Dia: Dimecres, 7 de setembre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Laurent Stolovich, Université de Nice

Títol: Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Resum: We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n≥3. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Stróżyna-Żoładek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and 𝔰𝔩2(ℂ)-representations

Dia: Dimecres, 21 de setembre de 2016

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Daniel Sánchez-Taltavull, Ottawa Hospital Research Institute

Títol: Bioinformatics modelling vs Mathematical modelling of red blood cell development

Resum: Haematopoietic Stem cells have the ability to produce any cell type of the haematopoietic system. The path the stem cell follows to become a specialized cell type is characterized at the molecular level by the mRNA and the proteins that are expressed during differentiation. We are going to focus on red blood cell diferentiation. The structure of the Gene Regulatory Network (GRN) that determines this process is not known and it is crucial to determine it to deal with genetic diseases.

To determine GRNs biologists and bioinformaticians have used data driven models (in general based in correlations). However, data driven models have a lot of limitations and most of the times can not explain the mechanisim behind the process which hinders obtaining accurate predictions. For this reason, seems natural to use mathematical models (e.g. ODEs or PDEs models) to complement them.

In this talk we explain the different experimental and bioinformatics techniques to study red blood cell development, their limitations and how to incoporate mathematical modelling techniques in the study of red blood cell development.

Dia: Dimecres, 28 de setembre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Alejandro Luque, ICMAT

Títol: Beltrami flows looking like Hamiltonian flows

Resum: Beltrami fields (BFs) have long played a fundamental role in fluid mechanics. BFs satisfy the equation ∇×u = λu and are stationary solutions of the Euler equations.

Motivated by Hénon's numerical simulations, Arnold conjectured that the dynamics of BFs has an arbitrarily complicated topology, with the same complexity as a mechanical system with 2 DOF. It is natural to interpret this conjecture as the existence of invariant tori of complicated topology that enclose chaotic regions. The first condition has been recently established by A. Enciso and D. Peralta-Salas: they proved that for every finite collection of tori embedded in ℝ³, there exist BFs having these tori (an arbitrarily small perturbation of them) as invariant objects with quasi-periodic dynamics.

The goal of this talk is to corroborate Arnold's vision by showing the existence of BFs with invariant tori of arbitrary topology that enclose chaotic regions. Indeed, we construct BFs with positive topological entropy in the interior of each torus. The important factor that accounts for the difficulty of the problem is that BFs are strongly non-generic, and we do not have explicit expressions for them (we need to consider general solutions of a PDE). This is a joint work with A. Enciso and D. Peralta-Salas.

Dia: Dimecres, 5 d'octubre de 2016

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Andreas Knauf, Universitat d'Erlangen-Nürnberg

Títol: From the n-center to the n-body problem

Resum: The n-center problem of celestial mechanics describes the motion of a particle in the force field of n fixed gravitational centers. For n≤2 this is integrable. For n≥3 and positive energy, a rather complete understanding of the dynamics has been achieved, using geodesic motion on surfaces of negative curvature (d=2 dimensions), and also for d=3.

In an ongoing collaboration with J. Fejoz (Paris Dauphine) and R. Montgomery (UC Santa Cruz), we try to derive related results for the n-body problem.

Dia: Dimecres, 19 d'octubre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Bastien Le Bihan, ISAE-SUPAERO, Toulouse, France

Títol: Systematic study of the dynamics about and between the libration points of the Sun-Earth-Moon system

Resum: Libration points are the five equilibrium points of the Circular Restricted Three-Body Problem (CRTBP), which models the motion of a spacecraft in the gravitational field of two massive bodies (e.g. the Earth and the Moon or the Sun and the Earth). Non-keplerian families of orbits exist about these points.

For the last three decades, the dynamics about specific libration points of the Sun-Earth (denoted SELi, i = 1,...,5) and Earth-Moon systems (denoted EMLi) have been increasingly studied and used as the backbone of numerous space missions, both in terms of transfer trajectory and nominal orbit design. Famous successful applications include the ARTEMIS and SMART-1 probes in the Earth-Moon system as well as the SOHO, DSCOVR, and Gaia spacecrafts at SEL1,2.

Besides these practical examples in each system, the dynamics of both problems can be combined to produce efficient transfers in the extended Sun-Earth-Moon (SEM) system. In this approach, the SEM system is seen as two coupled CRTBPs, the Sun-Earth and Earth-Moon systems, with their associated libration points. The invariant manifolds of the orbits about EML2 and SEL1,2 provide dynamical channels that can be suitably combined to produce efficient transfers. Since then, the coupled CRTBPs have been used to design Earth-to-EML2 and SEL1,2-to-EML2 transfers. All these applications uncovered the low-energy network that interconnects the EML1,2 points, the Moon, the SEL1,2 points, and the Earth.

In this framework, we aim to provide a systematic or near-systematic preliminary analysis tool for the motion of a spacecraft in the SEM low-energy network, with a particular focus on connections between SEL1,2 and EML2. The foundation stone of this project is the implementation of a new semi-analytical tool for the computation of invariant manifolds about the libration points in a single coherent SEM four-body problem, seen as a perturbed three-body problem: either the Sun-perturbed Earth-Moon system or the Moon-perturbed Sun-Earth system.

In this talk, we will first present a time-dependent coherent model of the Sun-Earth-Moon system that fits our purpose. Then we will describe how a particular semi-analytical tool, called the parameterization method, can be adapted to the computation of invariants manifolds in such a perturbed model. Then, numerical results will allow us to estimate the limits of the domain of validity of our semi-analytical approximations. Finally, we will briefly present some global applications of such parameterizations for the SEL2-to-EML2 transfers

Dia: Dimecres, 26 d'octubre de 2016

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Josep Sardanyés, Complex Systems Lab, Universitat Pompeu Fabra (Barcelona)

Títol: Abrupt transitions to tumor extinction via a trans-heteroclinic bifurcation

Resum: In this seminar we will analyze an ODEs model describing the dynamics of a heterogeneous population of tumor cell phenotypes competing with healthy cells. By means of Eigen's quasispecies model, we will show the presence of an abrupt transition separating a phase of tumor persistence from a phase of tumor extinction as some relevant biological parameters are tuned. The transition between these phases is governed by a trans-heteroclinic bifurcation. Such a bifurcation, with infinite codimension, involves the interchange of stability between two distant fixed points that have an heteroclinic connection. This stability exchange that occurs once the bifurcation value is surpassed together with the no collision of the two equilibria involves a discontinuous transition. We will comment on the possible novelty of this type of bifurcation in smooth dynamical systems. Finally, we will also discuss the implications of our results in potential targeted cancer therapies.

This work is the result of a collaboration with Carles Simó, Regina Martínez, and Ricard Solé.

Dia: Dimecres, 2 de novembre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Carles Bonet, Universitat Politécnica de Catalunya

Títol: A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

Resum: It is known that there is no canonical way to define piecewise dynamical systems in the switching manifold. The most widely used definitions at the switching manifold are the Filippov and Utkin conventions. Hysteresis and smoothing parameter regularizations are defined and we prove that they give respectively, in the limit when the parameter goes to zero, the Filippov and the Utkin conventions. Then, a unified approach is proposed to treat simultaneously both regularizations by its embedding in a higher dimensional singular perturbed system. Estimations of the order of approximation to both conventions are derived in terms of the parameters involved.

Dia: Dimecres, 23 de novembre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Sergey Gonchenko, Nizhny Novgorod Univ., Russia

Títol: Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps

Resum: We focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has a positive maximal Lyapunov exponents and this property is robust, i.e. it holds for all close systems. We restrict attention to the study of (pseudohyperbolic) attractors that contain only one fixed point. Then we show that three-dimensional orientable maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized Henon maps of form x = y, y = z, z = Bx + Az + Cy + g(y,z), where A,B and C are parameters (B is the Jacobian) and g(0,0) = g'(0,0) =0.

We observe also a peculiarity of chaotic dynamics of such maps with B<0 (nonorientable ones).

Dia: Dimecres, 30 de novembre de 2016

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Chara Pantazi, Universitat Politècnica de Catalunya

Títol: On the non-integrability of some families of planar vector fields.

Resum: We study the integrability/non integrability of planar polynomial vector fields using tecnics of differential Galois Theory. We apply Ziglin-Morales-Ramis approach in order to decide the non-integrability of some families of foliations with rational coefficients coming from planar vector fields. We present an approach which relate the order of the poles of the variational equations with the integrability of the foliation associated to a planar vector field. Kaltofen's algorithm also applied in our study.

This is a work in progress and in colaboration with P. Acosta-Humánez, T. Lázaro and J.J. Morales-Ruiz.

Dia: Dimarts, 20 de desembre de 2016

Lloc: Aula 103, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Jacques Féjoz, Université de Paris Dauphine - Observatoire de Paris

Títol: Symmetry and degeneracy in KAM theory

Resum: In this talk some symmetry and transversality conditions on a vector field will be described, which ensure the preservation of quasi-periodic solutions, with a view to applications to celestial mechanics.

Dia: Dimecres, 21 de desembre de 2016

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 17h00m Café i galetes
• Hora: 17h15m

A càrrec de: Dmitry Treschev, Lomonosov State University in Moscow and Steklov Mathematical Institute of the Russian Academy of Sciences

Títol: Transformations of parallel systems of rays

Resum: I will explain how it is possible to realize a diffeomorphism between two parallel bundles of light rays by a finite system of mirrors.

Dia: Dimecres, 25 de gener de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Pablo Cincotta, Universidad Nacional de La Plata & IALP-CONICET, Argentina

Títol: Characterization of diffusion in N > 2 conservative dynamical systems. Applications to Planetary and Galactic dynamics

Resum: We characterize diffusion in a 2.5 degrees of freedom Hamiltonian system and a 4D simplectic map. We show the the usual approximation of free or normal diffusion fails and therefore, any attempt to derive a reliable diffusion coefficient should take into account phase correlations, the latter being still a completely open problem. The diffusion is clearly inhomogeneous and anisotropic, several dynamical objects in phase space seriously affect its rate, for instance the well known stickiness phenomena. All these results will be illustrated and discussed with several numerical experiments. Anyway, we argue that diffusion experiments would help us to guess about stability/instability within chaotic domains over finite (or physical) timescales.

Then we investigate the dynamical structure of the (1,-3,2) Laplace resonance in the planetary system Gliese 876 by means of diffusion experiments. The results show that there are two main regions in the surroundings of the Laplace resonance: the inner resonant region is characterized by large Lyapunov times and very slow diffusion. This multi-resonant configuration seems to be responsible for its long-term stability. The outer resonant region is dominated by a extremely chaotic dynamics, exhibiting a fast diffusion. Although these results correspond to a specific planetary system, it seems reasonable that the main characteristics of any system representing similar multi-resonant configurations could share all these main features. Another application of diffusion concerns the relevance of chaos for halo stars in the solar neighborhood.

Using a very realistic potential for the DM halo (Aquarius project), we find that chaotic mixing, although non-negligible, is not a significant factor in erasing for instance, local signatures of accretion events at least within a physically meaningful timescale in the Solar Neighborhood.

Dia: Dimecres, 1 de febrer de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Yuri Fedorov, Universitat Politècnica de Catalunya

Títol: A shortcut to the Kovalevskaya curve

Resum: There have already been numerous studies and interpretations of the celebrated separation of variables in the integrable top of S.Kovalevskaya.

In the talk it will be shown how the known Kovalevskaya curve of separation can be obtained, by a simple one-step transformation, from the spectral curve of the corresponding Lax representation found by Reimann and Semonov-Tian-Shanski. The algorithm works for the general constants of motion of the top and is based on W. Barth's description of Prym varieties via pencils of genus 3 curves. It can be used for separation of variables of various generalizations of the Kovalevskaya top. Some examples will be considered.

Dia: Dimecres, 15 de febrer de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Josep Sardanyés, Centre de Recerca Matemàtica (CRM)

Títol: Dynamical systems for cooperation: from viruses to origins of life and beyond

Resum: In this seminar we will discuss mathematical models used to analyze the dynamics of biological cooperation, based on autonomous ordinary differential equations. Cooperation is an intrinsic feature of multitude of biological dynamical systems. For instance, viruses, prebiotic replicators at the origins of life, or complex ecosystems. We will illustrate important dynamical systems displaying cooperation within the context of viral complementarion, hypercycles, and ecological metapopulations. The asymptotic dynamics and the bifurcations for these systems will be commented within the framework of catastrophes and phase transitions.

We will close the seminar discussing potential frameworks to increase further complexity in such systems to evaluate the impact of relevant dynamical phenomena such as time lags, external forcing, or cooperation in antagonistic eco-coevolutionary systems within the framework of the Red Queen dynamics.

Dia: Dimecres, 22 de febrer de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Warwick Tucker, Uppsala University

Títol: In search for the Hénon attractor

Resum: By performing a systematic study of the Hénon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Hénon attractor is a strange attractor, or simply a stable periodic orbit. Using results from our study, we conclude that even if the latter were true, it would be practically impossible to establish this by computing trajectories of the map. This is joint work with Zbigniew Galias, AGH University, Krakow, Poland.

Dia: Dimecres, 1 de març de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Aleksandr Gonchenko, University of Nizhny Novgorod

Títol: Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps

Resum: We focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has positive maximal Lyapunov exponent and this property is robust, i.e., it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized HÃ©non maps. This is a joint work with Sergey Gonchenko.

Dia: Dimecres, 8 de març de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Tere Seara, Universitat Politècnica de Catalunya

Títol: A General Mechanism of Instability in Hamiltonian Systems

Resum: We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the 'outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the 'scattering map'. We find pseudo-orbits of the scattering map that keep moving in some privileged direction. Then we use the recurrence property of the 'inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics.

This method differs, in several crucial aspects, from earlier works. Unlike the well known 'two-dynamics' approach, the method relies heavily on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all.

The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. This is a joint work with M. Gidea and R. de la Llave.

Dia: Dimecres, 15 de març de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Maciej Capinski, AGH University of Science and Technology

Títol: A topological mechanism for diffusion in a priori chaotic dynamical systems, with application to the Neptune-Triton elliptic restricted three body problem.

Resum: We present a topological mechanism of diffusion in a priori chaotic systems. The method leads to a proof of diffusion for an explicit range of perturbation parameters. The assumptions of our theorem can be verified using interval arithmetic numerics, leading to computer assisted proofs. As an example of application we prove diffusion in the Neptune-Triton planar elliptic restricted three body problem.

Dia: Dimecres, 22 de març de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Stefan Fleischer, Friedrich-Alexander-Universität Erlangen-Nürnberg

Títol: Improbability of (non-)collision singularities in the N-body problem with centers

Resum: In the 1970's, D. Saari showed the improbability of collision orbits in the N-body problem of celestial mechanics. When generalizing his results to a family of homogeneous force fields, however, his bounds are not optimal. We introduce techniques from symplectic geometry to approach the problem, allowing us to generalize the results to an even bigger class of two-body interactions.

In this talk, we explain how to use those techniques to show the improbability of collision orbits in the N-body problem including some fixed centers. Time permitting, we also take a look at non-collision singularities in the case of four particles.

Dia: Dimecres, 29 de març de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Marc Jorba, Universitat de Barcelona

Títol: A fractalization route for affine skew-products on the complex plane

Resum: Consider an affine skew product of the complex plane

ω → θ + ω
z → a(θ,μ) z + c,

where θ ∈ 𝕋, z ∈ ℂ, ω is Diophantine, and μ and c are real parameters. In this talk we show that, under suitable conditions, the affine skew product has an invariant curve that undergoes a fractalization process when μ goes to a critical value. The main hypothesis needed is the lack of reducibility of the system. A characterization of reducibility of linear skew-products on the complex plane is provided. We also include a linear and topological classification of these systems.

Join work with: Núria Fagella, Àngel Jorba and Joan Carles Tatjer

Dia: Dimecres, 5 d'abril de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Jianlu Zhang, ETH Zurich

Títol: On Siegel's question and density of collisions in the Restricted Planar Circular Three Body Problem

Resum: K. Siegel asked if there is an open set of initial conditions for a Three Body Problem which has a dense subset of collision solutions, i.e. initial conditions in the phase space leading to a double collision. For the Restricted Circular Planar 3-Body Problem with the mass ratio of the primaries μ, we present an open set such that the collision solutions form a O(μ^(1/25)) dense set. As μ → 0 this set is asymptotically dense. This is a joint work with M. Guardia and V. Kaloshin.

Dia: Dimecres, 19 d'abril de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Alexandre Rodrigues, Universidade do Porto

Títol: On Taken's Last Problem: times averages for heteroclinic attractors

Resum: In this talk, after introducing some technical preliminaries about the topic, I will discuss some properties of a persistent family of smooth ordinary differential equations exhibiting tangencies for a dense subset of parameters.

We use this to find dense subsets of parameter values such that the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33-T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions.

We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. This is a joint work with I. Labouriau (University of Porto).

Dia: Dimecres, 26 d'abril de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Sinisa Slijepcevic, Universitat de Zagreb

Títol: Variational approach to description of instabilities in Hamiltonian dynamics

Resum: We propose a new variational tool for construction of invariant sets of Lagrangian dynamical systems with positive Lyapunov exponents, based on recent advances in the theory of evolutionary PDEs on unbounded domains. Examples include: 1) a measure of instability of a Mather set of a twist map which generalizes both the angle of splitting of separatrices and the flux through a gap in a Cantori; 2) construction of uncountably many positive entropy invariant measures in the Arnold's 2½ degrees of freedom flow (the Arnold diffusion example).

Dia: Dimecres, 3 de maig de 2017

Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Carles Simó, Universitat de Barcelona

Títol: Anomalous diffusion in conservative low dimensional problems

Resum: The classical standard map can be considered as defined on a cylinder 𝕊 × ℝ, say an angle and action. Up to the critical Greene's value of the parameter there exist invariant rotational curves giving bounds on the action. Beyond that value do not exist bounds one can observe diffusion. But there are some ranges of the parameter for which elliptic-hyperbolic bifurcations give rise to small islands known as accelerator modes. The stickiness and the role of Cantor sets produce anomalous diffusion with a rate of increase of the standard deviation much larger that the typical square root of the number of iterates.

Similar phenomena occur in volume preserving maps in 𝕊 × 𝕊 × ℝ (2 angles and 1 action) when 2D invariant rotational tori are no longer existing. The related bifurcation giving rise to generalized islands 3D is the Hopf-saddle-node one.

Massive long term simulations allow to detect these phenomena. Different sources of discrepancy with respect to typical diffusion are identified, the individual roles of them are compared and explained in terms of suitable limit models.

This is a work in collaboration with J. Meiss, N. Miguel and A. Vieiro.

Dia: Dimecres, 10 de maig de 2017

Lloc: Sala de Graus, E.P.S. Edificació, UPC.

• Hora: 16h00m Café
• Hora: 16h15m

A càrrec de: Jacques Féjoz, Université de Paris Dauphine-Observatoire de Paris

Títol: On Linear Point Billiards

Resum: Motivated by the high-energy limit of the N-body problem we construct a non-deterministic billiard process, whose table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed, piecewise linear curve with vertices on the subspaces and changes of directions upon hitting a subspace constrained by `conservation of momentum' (mirror reflection). The itinerary of a trajectory is the sequence of subspaces it hits. We will describe a fundamental theorem of Burago-Ferleger-Kononenko on the finiteness of itineraries, and a description of the space of trajectories having a fixed itinerary, which has emerged recently in a joint work with Andreas Knauf and Richard Montgomery, as a Lagrangian relation on the space of lines in the Euclidean space. Our method combines those of BFK in non-smooth metric geometry, with generating families.

Aquesta xerrada serà part de la inauguració del Lab de Geometria i Sistemes Dinàmics. El programa complet de la inauguració és:

• 12.30 Francisco Presas (ICMAT-CSIC), Symplectic topology and Hamiltonian dynamics: a love story.

Historically, symplectic geometry appeared as a geometric setup for Hamiltonian dynamics. The Hamiltonian side influenced the geometric aspects at the beginning of the story. In the eighties, however, the discovery of rigidity phenomena in Symplectic Geometry turned around the table. We want to describe in this talk the conjectures, nurtured by Symplectic rigidity, that have grown up in the last thirty years. Most of them have to do with the existence/non-existence of periodic orbits in general Hamiltonian systems. Finally, we would address some more recent results addressing the existence of minimal flows in Hamiltonian dynamics.

• 13:30 - 15:00 Welcome Speeches and Cava
• 16:00: Jacques Féjoz, Université de Paris Dauphine-Observatoire de Paris. On Linear Point Billiards
• 17:30: Closing
• Web de la inauguració: https://dynamicalsystems.upc.edu/en/events/inauguration-of-the-b-lab

Dia: Dimecres, 17 de maig de 2017

Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Inmaculada Baldomà, Universitat Politècnica de Catalunya

Títol: Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

Resum: We present some results related with parabolic stable manifolds for analytic maps. We put special interest to the Gevrey character of these manifolds in the one dimensional case. We also provide examples which assure the optimality of the results when general maps are considered.

This is a joint work with Ernest Fontich and Pau Martin

Dia: Dimecres, 24 de maig de 2017

Lloc: Aula 002, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Mercè Ollé, Universitat Politècnica de Catalunya

Títol: To and from motion for the hydrogen atom in a circularly polarized microwave field

Resum: We consider the problem of the hydrogen atom interacting with a circularly polarized microwave field. We are particularly interested in the so called to and fro motion, that is, the erratic trajectories described by the electron making several far/close excursions from/to the nucleus. The skeleton of such trajectories is based on the so called ejection-collision orbits (ECO) which are computed and analysed

Dia: Dimecres, 31 de maig de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Jeroen Lamb, Imperial College, London

Títol: Bifurcation in random dynamical systems

Resum: Despite the obvious relevance to applications, a bifurcation theory for random dynamical systems is still only in it's infancy. I will discuss some progress on the understanding of the dynamics of a random dynamical systems near a (deterministic) pitchfork and Hopf bifurcations which illustrate recent advances and remaining challenges.

Dia: Dimecres, 7 de juny de 2017

Lloc: Aula S02, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: S.V.Gonchenko, Lobachevsky University of Nizhny Novgorod, Russia

Títol: On three types of dynamics, and the notion of attractor

Resum: We propose a theoretical framework for an explanation of the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes which are observed in dynamical systems with attractors as defined in a work by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior - conservative and dissipative, while the attractors of the third type, the reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core.

In colloboration with D.Turaev (Imperial College, London)

Dia: Dimecres, 21 de juny de 2017

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Renato Calleja, Departamento de Matemáticas y Mecánica, IIMAS-UNAM

Títol: KAM tori in mechanical systems with friction: the limit of small friction

Resum: Many problems in Physics are described by conformally symplectic systems (e.g. mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems.

I will present a study of the limit of small dissipation in conformally symplectic systems. Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. I will present a conjecture about the geometry of the domains and numerical computations that support it. The numerical computations are based on the parameterization method for conformally symplectic systems. This is joint work with Alessandra Celletti, Rafael de la Llave, and Adrián P. Bustamante.

Dia: Dimecres, 12 de juliol de 2017

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

• Hora: 16h00m Café i galetes
• Hora: 16h15m

A càrrec de: Yutaka Ishii, Kyushu University, Japan

Títol: Boundary of the hyperbolic horseshoe locus for the Hénon family

Resum: The purpose of this talk is to investigate geometric and topological properties of the boundary of the parameter locus for the Hénon family where the maps become hyperbolic horseshoes. Our main result states that the boundary of the hyperbolic horseshoe locus forms a piecewise real analytic curve in the parameter space. As a consequence of this result, we show that the hyperbolic horseshoe locus is connected and simply connected, which indicates, in some sense, a weak form of monotonicity. As another consequence, we give a variational characterization of equilibrium measures "at temperature zero" for maps at the boundary parameters. The proofs of these results are based on the the complexification of both the dynamical and the parameter spaces of the Héenon family and employ complex dynamics and complex geometry together with computer assistance. Joint with Zin Arai (Chubu) and Hiroki Takahasi (Keio).

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Last updated: Sun Apr 14 09:38:41 2024