing University .
Títol: Variational construction of diffusion orbits in Hamiltonian systems .
Resum: Veure a mparc:
04-265 Chong-Quin Cheng, Jun Yan Arnold diffusion in Hamiltonian systems: a priori unstable case
A càrrec de: Rafael de la Llave, Dept. of Mathematics, U. of Texas at Austin .
Títol: Critical Points which are not minimizers in periodic variational problems: Dynamical Systems and Partial Differential Equations .
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Adrien Douady, ENS Paris .
Títol: Julia sets of positive measure, following Cheritat and Buff .
A càrrec de: Robert Devaney, Boston University .
Títol: Parameter plane structures for families of singularly perturbed rational maps .
Resum: For families of rational maps of the complex plane of the form $z^n + \lambda / z^d$, there are a number of different structures in both the dynamical and parameter planes of these families. In this talk we describe some of these objects, including Cantor necklaces, satellites around the McMullen domain, and circles of Sierpinski holes.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Lubomir Gavrilov, Laboratoire de Mathematiques Emile Picars, Univ. Paul Sabatier, Toulouse .
Títol: Families of Painleve VI equations having a common solution .
Resum: It is known that the solutions of the Painlevé sixth equation:
| d2λ/dt2 = | 1/2(1/λ+ 1/(λ-1)+ 1/(λ-t))(dλ/dt)2 | |
| -(1/t+1/(t-1)+1/(λ-t)) dλ/dt+ | ||
| λ(λ-1)(λ-t) t2(t-1)2 |
[α0-α1t/λ2+α2 (t-1)/(λ-1)2+ (1/2-α3) t(t-1)/(λ-t)2] |
appear in an universal way in many branches of mathematics. For generic values of αi, solutions of this equation are what Painlevé called "transcendantes essentiellement nouvelles", so to write down any solution is a difficult task.
In this talk we shall discuss algebraic solutions -- where λ is defined implicitly as a function of t by a polynomial equation P(λ,t)=0.
We classify, in fact, all algebraic functions satisfying non-trivial families of Painlevé sixth equations. It turns out that these algebraic solutions are closely related to deformations of appropriate Picard-Fuchs equations.
Conjecturaly, one finds in this way, up to an equivalence, all one-parameter families of solutions of Painlevé sixth equation.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Andrew Hone, University of Kent at Canterbury .
Títol: An integrable perturbation of an Henon-Heiles system .
Resum: We consider a fourth order partial differential equation (PDE) which appeared in a classification of integrable PDEs of Boussinesq type (with a second order time derivative). Using the prolongation algebra method of Wahlquist-Estabrook, we obtain a 2 x 2 Lax pair for the PDE, and associate it to a Schrodinger operator with energy-dependent potential. By making a reduction to the stationary flow of a higher symmetry in the associated integrable hierarchy, we find a perturbation of a type (ii) Henon-Heiles system that has the weak Painleve property i.e. its general solution has algebraic branching. Separating variables in this integrable Hamiltonian system, we obtain a class of solutions for the PDE that include the interaction of two solitons on a constant background. This is joint work with Caroline Verhoeven and Vladimir Novikov.
Lloc: Aula Magna (1er pis), Facultat de Matemàtiques, UB.
A càrrec de: Antonio Bru Espino, del Departamento de Matemática Aplicada, Universidad Complutense de Madrid .
Títol: El análisis de escalas aplicado al crecimiento tumoral .
Resum: Desde el establecimiento definitivo de la Geometría Fractal en 1982, la Física Estadística ha experimentado un importante avance en el estudio de los sistemas que evolucionan bajo la acción de un ruido. Una característica de dichos sistemas es que desarrollan interfases rugosas, fractales. Por medio de las invariancias tanto espaciales como temporales de las mismas, se puede determinar una ecuación que describe el crecimiento. Estas herramientas aplicadas al crecimiento tumoral han permitido obtener una nueva descripción del mismo compatible con la experimentación biológica y clínica existente a dia de hoy. La determinación de la dinámica del crecimiento de los tumores sólidos ha permitido la obtención de un resultado sorprendente: todos los tumores sólidos crecen según la misma dinámica y su crecimiento posee un mecanismo principal responsable del crecimiento, que involucra el movimiento de difusión de las células tumorales en el borde del tumor.
A càrrec de: Antonio Bru Espino, del Departamento de Matemática Aplicada, Universidad Complutense de Madrid .
Títol: Dinámica tumoral y propuesta terapéutica basada en la neutrofilia .
Resum: El mecanismo mencionado anteriormente implica que existe una competición por el espacio entre las células tumorales y las células del organismo, tanto las células del tejido del órgano en el que crece el tumor como las células de la respuesta inmunológica. Ésta se caracteriza por el papel fundamental que desempeña un tipo de leucocitos: los neutrófilos. Mediante una neutrofilia en sangre se consigue una inflamación peritumoral por la acción de los neutrofilos que anula el mecanismo fundamental de crecimiento del tumor e impide el desarrollo del tumor, induciéndole una necrosis. Dicho resultado ha sido demostrado tanto en experimentación animal como en experimentación clínica, obteniendo unos resultados muy prometedores, que permiten considerar el establecimiento de una nueva estrategia terapéutica, actualmente en fase de estudio.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Simple equations with complex bifurcation diagram: Hill's equations with periodic coefficients .
Resum: Equations of Hill type x''+(a+b p(t))x=0, p being a periodic function and a,b real parameters appear in many applications, mainly in the stability of periodic orbits of Hamiltonian systems. They can also be considered as simple Schroedinger operators with periodic potential. The simplest one is Mathieu equation, when p(t)=cos(t).
Despite the equation is linear, the dependence of the solutions and, in particular the stability properties IS NOT. This is a source of a very interesting behaviour which can be read off in the bifurcation diagram. Has immediate consequences for the spectrum in Schroedinger.
The perturbative case (b small) is elementary and can be studied using normal forms.
The goal of the talk is to present the results for a,b arbitrarily large, to describe the asymptotic behaviour and to sketch the tools which are required in the proofs. Some numerical examples will be shown.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alexander Komech, Mechanics & Mathematics Department Moscow State University.
Títol: Nonlinear Hyperbolic PDEs in Quantum and Statistical Physics .
Resum: Most problems of modern Quantum and Statistical Physics are formulated in terms of Hamiltonian PDEs: the Schroedinger, Dirac, Maxwell, Yang-Mills equations, the coupled equations, etc. The nonlinear nature of the coupled equations is crucial for a consistent description of the fundamental physical phenomena: Bohr's transitions to quantum stationary states, de Broglie's wave particle duality, and the Maxwell-Boltzmann-Gibbs equilibrium statistics.
At the same time, it is this nonlinear nature that creates tremendous difficulties for a rigorous investigation of the fundamental phenomena. This fact leads to open mathematical problems for the nonlinear coupled PDEs, related to the Heisenberg Program that was formulated in 1961.
We suggest that these fundamental phenomena are intrinsic mathematical features of general nonlinear Hamiltonian PDEs. We state the related open problems and describe recent results in these directions.
A càrrec de: Sonia Carvalho, Universidade Federal de Minas Gerais.
Títol: Instability Zone for Billiards on Ovals .
Resum: The billiard problem consists in the free motion of a point particle in the plane region enclosed by an oval, being reflected elastically at the impacts with the boundary. Since the particle moves with constant velocity inside the region, the motion is completely determined by the point of reflection and the direction of movement immediately after each reflection. This billiard model defines a conservative two dimensional discrete dynamical system on an annulus. If the oval is sufficiently differentiable then there are invariant non trivial curves near the boundary of the annulus. The region between two consecutive invariant curves was called instability zone by Birkhoff. Together with M.J. Dias Carneiro and S. Oliffson Kamphorst, we investigate the generic dynamics on the instability zone that contains the 2-periodic orbits and show that the the closure of the invariant manifolds of the largest diameter of the oval together with a countable union of disks sorrounding periodic orbits fills this instability zone.
Lloc: Aula Magna (1er pis), Facultat de Matemàtiques, UB.
ACTE ACADÈMIC EN MEMÒRIA D'EN PERE MUMBRÚ
Presentació: Joaquin Ortega Aramburu (Degà de la Facultat de Matemàtiques de la UB)
Remembrances d'en Pere: Jaume Llibre, Francesc Mañosas, Ernest Fontich, Montse Navarro.
La recerca d'en Pere en Sistemes Dinàmics: Lluís Alsedà.
PAUSA
Xerrada divulgativa a càrrec de David Juher: Claus secretes i missatges ocults.
Organitza:
Facultat de Matemàtiques de la UB,
Grup de Sistemes Dinàmics de la UAB.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Tobias Jäger, Mathematisches Institut, Erlangen Universität.
Títol:The structure of strange non-chaotic attractors in pinched skew products.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Andrea Venturelli, Dept. de Mathematiques, Univ. Avignon.
Títol: Existence of choreographies in a rotating frame for the Newtonian N-body problem.
Resum: We consider the Lagrangian action functional for the Newtonian N-body problem in a rotating frame and we minimize the functional under a choreography constraint. We show that for some values of the angular velocity the minimizer is a relative equilibrium, while for some others values it is a non-trivial choreograpy. This result is variational in nature and has been obtained in 2004 by V. Barutello and S. Terracini.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Renato Vitolo, Università di Camerino (Italia) .
Títol: Extreme Value Statistics of the Total Energy in an Intermediate Complexity Model of the Mid-latitude Atmospheric Jet.
Resum: A baroclinic model for the atmospheric jet at middle-latitudes is used as a stochastic generator of time series. The total energy of the system is used as observable to compute the time series. Statistical inference of extreme values is applied to sequences of yearly maxima of the total energy, in the rigorous setting provided by extreme value theory. In particular, the Generalized Extreme Value (GEV) family of distributions is used here. Several physically realistic values of the parameter TE, descriptive of the forced equator-to-pole temperature gradient and responsible for setting the average baroclinicity in the atmospheric model, are examined. The location and scale GEV parameters are found to have a piecewise smooth, monotonically increasing dependence on TE. This is in agreement with the similar dependence on TE observed in the same system when other dynamically and physically relevant observables are considered.
The GEV shape parameter also increases with TE but is always negative, as a priori required by the boundedness of the total energy of the system. The sensitivity of the statistical inferences is studied with respect to the selection procedure of the maxima: the roles of both the length of sequences of maxima and of the length of data blocks over which the maxima are computed are critically analyzed. Issues related to model sensitivity are also explored by varying the resolution of the system.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Yannick Sire, INSA Toulouse and U. of Texas at Austin .
Títol: Existence of a center manifold and travelling breather solutions for Klein-Gordon lattices.
Resum: In this talk, I am interested in constructing travelling breather solutions in Klein-Gordon lattices. These solutions are spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. Thanks to an appropriate formulation of the problem via a system of advance-delay differential equations, I will construct an invariant center manifold and prove a reduction result. The study of (small amplitude) travelling breather solutions is then performed via the study of the dynamics on the center manifold. I will also present some numerical computations (opening the possibility of a study in the large amplitude regime) and some open questions concerning this problem.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Sergi Simón, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: On the non-integrability of some N-body problems.
Resum: A series of non-integrability results for the N-Body Problem in Celestial Mechanics have been obtained. Namely, we give a new, simpler non-integrability proof for the Three Body Problem with arbitrary masses and a further result for N>3 equal masses. The main tools used for such results come from Morales-Ramis theory.
Joint work with J.J.Morales
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Víctor Sirvent, Dept. Mat., U. Simón Bolivar, Caracas .
Títol: Space-filling curves, symbolic dynamics and geodesic laminations.
Resum: In this talk we will study dynamical systems on the hyperbolic disc as models of some self-similar structures on Eucledian spaces and of a class of symbolic dynamics showing certain symmetries.
First we shall associate space filling curves to connected fractals, obtained as the fixed point of an iterated function systems (IFS) satisfying the common point property and the open set condition. These curves are Hölder continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.
In the second part of the talk we shall consider symbolic systems coming from a family of minimal sequences on a 3-symbol alphabet with complexity 2n+1, which satisfy a special combinatorial property. These sequences are a generalization of the binary sturmian sequences. To these systems we will associate a minimal dynamical systems on the hyperbolic disc.
We show some applications of these results and in particular when the two constructions coincide.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Carlos Villegas-Blas, IMATE, unidad Cuernavaca, UNAM .
Títol: Asymtotics of cluster of eigenvalues for perturbations of the hydrogen atom Hamiltonian.
Resum: We present in this talk a limiting eigenvalue distribution theorem for the Schrödinuer operator of the hydrogen atom (with the Planck parameter $\\hbar$ included) plus $\\epsilon$ times a bounded continuous function $Q$. By considering suitable dilation operators , we prove that taking $\\epsilon=O(\\hbar^2)$ we obtain well defined clusters of eigenvalues around the energy $E=-1/2$ whose limiting distribution involves the Radon transform of the function $Q$ along the classical orbits of the Kepler problem with energy $E=-1/2$ with respect to an integration over the space of geodesics of the 3-sphere $S^3$. The idea of the proof involves a well known unitary transformation from the Hilbert space generated by the bound states of the hydrogen atom onto $L^2(S^3)$ and coherent states on the sphere $S^3$. We will comment on the generalization of the theorem above to the n-dimensional case and when $Q$ is a pseudodifferential operator of order zero.
Lloc: Aula B1 (P.baixa), Facultat de Matemàtiques, UB.
A càrrec de: Maria Jose Pacifico, IMPA, Rio de Janeiro, Brasil .
Títol: Singular-hyperbolicity, a new theory for chaotic attractors .
Resum: Inspired by the Lorenz famous attractor we introduce the notion of singular-hyperbolic attractors: they have dense orbits and are partially hyperbolic with central direction volume expanding. I shall present some interesting dynamics of such attractors, showing that they display many of the properties presented by the chaotic Lorenz attractor.
A càrrec de: Lluis Benet, Depto. de Fisica, UNAM, Mexico .
Títol: Clumps in narrow rings: Results on a toy model and questions .
Resum: In this talk I will address the problem of arcs and clumps in narrow rings using a toy model. The toy model shows the appearance of such features as corrotation resonances, which agrees with the (second) explanation by Namouny and Porco. Yet, the phase space structure that gives rise to them is not clear.
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