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, 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.
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).
Dia: Dimecres, 23 de gener de 2019
Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Michela Procesi, University of Roma Tre
Títol: Almost-periodic solutions for the NLS with parameters
Resum: I shall discuss a recent result with L. Biasco and J. Massetti on the existence of almost-periodic solutions for the NLS on the circle with external parameters. After discussing the (very few) known results I shall describe our strategy, which is quite flexible and can be applied also for the construction of non maximal tori. Time permitting I shall discuss also the connection with exponential/subexponential stability for small initial data.
Dia: Dimecres, 30 de gener de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marc Jorba, Facultat de Matemàtiques i Informàtica, UB
Títol: On recollisions of electrons under the influence of a laser field
Resum: This talk is concerned about recollisions, a mechanism in which an electron gets expelled away from an atom and returns back to its ionic parent, caused by the excitation of a strong laser field. The interaction between the Coulomb potential and the laser is modeled by means of a Hamiltonian system with periodic time dependence.
A dynamical scenario to explain recollisions is well established in the one dimensional case: The stable and unstable manifolds of a key periodic orbit drive regions of the phase space to recollide many times. The fact that the system is far from integrable and that the manifolds display a very intricate behaviour is crucial for trajectories displaying a large number of recollisions to occur.
After a suitable introduction to the problem we will explain how the well known scenario for the one dimensional case is extended to the two dimensional one. The scenario is changed substantially as higher dimensional invariant objects must be involved.
This is a joint work with Jonathan Dubois, Àngel Jorba, Cristel Chandre, Turgay Uzer and Simon Berman.
Dia: Dimecres, 6 de febrer de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Stefanella Boato, Depto Matemática Aplicada. IM, Universidade Federal de Rio de Janeiro, Brasil
Títol: N-body dynamics on an infinite cylinder: the topological signature and the stability of a ring
Resum: The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. For such purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following what done in Boatto, Dritschel and Schaefer [1], we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas.
Furthermore, when focusing on the case of two bodies’ motion, Poincaré sections indicate that the dynamics is non integrable. Moreover, for very low energies, when the bodies are restricted to a very small region of the cylinder, the topological signatures of the cylinder and of the plane are still present in the dynamics. A perturbative expansion is founded for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation.
Joint work with Gladston Duarte, Teresa Stuchi and Jaime Andrade.
References:
Dia: Dimecres, 13 de febrer de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Stefano Pasquali, Centre de Recerca Matemàtica, Universitat Politècnica de Catalunya
Títol: Birkhoff Normal Form results for singular limits of nonlinear Hamiltonian PDEs
Resum: I shall discuss an approach that allows to combine results comings from Birkhoff Normal Form theory and results from the theory of dispersive PDEs in order to study the dynamics of nonlinear Hamiltonian PDEs. As a main application, I shall describe how this techniques applies to the non-relativistic limit of the nonlinear Klein–Gordon (NLKG), in order to approximate its solutions for long timescales. Time permitting, I shall discuss how this technique can be applied to the continuous approximation of lattice dynamics with nearest-neighbours interaction.
Dia: Dimecres, 20 de febrer de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Roberto Feola, University of Nantes
Títol: Birkhoff Normal Form and long time existence for periodic gravity water waves
Resum: We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This prove a conjecture of Zakharov-Dyachenko based on the formal Birkhoff integrability of the waver waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3
Dia: Dimecres, 27 de febrer de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Ricardo Pérez-Marco, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris-Diderot
Títol: On Briot and Bouquet problem on singularities of analytic differential equations
Resum: We solve Briot and Bouquet problem (1856) on the existence of non-monodromic (multivalued) solutions for singularities of differential equations in the complex domain. The solution is an application of hedgehog dynamics for indifferent irrational fixed points. We present an important simplification by only using a local hedgehog for which we give a simpler and direct construction of quasi-invariant curves which does not rely on complex renormalization.
Dia: Divendres, 1 de març de 2019
Lloc: Aula B2 , Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Ariadna Farrés, University of Maryland Baltimore County, NASA Goddard Space Flight Center.
Títol: Optimal aerobrake maneuver estimation for MAVEN
Resum: The Mars Atmosphere and Volatile EvolutioN (MAVEN) mission was designed to determine the role that loss of volatiles from the Mars atmosphere to space has played through time, giving insight into the history of Mars' atmosphere and climate. The spacecraft has been orbiting Mars for four years in a highly-inclined and highly-elliptic orbit. As part of a proposed extended relay mission, an apogee decrease is required which will be achieved through a combination of small impulsive maneuvers and aerobraking.
Aerobrake maneuver schemes are challenging for many reasons, perhaps the most important from a flight dynamics perspective is the need to accurately account for orbital perturbations. Particularly for MAVEN, higher-order gravitational perturbations from Mars, the gravitational perturbations from third bodies, as well as atmospheric drag must be accounted for in order to accurately predict the evolution of the spacecraft's orbit.
This study presents a method to calculate an optimal aerobrake maneuver scheme that minimizes the propellant consumption. The proposed method uses the spacecraft's state transition matrix, propagated on a high-fidelity model, evaluating the variation of the spacecraft's final position with respect to the velocity variations throughout the trajectory. The use of Poincaré maps allows us to determine, the location for the most efficient maneuver as well as the direction and magnitude of the impulse maneuver needed in order to target the desired orbital parameters. The method has shown to reasonably predict the optimal aerobrake maneuvers (time, direction and magnitude) when compared to an indirect optimization method with all the perturbations. Finally, its software implementation permits orders of magnitude faster calculation allowing it to be used in large grid searches.
Defensa de la tesi doctoral de Marc Jorba titulada Periodic time dependent Hamiltonian systems and applications, dirigida per Àngel Jorba i Ariadna Farrés
Dia: Dimecres, 13 de març de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marcel Guàrdia, Universitat Politècnica de Catalunya
Títol:Growth of Sobolev norms for the cubic nonlinear Schrödinger equation near 1D quasi-periodic solutions
Resum: The study of solutions of Hamiltonian PDEs undergoing growth of Sobolev norms Hs (with s≠ 1) as time evolves has drawn considerable attention in recent years. The importance of growth of Sobolev norms is due to the fact that it implies that the solution transfers energy to higher modes. Consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of invariant quasiperiodic tori. These are inherited from the 1D cubic NLS (on the circle) by considering solutions that depend only on one variable. We show that, under certain assumptions, these tori are transversally unstable in Sobolev spaces Hs (0<s<1). More precisely, we construct solutions of the 2D cubic NLS which start arbitrarily close to such invariant tori in the Hs topology and whose Hs norm can grow by any given factor. This is a joint work with Z. Hani, E. Haus, A. Maspero and M. Procesi.
Dia: Dimecres, 20 de març de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Joaquim Puig, Universitat Politècnica de Catalunya
Títol: Lyapunov exponents for Quasi-Periodic Schrödinger Operators and Cocycles
Resum: We will review some recent developments in the understanding of the upper Lyapunov exponent for quasi-periodic Schrödinger operators and their cocycles. We will be interested in its regularity as a function of the energy and the existence of lower bounds, which produce non uniformly hyperbolic dynamics. For trigonometric potentials with one frequency, we will present a Thouless formula and an approach based on Aubry duality which allows a fairly accurate description of the general mechanisms in this setting.
This talk is part of a joint research project with Alex Haro.
Dia: Dimecres, 27 de març de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Marina Gonchenko, Universitat Politècnica de Catalunya
Títol: On local global aspects of 1:4 resonances of the conservative cubic Hénon map
Resum: We study the 1:4 resonance for the conservative cubic Hénon maps with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues ±i and for 4-periodic orbits. While for the map with negative cubic term the 1:4 resonance unfolding has the so-called Arnold degeneracy (the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient), the map with positive cubic term has a different type of degeneracy because the resonant term can vanish. This new type of degeneracy has not been studied before. In this case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by π/4.
This is a joint work with S. Gonchenko, I. Ovsyannikov and A. Vieiro.
Dia: Dimecres, 3 d'abril de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Joan Carles Tatjer, Universitat de Barcelona
Títol: Dynamics of the QR-flow for real matrices
Resum: The QR-flow can be seen as the continuous analogous of the QR-iteration for computing the eigenvalues of a squared matrix. Using basic ideas of dynamical systems we investigate phase space properties of the QR-flow when restricted to upper Hessenberg matrices. After the analysis of the linear stability of the equilibrium matrices, we shall describe the α and ω-limits sets of orbits of this system. In particular, some of the well-known convergence results of the QR-iteration, properly adapted to the continuous case, are recovered. Several numerical examples will be given to show the different limit behaviour of the orbits and to illustrate the theory. Joint work with A. Vieiro.
Dia: Dimecres, 10 d'abril de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Nikolaos Karaliolios, Imperial College London
Títol: KAM theory for quasi-periodic cocycles in compact Lie groups
Resum: The talk will focus on perturbative results for quasi-periodic cocycles in compact Lie groups. We will define a fibered rotation vector for such systems and present the different possible dynamical phenomena associated with the arithmetics of the rotation vector: foliation of the phase space into smooth, differentiable or measurable KAM tori, weak mixing fiberwise, distributional unique ergodicity.
Dia: Dimecres, 24 d'abril de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Josep Sardanyés, Centre de Recerca Matemàtica (CRM) and Barcelona Graduate School of Mathematics (BGSMath)
Títol: Some intimacies about the saddle-node bifurcation in one and more dimensions
Resum: In this seminar we will discuss properties and dynamic behaviours tied to local bifurcations. We will specifically focus on the saddle-node (S-N) bifurcation of equilibrium points. First, we will start with a three-dimensional model for semi-arid ecosystems dynamics displaying a S-N bifurcation as the soil degradation rate increases. We will discuss the so-called ghost ecosystems, proposing a simple intervention method to avoid catastrophic extinctions for these ecosystems. Then, we will move to investigate properties of ghost phenomena due to S-N bifurcations in low-dimensional, time-continuous dynamical systems. Specifically, we will introduce analytical results on the so-called inverse-root scaling law obtained from dynamics occurring in the complex phase space which affect dynamics in the real numbers phase space. Finally, we will explain how time lags affect ghost transients properties by investigating S-N bifurcations with delay differential equations.
The results explained in this seminar are the result of joint collaborations with: Blai Vidiella, Ricard Solé, Ernest Fontich, J. Tomás Lázaro, Tomás Alarcón, Àngel Jorba, and Joan Gimeno.
Dia: Dimecres, 8 de maig de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Florian Mintert, Imperial College, London
Títol: Quantum simulations with periodically driven quantum systems
Resum: The dynamics of synthetic quantum systems can be modified such that they mimmic the dynamics of many-body systems that we can not simulate with existing classical computers. A common way of controlling such dynamics is via periodic driving, such that Floquet theory allows us to identify an effective, time-independent model.
I will discuss how suitably chosen driving can be used to control tunnelling dynamics and interactions between atoms that are trapped in light fields, and what type of physical processes can be simulated in this fashion.
Dia: Dimecres, 22 de maig de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Alberto Pérez-Cervera, Universitat Politècnica de Catalunya
Títol: On the role of Oscillatory Dynamics in Neural Communication
Resum: Oscillations are an ubiquitous phenomena in biology but also in a wide set of complex systems, ranging from ecosystems to cells. Brain is a particularly interesting example merging oscillations and function. Fast oscillations in the brain are thought to be caused by the interaction of excitatory (E) and inhibitory (I) neurons. The rhythmic alternation of excitation and inhibition generates rhythmic changes in the excitability of the cycle, so the effect of a given input into the cycle shows a phase dependency. Indeed, inputs arriving when excitation is ready to fire (most excitable phase) are expected to generate a noticeable effect whereas inputs arriving during the inhibitory phase may be suppressed. This phase dependence for the effect of an input has been proposed as a feasible mechanism underlying neural communication. To gain insight about this communication channel, we take a rate model describing the mean activity of a E-I network, we set parameters for the model to show oscillations, and then we perturb it to study how the different phase excitability properties of the cycle shape the perturbed dynamics. The different cases that we show generate very rich dynamics (including bistability between different oscillatory states) whose implications in neural communication are discussed. To conclude, this work aims to show a set of techniques whose applicability is not reduced to neuroscience but to any complex system, provided it can be modelled by a reduced set of ODEs.
Dia: Dimecres, 22 de maig de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Alejandro Luque, Matematiska Institutionen, Uppsala Universitet
Títol: Effective bounds for the measure of rotations
Resum: A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). In this talk we discuss the following problem: given a (non-perturbative) family of analytic circle diffeomorphisms, obtain quantitative lower bounds of the measure of the set of parameters that are conjugated to a rigid rotation. This is joint work with Jordi Lluis Figueras and Alex Haro.
Dia: Dimecres, 5 de juny de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Andrew Clarke, Imperial College London
Títol: Arnold Diffusion in Multi-Dimensional Convex Billiards
Resum: Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved in the 70’s that in two dimensions, it is impossible for this angle to tend to zero. Using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, this phenomenon is generic in the real-analytic category.
Dia: Dimecres, 3 de juliol de 2019
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Jonathan Jaquette, Brandeis University
Títol: Validated Approximations of Infinite Dimensional Stable Manifolds in Parabolic PDE
Resum: In this work we present a novel method to represent infinite dimensional stable manifolds in nonlinear parabolic partial differential equations on a large neighborhood of the equilibrium, that is amenable to validated computation. Namely, we obtain a parametrization that is valid on an explicitly described neighborhood with explicit rigorous error control. This goes far beyond using the stable subspace as a linear approximation of the manifold, by providing theorems that guarantee explicit estimates on the accuracy of the computed approximation. Moreover, a feature of the method is that one can use approximations that improve on the tangent subspace in certain direction, namely those along which one expects connecting orbits to approach the equilibrium. The main difficulty is to overcome is the nonuniformity of the decay in the stable manifold. We need to control the interaction of error terms corresponding to starkly different decay rates, which we accomplish by iteratively bootstrapping Gronwall's inequality for systems of inequalities. As an example, we apply this technique to the 1D Swift-Hohenberg equation.
Joint work with Jan Bouwe van den Berg and Jay Mireles James.
Dia: Dimecres, 17 de juliol de 2019
Lloc: Aula 1.3 (Edifici Josep Carner), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Victor Vilaça Da Rocha, Basque Center of Applied Mathematics
Títol: Construction of unstable quasi periodic solutions for a system of coupled NLS equations.
Resum: The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...). From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013).
In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see how this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.
This is a work in collaboration with Benoît Grébert (Université de Nantes).
Last updated: Monday, 22-Jul-2019 11:21:40 CEST