Dia: Dimecres, 10 de setembre de 2014
Lloc: Aula 102, FME, UPC.
A càrrec de: Marco Antonio Teixeira, Universidade Estadual de Campinas
Títol: Stability and Periodic orbits in Piecewise Smooth Dynamical Systems
Resum: we discuss some properties of some models in Piecewise Smooth Dynamical Systems, such as asymptotic stability and birth of periodic orbits.
Dia: Dimecres, 8 d'octubre de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marina Gonchenko, Technische Universitat Berlin
Títol: Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequencies
Resum: We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in nearly-integrable Hamiltonian systems such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus whose frequency ratio Ω is a quadratic irrational number. Applying the PoincarĂ©-Melnikov method, we provide an asymptotic estimate for the maximal splitting distance, and show the existence of 4 transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for their transversality of the splitting. Such estimates are exponentially small in the perturbation parameter, and the functions in the exponents satisfy a periodicity property. This requires to carry out a careful study of the dominant harmonics of the Melnikov potential, which are strongly related to the arithmetic properties of Ω, given by its continued fraction. The estimates of the maximal splitting distance are valid for all the sufficiently small values of the parameter, and the transversality can be established for a majority of values of the parameter, excluding some small intervals where some changes in the dominance of the harmonics take place, and bifurcations could occur. This is a joint work with Amadeu Delshams and Pere Gutiérrez.
Dia: Dimecres, 15 d'octubre de 2014
Lloc: Aula 103, FME, UPC.
A càrrec de: Luis Garcia Naranjo, UNAM (México)
Títol: Preservación de volúmenes en mecánica no-holónoma
Resum: En mecánica, las restricciones en las configuraciones de un sistema se denominan "holónomas". Un ejemplo sencillo es la longitud constante del péndulo. Sistemas mecánicos con restricciones en las velocidades que no pueden reducirse a restricciones en las posiciones se llaman "no-holónomas". Un ejemplo clásico es una esfera que rueda sin resbalar en una mesa.
El reto en el estudio de los sistemas mecánicos no-holónomos aparece debido a que las ecuaciones de movimiento no poseen una estructura Hamiltoniana. Sin embargo, la dinámica del sistema puede ser descrita en términos de un corchete de funciones que no satisface la identidad de Jacobi. Hablamos entonces de un "corchete casi-Poisson".
La pérdida de la identidad de Jacobi da lugar a fenómenos que no son posibles en los sistemas Hamiltonianos clásicos. Algunas preguntas abiertas en el área de mecánica no-holónoma incluyen determinar condiciones para la existencia de una medida conservada, existencia de equilibrios asintóticos, relación entre simetrías y leyes de conservación, reducción e integrabilidad.
En la primera parte de la charla presentaré una introducción básica a los sistemas no-holónomos rica en ejemplos. Después procederé a presentar un trabajo reciente en conjunto con Y. Fedorov y J. C. Marrero en donde estudiamos el problema de la preservación de una medida para sistemas no-holónomos con simetrías de una manera sistemática. Nuestro método nos permite identificar valores de los parámetros para los cuales existe una medida invariante para sistemas mecánicos concretos.
Dia: Dimecres, 22 d'octubre de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alejandro Luque, Universitat de Barcelona
Títol: Otra charla más de teoría KAM... esta vez con ordenador
Resum: ¿Cuántas veces se ha hablado de teoría KAM en este seminario? Unas cuantas. ¿Cuántas veces se ha hablado de métodos de la parametrización? Unas cuantas también. ¿Hay razones para retomar el tema de nuevo? Esperamos convencer de que sí.
En diciembre del año pasado, bajo el título "The parametrization method in KAM theory towards CAPs", ya hablamos en este seminario de las principales ideas y del projecto (entonces en fase preliminar) para obtener rigurosamente toros invariantes con la ayuda de un ordenador. El objetivo de este seminario es presentar los primeros frutos del trabajo.
Presentaremos a fondo una demostración asistida por ordenador de curvas invariantes en aplicaciones simplécticas del anillo. Como la demostración se basa en el método de la parametrización, presenta todas las cosas buenas que se han ido anunciando en los últimos años [y citamos: R. de la Llave, A. González, iÀ. Jorba, J. Villanueva, E. Fontich, Y. Sire, G. Huguet, A. Haro, H. N. Alishah, R. C. Calleja, A. Celletti, etc]. En la charla, haremos énfasis a los resultados analíticos que conectan cierto teorema KAM con una serie de numeritos que aparecen en una pantalla y que nos permiten afirmar la existencia de curvas invariantes.
Este es un trabajo conjunto con Alex Haro y Jordi-Lluís Figueras.
Dia: Dimecres, 29 d'octubre de 2014
Lloc: Aula 103, FME, UPC.
A càrrec de: Richard Montgomery, Dept. of Mathematics, UC Santa Cruz
Títol: The Syzygy Program
Resum: A syzygy in the N-body problem is the same as a collinearity, or eclipse. In the three body problem syzygies come in three flavors, 1,2, and 3, depending on which body lies between the other two. A syzygy sequence is a word, finite or infinite, in 1, 2 and 3. For the Newtonian planar three body problem with zero angular momentum is it true that a syzygy sequence is realized by some solution? Is every periodic syzygy sequence realized by a periodic solution? If not, which sequences are realized? I have worked on these questions for 16 years, with minimal success. My lack of success persists, despite the fact that I know the complete answer to these questions if I change the Newtonian potential to a 1/r2 potential while taking all three masses equal. In this talk I will describe the origin of the questions, a few results which have flowed out of work on these questions, and perhaps some failed attempts at answering them. Lately, I have begun to pursue the question through numerical experimentation. I end with a plea for help in this pursuit.
Dia: Dimecres, 5 de novembre de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Edriss S. Titi, Dpt. of Computer Science and Applied Mathematics, The Weizmann Institute of Science & Dpt. of Mathematics, Mechanical and Aerospace Engineering, University of California - Irvine
Títol: An Algorithm for Advancing Slow Features in Fast-Slow Systems without Scale Separation - A Young Measure Approach
Resum: In the first part of the talk, and in order to set the stage, we will offer a multi-scale and averaging strategy to compute the solution of a singularly perturbed system when the fast dynamics oscillates rapidly; namely, the fast dynamics forms cycle-like limits which advance along with the slow dynamics. We describe the limit as a Young measure with values being supported on the limit cycles, averaging with respect to which induces the equation for the slow dynamics. In particular, computing the tube of the limit cycles establishes a good approximation for arbitrarily small singular parameters. We will demonstrate this by exhibiting concrete numerical examples.
In the second part of the talk we will examine singularly perturbed systems which may not possess a natural split into fast and slow state variables. Once again, our approach depicts the limit behavior as a Young measure with values being invariant measure of the fast contribution to the flow. These invariant measures are drifted by the slow contribution to the value. We keep track of this drift via slowly evolving observables. Averaging equations for the latter lead to computation of characteristic features of the motion and the location the invariant measures. To demonstrate our ideas computationally, we will present some numerical experiments involving a system derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion.
This is a joint work with Z. Artstein, W. Gear, I. Kevrekidis, J. Linshiz and M. Slemrod.
Dia: Dimecres, 19 de novembre de 2014
Lloc: Aula 103, FME, UPC.
A càrrec de: Josep Sardanyés (Complex Systems Lab, Universitat Pompeu Fabra (UPF) and Institute of Evolutionary Biology CSIC-UPF)
Títol: Variability in mutational fitness effects prevents full lethal transitions in large quasispecies populations
Resum: The distribution of mutational fitness effects (DMFE) is crucial to the evolutionary fate of quasispecies. In this article we analyze the effect of the DMFE on the dynamics of a large quasispecies by means of a phenotypic version of the classic Eigen's model that incorporates beneficial, neutral, deleterious, and lethal mutations. By parameterizing the model with available experimental data on the DMFE of Vesicular stomatitis virus (VSV) and Tobacco etch virus (TEV), we found that increasing mutation does not totally push the entire viral quasispecies towards deleterious or lethal regions of the phenotypic sequence space. The probability of finding regions in the parameter space of the general model that results in a quasispecies only composed by lethal phenotypes is extremely small at equilibrium and in transient times. The implications of our findings can be extended to other scenarios, such as lethal mutagenesis or genomically unstable cancer, where increased mutagenesis has been suggested as a potential therapy.
Subject terms: Evolutionary theory, Virology, Applied mathematics, Dynamical systems
Publication available at Nature
Dia: Dimecres, 26 de novembre de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Anna Tamarit, Universitat Politècnica de Catalunya
Títol: On the length and area spectrum of analytic convex domains
Resum: Area-preserving twist maps have at least two different (p,q)-periodic orbits and every (p,q)-periodic orbit has its (p,q)-periodic action for suitable couples (p,q). We establish an exponentially small upper bound for the differences of (p,q)-periodic actions when the map is analytic on a (m,n)-resonant rotational invariant curve (resonant RIC) and p/q is "sufficiently close'' to m/n. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the n-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle.
We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the (1,q)-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period q. This improves some classical results of Marvizi, Melrose, Colin de Verdière, Tabachnikov, and others about the smooth case.
This is a joint work with Pau Martín and Rafael Ramírez-Ros.
Dia: Dimecres, 14 de gener de 2015
Lloc: Aula 103, FME, UPC.
A càrrec de: Stefanella Boatto, Depto de Matematica Aplicada, Instituto de Matematica, Universidade Federal de Rio de Janeiro, Brazil
Títol: The dynamics of vortices and masses over surfaces of revolution: an equivalence principle
Resum: One of the today's challenges is the formulation of the N-body and N-vortex dynamics on Riemann surfaces. In this talk we show how the two problems are strongly related one another when looking at them from the point of view of the intrinsic geometry of the surface where the dynamics takes place. Given a surface M of metric g, the distribution of matter S on M, we deduce the dynamics of the masses and some of its properties. In particular we observe that some of the known physics laws depend upon the dimension and the intrinsic geometry of the space.
Joint work with Rodrigo Schaefer (UPC), David Dritschel (Univ. St-Andrews, UK)
Dia: Dimecres, 21 de gener de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marcel Guàrdia, Universitat Politècnica de Catalunya
Títol: Nearly integrable Hamiltonian systems with orbits accumulating to KAM tori
Resum: The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.
Dia: Dimecres, 28 de gener de 2015
Lloc: Aula 103, FME, UPC.
A càrrec de: Anna Kiesenhofer, Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya
Títol: Action-angle coordinates and KAM theory for b-Poisson manifolds
Resum: I will talk about an action-angle coordinate theorem for integrable systems on b-Poisson manifolds [GMP] which improves the action-angle theorem contained in [LMC] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds. This is joint work with Eva Miranda and Geoffrey Scott.
References:
[GMP] Guillemin, Victor; Miranda, Eva; Pires, Ana Rita: Symplectic and Poisson geometry on b-manifolds. Adv. Math. 264 (2014), 864-896.
[LMC] Laurent-Gengoux, Camille; Miranda, Eva; Vanhaecke, Pol: Action-angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Not. IMRN 2011, no. 8, 1839-1869.
[MKS] Miranda, Eva; Kiesenhofer, Anna; Scott, Geoffrey: An action-angle theorem for b-Poisson manifolds and applications to KAM theory, preprint (2015).
Dia: Dimecres, 4 de febrer de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Piotr Zgliczynski (Jagiellonian University, Krakow, Poland)
Títol: Averaging for dissipative PDEs
Resum: We study the effect of fast movement in dissipative PDEs with the forcing term, for example Burgers equation or Navier-Stokes system in 2D. By the fast movement we mean that the average speed of the fluid is large and is preserved during the evolution.
In fact we describe a method to study dynamics with rapidly oscillating vector fields (some version of averaging with an eye toward a prori bounds). As an example we apply the technique to the Burgers equation with nonautonomous forcing and the periodic boundary conditions. We prove that for large initial condition integral the equation admits a globally attracting solution defined on the real line. We show that the technique applies to 2D Navier-Stokes equations.
This is a joint work with Jacek Cyranka.
Dia: Dimecres, 11 de febrer de 2015
Lloc: Aula 103, FME, UPC.
A càrrec de: Maisa de Oliveira Terra, Instituto Tecnológico de Aeronáutica, São José dos Campos,Brasil
Títol: Evidences of Diffusion Related to the Center Manifold of L3 of the SRTBP
Resum: In this talk we present evidences of diffusion of trajectories in the framework of the Spatial Restricted Three-Body Problem (SRTBP) and introduce a methodology to quantify and to examine the diffusion process. We focus our analysis in the vicinity of the center manifold of the equilibrium L3, WCL3, which is four-dimensional. We compute the invariant structures inside WCL3, in particular, we obtain Fourier representations of invariant curves, corresponding to Poincaré sections of bidimensional invariant tori inside WCL3. Then, we report on the diffusion of trajectories with initial conditions very close to these invariant curves. We introduce a methodology for diffusion analysis which provides statistics of the process and shows that the diffusion rate of trajectories is not constant in the phase space. The diffusion rate increases as trajectories go away from the vertical periodic orbit and then decreases when they approach the planar periodic orbit, wandering across the hyperbolic manifolds of distinct tori. Besides the analysis of a large ensemble of trajectories, specific cases are also studied to illustrate the process. In some cases, stickiness to tridimensional tori is observed. Eventually, the trajectories escape due to approximation to the secondary. Finally, we show that the diffusion mechanism is associated to the existence of transition chains of heteroclinic connections and relate the rate of diffusion with the splitting of the stable and unstable hyperbolic manifolds of the two-dimensional invariant tori. Our results are based on numerical experiments rather than analytical tools.
This is a joint work with Carles Simó and Priscilla A. Sousa-Silva.
Dia: Dimecres, 18 de febrer de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Ariadna Farrés, Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona
Títol: High-Order Symplectic Splitting Methods for Dynamical Astronomy
Resum: In this talk we will present new families of splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, with special interest on the planetary N-body problem. We use a Newtonian model for the motion of the planetary system, which can be written as HA + εHB with an appropriate set of coordinates. We will perform an extensive comparison between different high-order symplectic splitting schemes, trying to determine the best scheme for the particular case of the solar system.
This is a joint work with: Jackes Laskar (IMCCE), S. Blanes (UJI), F. Casas (UJI) and A. Murua (UPV)
References:
Dia: Dimecres, 25 de febrer de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Alexander Wittig, Politecnico di Milano (Department of Aerospace Science and Technology)
Títol: Rigorous Lower Bounds of the Topological Entropy in the Lorenz System
Resum: The Lorenz system is probably the most common example of chaotic behavior in a differential equation. Despite having been studied extensively for decades, surprisingly little is known rigorously about the properties of this system, particularly away from the standard parameters of ρ=28, σ=10 and β=8/3. Non-verified numerical simulations provide plenty of insight into the behavior of the system, but many numerically observed properties commonly accepted as true have not yet been proved rigorously. In this talk, an algorithm will be presented to automatically prove the existence of chaos (in the sense of positive topological entropy) in the Lorenz system over a large range of ρ values from 25 up to 85. The proof is computer assisted, meaning it employs rigorous numerical integration methods to verify certain topological properties of numerically constructed sets. More specifically, ρ-dependent candidate sets are constructed based on a numerical analysis of the system dynamics in a Poincare map. The sets thusly obtained are then integrated rigorously using the Taylor Model based COSY VI verified integrator and their return to the Poincare section is computed. Finally, using the rigorous bounds provided by the verified integration, certain topological constraints are verified in order to ensure a Markov-type crossing, which then give rise to simple symbolic dynamics with positive entropy.
Dia: Dimecres, 4 de març de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Adrià Simon, Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya
Títol: Diffusion through non-transversal transition chains: A long-time instability for the NLS
Resum: This is a joint work with Amadeu Delshams and Piotr Zgliczcynski.
The main goal of our work is to understand the geometric mechanism that gives rise to the instability shown by Colliander et al (2010) in the Nonlinear Schrödinger Equation with cubic defocusing. It can be seen as a diffusion mechanism, but it appears that the geometric skeleton of the system is not the standard for Arnold Diffusion: instead of having a sequence of non-resonant invariant tori connected along transverse heteroclinic orbits we have a non-transversal situation.
We expose that the instability (diffusion) can be achieved due to the large dimension of the system, while we try to generate a scheme for this new kind of diffusion that could be applied to other infinitely dimensional Hamiltonian systems.
Dia: Dimecres, 11 de març de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Jordi Canela Sánchez, Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona
Títol: On a Family of Degree 4 Blaschke Products
Resum: We investigate the parameter plane of the rational family of perturbations of the doubling map given by the Blaschke products Ba(z)=z³(z-a)/(1-āz). We study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter a. We also classify and study the different hyperbolic components of the parameter plane according to the critical orbits.
The non holomorphic dependence on the parameters of the family gives rise to the existence of phenomena which cannot occur otherwise. For |a|≥ 2, there appear tongues as sets of parameters with attracting cycles in 𝕊¹. We study the connectivity of these tongues and how bifurcations take place along their boundaries. Also for parameters with |a|>2, small "copies" of the connectedness locus of the antipolynomials pc(z)=z̅²+c, the Tricorn, appear contained in regions of parameters for which the free critical points enter and exit the unit disk. We use techniques of quasiconformal surgery to explore the relation between such Ba and the degree 4 polynomials pc²(z)=( z̅²+c)²+c
Dia: Dimecres, 25 de març de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Ernest Fontich, Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona
Títol: Stable manifolds of parabolic points through the parameterization method.
Resum: We consider maps with a fixed point whose linearization is the identity (parabolic point) and we study the existence, regularity and dependence of parameters of its stable set, under some appropriate conditions.
For that we use the parameterization method which first gives a suitable approximation and then provides the true invariant set.
Here the approximation may not be polynomial, but a sum of homogeneous functions.
The case of parabolic fixed points for differential equations is also considered.
As an application we consider the manifolds of the infinity of the elliptic spatial RTBP.
This is joint work with Inma Badomà and Pau Martín.
Dia: Dimecres, 8 d'abril de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Martin Krupa, INRIA, Paris-Rocquencourt
Títol: Slow-fast dynamics and canards in mathematics and neuroscience
Resum: The subject of this talk is the dynamics arising in slow-fast systems arising as a combination of canard explosion, bifurcation delay, mixed-mode oscillations and bursting, with special focus on canard solutions. To date, canards have been associated with the transition from quiescence to relaxation oscillations, or with mixed-mode oscillations. However, as it will be shown in this talk, they are equally important in the transition from quiescence to bursting and in spike adding.
Slow-fast systems not only have an interesting mathematical mathematical structure, but also provide a flexible modelling tool. To illustrate this point a slow-fast model of delayed neurotransmitter release in CCK positive basket cells will be introduced.
Dia: Dimecres, 15 d'abril de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Alejandro Luque, ICMAT (CSIC)
Títol: Difusión de Arnold en el movimiento de cargas magnéticas bajo la acción del campo ABC
Resum: El campo ABC es un caso particular de campo de divergencia cero en 𝕋3 que tiene una gran tradición en física de plasma (el ABC es un campo de los llamados "force-free" que aparecen cuando la presión magnética es mucho mayor que la presión del plasma) así como en matemáticas (es solución de la ecuación de Euler estacionaria) y aparece en diversos contextos, tales como física de la materia condensada, física de aceleradores, magnetohidrodinámica, etc.
Consideramos el movimiento de una partícula cargada bajo la acción del campo ABC, dando lugar a un sistema Hamiltoniano de 3 grados de libertad que depende, sin pérdida de generalidad, de 2 parámetros. Cuando dichos parámetros se anulan, el sistema tiene una variedad normalmente hiperbólica de dimensión 4 cuyas variedades estable e inestable, de dimensión 5, coinciden. En jerga popular, nuestro problema es a-priori inestable.
En esta charla queremos estudiar el fenómeno de difusión de Arnold usando los métodos geométricos introducidos por A. Delshams, R. de la Llave, y T.M. Seara. Como nuestro problema no coincide exactamente con los diversos modelos considerados en la literatura, tendremos que ver si podemos adaptar la construcción a nuestro caso. Suponiendo que somos capaces de obtener condiciones suficientes para la existencia de cadenas de transición, entonces sería interesante ver si somos capaces de verificar dichas hipótesis para el sistema ABC. Suponiendo que somos capaces de hacerlo, sería interesante interpretar el fenómeno en el contexto físico del problema. A lo largo de la charla veremos si podemos responder alguna de las cuestiones anteriores.
Este es un trabajo conjunto con Daniel Peralta-Salas.
Dia: Dimecres, 22 d'abril de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Pau Martin, Universitat Politècnica de Catalunya
Títol: Arnold diffusion and oscillatory orbits in the restricted elliptic planar three body problem
Resum: The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of two bodies evolving in Keplerian ellipses.
The purpose of this paper is to analyze two different dynamical behaviors of this system: the so-called oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region, and orbits with arbitrary large (but finite) growth in angular momentum of the massless body.
Even if both types of dynamics are apparently unrelated, they can be obtained by the same mechanism.
Dia: Dimecres, 20 de maig de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Chara Pantazi, Universitat Politécnica de Catalunya
Títol: Some results about the integrability of planar polynomial vector fields
Resum: We are going to present some inverse problemes related to the integrability of planar polynomial vector fields using invariant algebraic curves. We will also pay atention to a special class of polynomial vector fields and we will apply results of Differential Galois Theory in order to decide about its integrability. We will relate these results with the notion of invariant algebraic curves and the existence of first integrals\integrating factors. Some qualitative properties will be commented.
This talk is a presentation of some results that have been obtained in several works with co-authors: P. Acosta-Humanez, C. Christopher, T. Lazaro, J.J. Morales, J. Llibre, S. Walcher.
Dia: Dimecres, 27 de maig de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Anna Tamarit, Universitat Politècnica de Catalunya
Títol: Exponentially small asymptotic formulas for the length spectrum in some billiard tables
Resum: Let q>2 be a period. There are at least two (1,q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1,q)-periodic trajectories as q tends to infinity.
Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q-3exp(-rq) times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1,q)-periodic trajectories.
Our experiments are focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.
This is a joint work with Pau Martin and Rafael Ramirez-Ros.
Dia: Dimecres, 3 de juny de 2015
Lloc: Aula S04, FME, UPC.
A càrrec de: Juan José Morales, Universitat Politécnica de Madrid
Títol: Solitons and Differential Galois Theory
Resum: This talk will be devoted to an application of the Differential Galois Theory to the so-called "pde's integrable evolution equations", i.e., equations with "solitonic" solutions, like KdV (Korteweg de Vries), Sine-Gordon, etc. After Lax, these equations are obtained trough Lax pairs of linear differential equations. More concretely, after a minimum of necessary definitions and results on the Galois Theory of linear differential equations, the talk will be centered around the following Conjecture: the Galois group of one of the Lax pairs doesn't depends on time, i.e., the temporal evolution of the solutions must be isogaloisian. This conjecture will be verified for a classical family of rational like-solitons solutions of the KdV hierarchy obtained by Adler and Moser. Moreover we shall illustrate the above with some explicit concrete computations. The content of the talk is part of a joint work (in progress) with Sonia Jiménez, Raquel Sánchez-Cauce and Maria-Ángeles Zurro.
Dia: Dimecres, 10 de juny de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Oriol Castejón, Universitat Politècnica de Catalunya
Títol: Exponentially small splitting in analytic unfoldings of the Hopf-Zero singularity
Resum: If one considers conservative (i.e. one-parameter) unfoldings of the so-called Hopf-zero singularity, one can see that the truncation of the normal form at any finite order possesses two saddle-focus critical points with a one- and a two-dimensional heteroclinic connection. The same happens for non-conservative (i.e. two-parameter) unfoldings when the parameters lie on a certain curve.
However, considering the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of C∞ unfoldings, this was proved by Broer and Vegter during the 80's, but for analytic unfoldings it has remained an open problem. Recently, under some assumptions on the size of the splitting of the heteroclinic connections, Dumortier, Ibáñez, Kokubu and Simó proved the existence of Shilnikov bifurcations in the analytic case.
Our study concerns the splitting of the one and two-dimensional heteroclinic connections. These cannot be detected in the truncation of the normal form at any order, and hence they are expected to be exponentially small with respect to one of the perturbation parameters. We shall present asymptotic formulas of these splittings, putting emphasis on the differences between the conservative and non-conservative cases. In particular, we prove that under generic conditions, the main assumptions made by Dumortier, Ibáñez, Kokubu and Simó hold.
This is a joint work with I. Baldomà and T. Seara.
Dia: Dimecres, 17 de juny de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Giannis Moutsinas, University of Warwick
Títol: Exponentially small splitting of separatrices in area-preserving maps close to 1:3 resonance
Resum: Area-preserving maps are important tools for analysing the stability of periodic orbits. Unfortunately the dynamics of resonant maps are not fully understood as in general the normal form is not a sufficient analysis tool. In this talk we will see how one can see, using the Borel-Laplace summation method, that the separatrices of a resonant map split. This turns to be exponentially small. After that it will be demonstrated that the splitting of the unfolding is dominated by the splitting at the resonance.
Dia: Dijous, 25 de juny de 2015
Lloc: Aula S04, FME, UPC.
A càrrec de: Zaher Hani, Georgia Institute of Technology
Títol: Energy dynamics across scales and wave turbulence in Hamiltonian PDE
Resum: Broadly speaking, we will be interested in the mathematical study of a central physical problem, namely that of energy transfer in Hamiltonian systems. More precisely, suppose energy is initially injected only in a fraction of all the possible degrees of freedom of the system, how will this energy be redistributed as time passes by? This problematic arises in several phenomena from heat transfer to wave dynamics in oceans and fluid plasma. We will focus our attention on dispersive Hamiltonian systems, and see how this problem translates into deep analytical questions about the long-time behavior of solutions to the corresponding nonlinear PDE. We will survey some attempts to understand these questions from a deterministic and statistical viewpoint, and report on recent progress in both directions.
A càrrec de: Jozsef Farkas, Univeristy of Stirling
Títol: On structured epidemiological models
Resum: In this talk we will introduce and discuss some structured partial differential equations to model infectious disease dynamics.
Dia: Dimecres, 8 de juliol de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Horacio Rotstein, New Jersey Institute of Technology
Títol: Frequency Preference Response to Oscillatory Inputs in Neuronal Models: A Geometric Approach to Subthreshold Resonance
Resum: Many neuron types exhibit preferred frequency responses to subthreshold oscillatory input currents reflected in a voltage amplitude peak (resonance) and a zero phase-shift (phasonance or phase-resonance). These phenomena may occur in the absence of intrinsic oscillations in the corresponding autonomous system. The dynamics principles that govern the generation of resonance and the effect of the biophysical parameters on the resonant properties are not well understood.
We propose a framework to analyze the role of different ionic currents and their interactions in shaping the properties of the impedance amplitude and phase profiles (graphs of these quantities as a function of the input frequency) in linearized and quadratic biophysical models. We adapt the classical phase-plane analysis approach to account for the dynamic effects of oscillatory inputs and develop a tool, the envelope-plane diagrams, that capture the role that conductances and time scales play in amplifying the voltage response at the resonant frequency band as compared to smaller and larger frequencies. We use envelope-plane diagrams in our analysis to explain why the resonance phenomena do not necessarily arise from the presence of imaginary eigenvalues at rest, but rather it emerges from the interplay of the intrinsic and input time scales. This interaction is based mostly on transient effects. We further explain why an increase in the time scale separation causes an amplification of the voltage response in addition to shifting the resonant and phase-resonant frequencies.
We extend this approach to explain the effects of nonlinearities on both resonance and phase-resonance. We demonstrate that nonlinearities in the voltage equation cause amplifications of the voltage response and shifts in the resonant and phase-resonant frequencies that are not predicted by the corresponding linearized model. The differences between the nonlinear response and the linear prediction increase with increasing levels of the time scale separation between the voltage and the gating variable, and they almost disappear when both equations evolve at comparable rates. In contrast, voltage responses are almost insensitive to nonlinearities located in the gating variable equation. The method we develop provides a framework for the investigation of the preferred frequency responses in three-dimensional and nonlinear neuronal models as well as simple models of coupled neurons.
Dia: Dimecres, 15 de juliol de 2015
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Vadim Kaloshin, University of Maryland
Títol: An integrable billiard close to an ellipse of small eccentricity is an ellipse
Resum: In 1927 G. Birkhoff conjectured that if a billiard in a strictly convex smooth domain is integrable, the domain has to be an ellipse (or a circle). The conjecture is still wide open, and presents remarkable relations with open questions in inverse spectral theory and spectral rigidity. In the talk we show that a version of Birkhoff's conjecture is true for small perturbations of ellipses of small eccentricity. This is joint work with J. De Simoi and A. Avila
Last updated: Monday, 28-Oct-2024 11:34:21 CET