Dia: Dimecres, 24 d'octubre de 2018
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Alexey Kazakov, National Research University Higher School of Economics, Nizhny Novgorod, Russia
Títol: On the scenarios of the appearance of a strongly dissipative mixed dynamics by an explosion
Resum: In this talk, a few scenarios of the sudden emergence of mixed dynamics in reversible diffeomorphisms will be presented. The key point of these scenarios is a sharp increase in the sizes of both strange attractor and strange repeller which appear due to heteroclinic bifurcations. Due to such bifurcations, a strange attractor collides with the boundary of its absorbing domain, while a strange repeller collides with the boundary of its "repulsion" domain and, as a result, the intersections between these two sets appear immediately. As a result of the scenarios, the dissipative dynamics associated with the existence of strange attractor and strange repeller (which are separated from each other) sharply becomes mixed, when attractors and repellers are principally inseparable.
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Dia: Dimecres, 17 d'octubre de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marco Fenucci, University of Pisa, Mathematics Department
Títol: Symmetric periodic motions in the N-body problem and (in)stability via computer-assisted approach
Resum: The existence of several periodic orbits of the Newtonian N-body problem has been proved by means of variational methods. In most cases these orbits are found as minimizers of the Lagrangian action functional and the bodies have all the same mass. There are two main difficulties in the variational approach: lack of coerciveness, and exclusion of collisions.
Besides the theoretical approach, also numerical methods have been used to search for periodic motions in a variational context. Several periodic motions with a rich symmetry structure can be found in (Simó, 2001), where the term choreography was first used to denote a motion of N equal masses on the same closed path equally shifted in phase. The introduction of rigorous numerical techniques, led to computer-assisted proofs of the existence of periodic orbits (i.e. Kapela & Zgliczynski, 2003), the linear and KAM stability of the Figure Eight in (Kapela & Simó, 2007) and (Kapela & Simó, 2017).
In this talk we review these two different approach in searching for periodic orbits of the N-body problem, and apply them to a concrete case. In particular, we take into account periodic motions whose existence has been proved in (Fusco et. al., 2011) using the variational technique. All these orbits share the symmetry of a Platonic polyhedron. After proving the existence, we describe the methods used to actually compute them, both non-rigorously and rigorously. Using a procedure similar to the one described in (Kapela & Simó, 2017), we were able to give a computer-assisted proof of the existence and instability of some of these orbits. At the end we will discuss a variant of the problem, using Gamma-convergence theory.
Dia: Dimecres, 31 d'octubre de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Sergey Gonchenko, Lobachevsky University of Nizhny Novgorod, Russia
Títol: Wild pseudohyperbolic attractors in a four-dimensional Lorenz system
Resum: In this talk we present an example of a new strange attractor. We show that it belongs to the class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential equations which can be represented as an extension of the Lorenz system. This is a joint work with A. Kazakov and D. Turaev.
Dia: Dimecres, 7 de novembre de 2018
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Josep M. López Besora, Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili
Títol: Use of biomechanical simulations to study aspects of deep venous thrombosis
Resum: Deep venous thrombosis is a common disease. Large thrombi in venous vessels cause bad blood circulation and pain; as well as pulmonary embolisms if thrombi are detached from the vessels walls. Biomechanical simulations are used to study the effects of walking or the implantation of different vena cava filters. A realistic geometric model build from data from a real patient is used.
Dia: Dimecres, 14 de novembre de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Filippo Giuliani, Universitat Politècnica de Catalunya
Títol: KAM for quasi-linear PDEs
Resum: The KAM for PDEs is the mathematical theory developed for the search of quasi-periodic (in time) solutions of partial differential equations on compact (spatial) domains.
Many notable PDEs (NLS, KdV, Klein-Gordon...) possess a Hamiltonian structure and behave, in a neighborhood of the origin, like an infinite chain of harmonic oscillators weakly coupled by the nonlinear terms. Then it is natural to look at these equations as infinite dimensional dynamical systems and use perturbative arguments to find finite dimensional invariant tori close to the origin.
The main issues arising in this kind of problems are related to the geometry/dimension of the spatial domain, the dispersive effects of the PDE and the number of the derivatives appearing in the nonlinearities. In this talk we will focus on PDEs on the circle and we will give an overview of the strategy for proving KAM results by using generalized implicit function theorems.
Although the KAM theory for PDEs on spatial 1-d domains is now well understood, the progress for quasi-linear cases, namely when the derivatives contained in the linear and nonlinear terms have the same order, is quite recent.
In this framework, we present a new result (in collaboration with R.Feola and M. Procesi) of existence and stability of quasi-periodic solutions for perturbations of the Degasperis-Procesi equation on the circle, which is a model for nonlinear shallow water phenomena. In this work we developed new techniques to deal with quasi-linear equations that have very weak dispersive effects and a complicated resonant structure.
Dia: Dimecres, 21 de novembre de 2018
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Gemma Huguet, Universitat Politècnica de Catalunya
Títol: Quasi-periodic perturbations of heteroclinic attractor networks
Resum: We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that Gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasi-periodic perturbations. For this purpose, we present a methodology based on the separatrix map to model the dynamics close to heteroclinic networks with quasi-periodic perturbations. Our methodology considers two different approaches, one based on Melnikov integrals and another one based on variational equations. We apply it to two models: first, to the Duffing equation, which comes from the perturbation of a Hamiltonian system and, second, to a heteroclinic attractor network for binocular rivalry. In both models, the perturbed system shows chaotic behavior while dominance times achieve good agreement with Gamma distributions. Moreover, the separatrix map provides a discrete model for bistable perception. This is joint work with A. Delshams and A. Guillamon
Dia: Dimecres, 28 de novembre de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Daniel Pérez, Escuela superior de ingeniería y tecnología, UNIR.
Títol: Indirect Optimization of Low-thrust Earth-Moon Transfers in the Sun-Earth-Moon System
Resum: In this talk we will present some Optimal Low-Thrust Earth-Moon Transfers computed through indirect optimization methods, i.e. Pontryagin's Maximum Principle (PMP). The setting of study is a Planar Bicircular Restricted Four-Body Problem (PBRFBP). First, we will summarize the PMP main results. Then, we will formulate a minimum-fuel optimal control problem and present the standard numerical difficulties related to its solution by means of indirect shooting methods. After that, we will present the difficulties that arises in the PBRFBP. To overcome these difficulties, continuation techniques are implemented, however it is not possible to find the desired solution. Then, a new robust indirect approach is developed. Instead of using a standard shooting method, based on a Newton-like scheme, a derivative-free algorithm is used to find the zeros of the shooting function.
We will finish the talk with a classification of the orbits that has been found according the dynamical counterparts of the manifolds in the Circular Restricted Three Body Problem.
Dia: Dimecres, 12 de desembre de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Joan Sánchez, Universitat Politècnica de Catalunya.
Títol: Torsional solutions of convection in rotating fluid spheres.
Resum: A numerical study of the torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres will be presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman, E, and small Prandtl, Pr, numbers in the region of parameters where the linear stability of the conduction state predicts that they can be preferred at the onset of convection.
The periodic torsional solutions are axisymmetric and not rotating waves, unlike the non-axisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time and space averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E=10-4 at higher Rayleigh numbers, Ra, show that this trend is also valid for the non-periodic flows.
The modulated oscillations bifurcated from the quasi-periodic torsional solutions reach a high amplitude compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic connections of unstable periodic solutions. We have seen that these complex solutions are able to generate magnetic fields by dynamo effect.
Dia: Dimecres, 19 de desembre de 2018
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Xavier Jarque, Universitat de Barcelona
Títol: The secant map as a plane dynamical system
Resum: We present some results on the dynamical system induced by the secant map on the real plane applied to real polynomials. As it is well known the secant method is a root finding algorithm which does not use the evaluation of the derivative of the corresponding map.
On the one hand we prove that there exist some points, called focal points, belonging to the boundary of the basins of attraction of all fixed points. On the other hand we prove the existence of polynomials for which there are open sets of the plane not convergeging to any of its roots. Finally we will discuss some further dynamical considerations by somehow adding infinity on the field.
Dia: Dimecres, 16 de gener de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Juan J. Morales-Ruiz, Universidad Politécnica de Madrid
Títol: Integrabilidad de procesos estocásticos de nacimiento-muerte vía Teoría de Galois diferencial
Resum: Algunos procesos estocásticos relevantes de Markov de dinámica de poblaciones en tiempo contínuo son regidos por sistemas de infinitas ecuaciones diferenciales ordinarias acopladas. Estos sistemas se transforman en ecuaciones en derivadas parciales de tipo difusión mediante la función generatriz asociada al sistema (transformada Z). EL objetivo de esta charla es estudiar la integrabilidad mediante la teoría de Galois diferencial de dos ecuaciones en derivadas parciales de ese tipo que modelizan dos procesos estocásticos de nacimiento y muerte en dinámica de poblaciones.
(Trabajo conjunto con Primitivo B. Acosta-Humánez y José A. Capitán).
Last updated: Thursday, 17-Jan-2019 14:49:28 CET