Dia: Dimecres, 16 de setembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Yuri Fedorov, Universitat Politècnica de Catalunya
Títol: Somos recurrences as integrable maps and their analytic solutions
Resum: As was observed, amongst others, by M. Somos, various bilinear recurrence relations generate sequences of integer numbers.
Such (Somos) sequences generalize the Fibonacci sequences and have appeared in number theory, statistical mechanics, as well as arising from reductions of bilinear PDE in the theory of discrete integrable systems.
In this talk we will show that the recurrences of orders 4,5,6 are integrable maps and that their invariant manifolds are Abelian varieties. Some properties of their solutions will be described.
Dia: Dimecres, 23 de setembre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Àlex Haro, Universitat de Barcelona
Títol: Singularity theory for nontwist tori: from rigorous results to computations
Resum: We present a method to find degenerate KAM tori in families of symplectic maps, i.e. KAM tori for which the Birkhoff normal form is singular. The method provides a function, the potential, whose critical points correspond to invariant tori, leading to a natural classification of KAM tori which is based on Singularity Theory, and providing a procedure to study bifurcations of KAM tori. The construction of the potential involves techniques of KAM theory and Symplectic Geometry. The method also leads to effective algorithms of computation, and we present some numerical results up to the verge of breakdown.
This is a joint work with Alejandra Gonzalez and Rafael de la Llave.
Dia: Dimecres, 30 de setembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marco Antonio Teixeira, Universidade Stadual de Campinas
Títol: Some directions in nonsmooth dynamical systems theory
Resum: In this talk we adress some qualitative and geometric aspects of non-smooth dynamical systems theory. The birth of typical limit cycles is focused for 2D-Systems.
Dia: Dimecres, 7 d'octubre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Alberto Maspero, Università di Roma La Sapienza
Títol: Freezing of energy of a soliton in an external potential
Resum: We study the dynamics of a soliton in the generalized NLS with a small external potential εV of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order ε -r. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order ε -r. This is a joint work with D. Bambusi.
A càrrec de: Emanuele Haus, Università di Napoli
Títol: Exact controllability for quasi-linear perturbations of KdV
Resum: We prove exact controllability for a class of quasi-linear perturbations of the Korteweg-de-Vries equation. The proof is based on a Nash-Moser scheme, in its version "à la Hörmander". The main ingredient needed is exact controllability for the linearized operator in a neighborhood of zero. We prove this by Hilbert uniqueness method, i.e. we deduce the controllability of the linearized operator by its observability, which in turn is proved via Ingham inequality. To this end, we use the techniques developed by Baldi-Berti-Montalto, allowing us to reduce the linearized operator to constant coefficients, up to a bounded remainder. This is a joint work with P. Baldi and G. Floridia.
Dia: Dimecres, 14 d'octubre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Albert Granados, Danmarks Tekniske Universitet
Títol: The period adding and incrementing bifurcations in piecewise-smooth maps and application
Resum: In this talk we study the bifurcation scenario of a piecewise-smooth (discontinuous) one-dimensional map near a co-dimension two bifurcation point. Such bifurcation occurs when two fixed points simultaneously collide with the boundary (the discontinuity of the map). Depending on the sign of the slopes of the map near the boundary, one may observe in the parameter space an infinite number of border collision bifurcation curves involving an infinite number of periodic orbits. When the slopes have different sign, these periodic orbits are organized by the so-called period incrementing scenario. When both are positive, the so-called period adding occurs. We focus on proving sufficient conditions for the latter case. The map can be seen as an orientation preserving (discontinuous) circle map whose lifts undergo positive gaps at the discontinuities. We will show that, with little modifications, classical theory for circle maps also holds for such discontinuous case (the rotation number is unique and well defined, existence and uniqueness of twist periodic orbits, devil's staircases, etc). We also show that symbolic sequences of periodic orbits are given by concatenation of the ones of the Farey parents.
Finally, we will provide an applied example consisting of a first order control system based on relays.
This is a joint work with Lluís Alsedà and Maciej Krupa.
Dia: Dimecres, 21 d'octubre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Daniel Pérez, Universitat de Barcelona
Títol: Lagrangian Coherent Structures and other dynamical indicators
Resum: In this talk we will compare the Lagrangian Coherent Structures (LCS) with the invariant manifolds of different known systems. The LCS were defined by G. Haller at the beginning of the 2000 to study the dynamical structure of non-autonomous dynamical systems. At a first glance, it seems that LCS are equivalent structures to the invariant manifolds of autonomous systems. The idea of the LCS is to detect when neighbouring points have different behaviours. This is done computing the Finite Time Lyapunov Exponents (FTLE) at each point in a grid, which makes the procedure slow. In order to speed up the LCS detection a division/selection algorithm has been implemented.
The second part of the talk will be devoted to the detection of indicators using the Jet Transport. The Jet Transport is a numerical tool that allows the propagation of neighbourhoods instead of points. New indicators are developed inspired in the idea of detecting different behaviours of neighbouring points exploiting the benefits of a Jet Transport integration.
To end the talk an application of the different algorithms developed will be given to the detection of practical stability regions on the Circular Restricted Three Body Problem.
This is a joint work with G. Gómez and J.J. Masdemont
Dia: Dimecres, 28 d'octubre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Universitat de Barcelona
Títol: Experiments looking for theoretical predictions
Resum: Some models have global properties, discovered by means of massive simulations, which occur for parameter values which seem hard to predict. We consider three simple cases: a) Non-integrable 2D symplectic maps with separatrix splitting such that the related standard-like maps contain a periodic function far from a sinusoidal-like shape; b) standard-like maps driven in a quasiperiodic way. In both cases one wants to study up to which values of the parameters there exist invariant curves/tori implying the existence of confined dynamics. The case c) deals with 2D dissipative systems and looks for the existence of "thick" attractors, with dimension close to 2.
Dia: Dimecres, 4 de novembre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Rubén Berengue, Universitat de Barcelona
Títol: The parametrization method for invariant manifolds of tori in skew-product systems with spatial decay in lattices
Resum: In this talk we will introduce a class of systems which are skew products of a lattice ℓ∞ (ℝⁿ) and a torus. The type of systems considered are perturbations of systems which are infinite copies of maps ƒ:ℝⁿ→ℝⁿ having a hyperbolic fixed point.
We will determine the torus of the perturbed system using the parametrisation method as well as find its regularity. Assuming certain spatial decay properties for the perturbation we will determine decay properties for the torus. We will also use the parametrisation method to find non-resonant invariant manifolds for it. We will also discuss Sternberg theorems in lattices for systems with spatial decay. To deal with these two problems we will introduce the notion of decay spectrum for linear maps.
Dia: Dimecres, 11 de novembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alberto Enciso, ICMAT
Títol: Toros invariantes anudados en fluidos en equilibrio
Resum: En esta charla discutiremos la demostración de una conjetura debida a Lord Kelvin sobre la existencia de soluciones estacionarias de las ecuaciones de Euler en dimensión 3 cuyo campo de vorticidad presenta toros invariantes anudados de cualquier topología. La charla está basada en trabajo conjunto con Daniel Peralta Salas.
Dia: Dimecres, 18 de novembre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Piotr Zgliczynski, Jagiellonian University, Krakow, Poland
Títol: Melnikov-type method for splitting of separatrices for an explicit range of small parameter
Resum: One of the main tools to determine the existence of (or non-existence of) chaos in a perturbed hamiltonian system is the Melnikov theory. In this theory, the distance between the stable and unstable manifolds of the perturbed system is calculated up to the first order term, hence the precise range of small parameter for which the transversal intersection exists is unknown.
We propose the method which allows to compute an explicit range of small parameter for which the intersection exists, with the hope that to obtain the size of the parameter from which the continuation with other direct geometric tools is possible. The method is computer assisted (a computer assisted proof). We applied it to the system
To compute the Melnikov distance our method combines two ingredients, both computer assisted
This is a joint work with Maciej Capinski.
A càrrec de: Jonathan Jaquette, Rutgers, The State University Of New Jersey
Títol: Computing epsilon-Approximations of Persistent Homology
Resum: Physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this talk, I will describe a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology using interval arithmetic. We study the filtration generated by sub-level sets of a function f:X ⟶ℝ, where X is a CW-complex. In the special case where X is a hypercube, I will discuss implementation of the proposed algorithms as well as a priori and a posteriori bounds of the approximation error introduced by our method.
Dia: Dimecres, 25 de novembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Juan José Morales, Universidad Politécnica de Madrid
Títol: Teoría de Galois Differencial y Simetrías de Lie
Resum: La motivación de Sophus Lie para formular una teoría de simetrías de las leyes naturales fue intentar encontrar para las ecuaciones diferenciales una teoría análoga a la teoría de Galois de las ecuaciones algebraicas. Hoy en día sabemos que la teoría de simetrías de Lie no es una teoría de Galois de ecuaciones diferenciales. La auténtica teoría de Galois de ecuaciones diferenciales se inicia a finales del siglo XIX, simultáneamente con la teoría de Lie, y puede ser considerada como algún tipo de teoría de simetrías "dual" de las simetrías de Lie. En los últimos cuarenta años, la teoría de Galois de ecuaciones diferenciales (y en diferencias) ha experimentado un renacimiento muy importante, con algoritmos de computación simbólica y aplicaciones a la física, ingeniería y matemáticas puras. En esta charla intentaremos clarificar la relación entre las simetrías de Lie y simetrías de Galois de las ecuaciones diferenciales lineales (trabajo conjunto con David Blázquez-Sanz y Jacques-Arthur Weil).
Dia: Dimecres, 2 de desembre de 2015
Lloc: Aula S01, FME, UPC.
A càrrec de: Zubin Olikara, IEEC & Universitat de Barcelona
Títol: Some astrodynamics problems and methods
Resum: In this talk we give an overview of topics worked on over the last few years. The work is motivated by the study of spacecraft motion, particularly near the collinear libration points.
We present some aspects of the implementation of the parameterization method for flows that could be useful for applications. In particular, we investigate the use of real (rather than complex) coordinates for parameterizing the center manifold. We also consider methods for integrating the reduced vector field on the center manifold. We mention the connection between Birkhoff normal form and the Poincaré-Lindstedt method.
In addition, we discuss various aspects of motion in the Sun-Earth-Moon system. We briefly present the safe disposal of spacecraft originating near a Sun-Earth libration point. We also look at the structure of the planar center manifold about the L1 and L2 points in a restricted four-body problem and connecting orbits between them.
Joint work with Gerard Gómez and Josep Masdemont.
Dia: Dimecres, 9 de desembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Juliana Fernandes Larrosa, Universidade Federal de Santa Maria
Títol: Regularization of a Filippov system near a fold-fold singularity
Resum: Consider a Planar Filippov System Z=(X,Y), where X and Y are sufficiently smooth vector fields defined in a neighborhood of the origin and whose discontinuity curve is given by the set of zeros of f(x,y)=y. In this work, we consider the Teixeira-Sotomayor regularization of Planar Filippov Systems having a fold-fold singularity and whose regularization has a critical point around the origin. In this context, we study the nature of its critical point as well the relations between the bifurcation diagram of the Planar Filippov System and the regularized system, provided a bifurcation occurs. We also use asymptotic methods and matching in order to prove the existence of a canard solution when the folds have opposite visibility.
Dia: Dimecres, 16 de desembre de 2015
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
Opening
A càrrec de: Carles Simó, Universitat de Barcelona
Títol: The invariant manifolds at infinity of the RTBP and the boundaries of bounded motion
A càrrec de: Rafael de Llave, Georgia Institute of Technology
Títol: Quasi-periodic solutions for state dependent delay equations
Coffee break
A càrrec de: Patrick Bonckaert, Hasselt University
Títol: Resonant planar saddle points
A càrrec de: Vassili Gelfreich, University of Warwick
Títol: Stokes Phenomenon, Singularly Perturbed Differential Equations and Bifurcations
A càrrec de: Xavier Cabré, Universitat Politècnica de Catalunya
Títol: Curves and surfaces with constant nonlocal mean curvature
A càrrec de: Yannick Sire, Université Aix-Marseille
Títol: KAM theory for ill-posed PDEs
Coffee break
A càrrec de: Amadeu Delshams, Universitat Politècnica de Catalunya
Títol: Global instability in the periodic cubic defocusing NLS equation using non-transverse heteroclinic chains
Dia: Dimecres, 13 de gener de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Rafael Ramírez Ros, Universitat Politècnica de Catalunya
Títol: Flujo de curvatura y billares
Resum: La charla tendrá dos partes, una primera muy bonita sobre contribuciones de otra gente y una segunda sobre nuestra modesta contribución. Esta segunda parte es una trabajo conjunto con Josué Damasceno (UFOP) y Mario Jorge Dias Carneiro (UFMG). Ver http://arxiv.org/abs/1511.01355i
Parte 1.- Describir las propiedades más elementales del flujo de curvatura, siguiendo los artículos clásicos de M. Gage, R. Hamilton y M. Grayson. Se puede acortar una curva plana moviéndola en la dirección de su vector normal con una velocidad proporcional a su curvatura. Esta evolución genera un flujo en el espacio de curvas suaves planas que coincide con el flujo por el gradiente negativo del operador longitud. Es decir, la curva se contrae de la forma más rápida posible usando tan sólo información local.
La longitud, el área encerrada por, la curvatura máxima, el número de puntos de inflexión y muchas otras cantidades geométricas nunca se incrementan a lo largo del flujo de curvatura. Esto es debido a que las "trayectorias" de este flujo satisfacen una EDP parabólica.
Parte 2.- Presentar, por contra, un ejemplo donde el flujo de curvatura incrementa la entropía topológica del billar. Concretamente, ver que el billar dentro de una elipse levemente deformada por el flujo de curvatura tiene entropía topológica positiva, pues rompe las separatrices y todas las cáusticas convexas del billar en una elipse. La estrategia consiste en comprobar, a partir del estudio de ciertas singularidades en el plano complejo, que los potenciales de Melnikov (tanto homoclínicos como subharmónicos) asociados a este problema no son constantes. Este estudio estuvo fuertemente inspirado por un ejemplo de Dan Jane [ETDS, vol. 27, pag. 1919-1932, 2007] donde el flujo de Ricci incrementa la entropía topológica del flujo geodésico en una superficie de Riemann.
Dia: Dimecres, 20 de gener de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Maria Aguareles, Universitat de Girona
Títol: On the asymptotic wavenumber of some λ-ω systems
Resum: We consider a general class of λ-ω systems which exhibit rigidly rotating spiral wave solutions. When studying these spiral wave solutions, the original system of partial differential equations becomes a singular system of ordinary differential equations with an unknown parameter that plays the role of a wavenumber at infinity. On the other hand, the functions λ and ω depend on a small parameter q, which, along with a set of prescribed boundary conditions, determines uniquely the value of the asymptotic wavenumber. In this talk we will show that the asymptotic wavenumber is indeed a C∞-flat function of the perturbation parameter q. This is a joint work with T. M-Seara and I.Baldomà
Dia: Dimecres, 3 de febrer de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marcel Guàrdia, Universitat Politècnica de Catalunya
Títol: Secular instability in the three-body problem
Resum: Consider the spatial three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential. This is a joint work with Jacques Fejoz.
Dia: Dimecres, 10 de febrer de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Tomás Alarcón, ICREA Research Professor, Centre de Recerca Matemàtica
Títol: Optimal path theory of stochastic processes and its applications to robustness of gene regulatory circuits
Resum: In this talk, I will introduce an asymptotic method to study Markov stochastic processes known as optimal path theory. This method, which is closely related to the semi-classical approximation in quantum mechanics and the WKB method, provides a variational description of the optimal path connecting an initial and a final state of the stochastic process. This optimal path is the solution of the Euler-Lagrange equations for the maximisation of an action functional associated with an "effective Hamiltonian". I will further explore the applications of this methodology to a particular biological problem, namely, robustness to fluctuations of steady state of gene regulatory networks.
Dia: Dimecres, 17 de febrer de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Esther Barrabés, Universitat de Girona
Títol: On the dynamics of the parabolic restricted three-body problem
Resum: We consider the motion of an infinitesimal mass under the gravitational influence of two equal masses (primaries) moving in two parabolic orbits, all of them in the same plane (planar parabolic problem). The flow of the system is described in terms of the final evolution of the solutions, forward and backward in time. The two main roles in the dynamics of the problem are the Hill's regions (which are non constant because the problem is like-gradient), and the invariant manifolds associated to the equilibrium points.
This model can be used to understand, at a basic level, the effect of a close encounter of two galaxies. Such a close encounter may cause a significant modification in the mass distribution. Taking into account just one particle within one galaxy, after the close encounter, the particle may jump to the other galaxy or escape. We study in the frame of the planar parabolic problem, the mechanisms that allow to explain that a particle escapes or its transfer from the neighborhood of one primary to the other (capture).
Dia: Dimecres, 24 de febrer de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Regina Martínez, Universitat Autònoma de Barcelona
Títol: "Schubart-like" periodic orbits in some n-body problems
Resum: We give sufficient conditions to ensure the existence of symmetrical periodic orbits for a class of Hamiltonian systems having some singularities. The results are applied to different subproblems of the gravitational n-body problem where singularities appear due to collisions. Also different families of these kind of orbits are shown.
Dia: Dimecres, 2 de març de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alejandro Luque, Instituto de Ciencias Matemáticas
Títol: Aplicación asistida por ordenador del teorema KAM
Resum: En esta charla afrontaremos varios problemas que aparecen al aplicar el teorema KAM en problemas concretos. Entre las cuestiones a tratar, plantearemos los siguientes problemas:
Este es un trabajo conjunto con Jordi-Lluís Figueras y Àlex Haro.
Dia: Dimecres, 9 de març de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: John Hogan, University of Bristol
Títol: Singularities in planar piecewise smooth systems
Resum: The study of piecewise smooth (PWS) systems has received a lot of attention recently, including the current research seminar at the CRM. This resurgence of activity has led to a renewed interest in the work of Filippov. In this talk, I will give a (brief) overview of PWS systems, followed by a longer look at the classification of isolated singularities in planar PWS systems. The aim will be to encourage more people to read Filippov and to present for the first time the complete unfolding of boundary equilibrium collisions. I shall also briefly outline recent work on the classification of Filippov's type 3 singularities and consider the challenges of extending this approach to higher dimensions.
This is joint work with Paul Glendinning (Manchester) and Martin Homer, Mike Jeffrey and Robert Szalai (Bristol)
Dia: Dimecres, 16 de març de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: David Chillingworth, University of Southampton
Títol: Geometry of grazing and chatter for a simple impact oscillator
Resum: A simple impact oscillator here means a second order ODE with one degree of freedom x and with T-periodic forcing, and such that whenever the value x = c (the clearance) is attained the velocity x' is replaced by -rx' with 0 < r < 1. A grazing orbit has x' = 0 when x = c, while a chattering orbit has infinitely many impacts with x=c in finite time. We give insight into these and other phenomena using a general framework that exploits tools from singularity theory in order to convert analysis into geometry.
Dia: Dimecres, 30 de març de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Laura Garcia, Universitat de Girona
Títol: Transfers from LEOs to GEOs visiting libration points of the Sun-Earth CRTBP
Resum: The objective of this work is to explore the use the invariant manifold dynamics of the Circular Restricted Three Body Problem to construct transfer trajectories from a Low Earth Orbit (LEO) to a Geostationary Earth Orbit (GEO). The underlying idea is to determine orbits that shadow the stable and unstable manifolds of the central manifold of the collinear libration points of the Sun-Earth system, and connect both kinds of orbits around the Earth. The resulting transfer orbits have two legs connected at the libration point region. After a first maneuver performed in the LEO, the spacecraft reaches the neighborhood of the equilibrium point, L1 or L2, driven by the stable manifold of the central manifold of the point. With a small impulse, the spacecraft is transferred back to a GEO shadowing an orbit of the unstable manifold of a libration point orbit. Once the GEO is reached, an insertion manoeuvre must be done.
Joint work with Esther Barrabés and Gerard Gómez.
Dia: Dimecres, 6 d'abril de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marcel Guàrdia, Universitat Politècnica de Catalunya
Títol: Growth of Sobolev norms for the defocusing analytic non-linear Schrodinger equation
Resum: Consider the completely resonant defocusing non-linear Schrodinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s>1. We show the existence of solutions of this equation which achieve arbitrarily large growth of Hs Sobolev norms. We also give estimates for the time required to attain this growth. This is a joint work with Emanuele Haus and Michela Procesi.
Dia: Dimecres, 13 d'abril de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Tere Seara, Universitat Politècnica de Catalunya
Títol: Regularization of the fold-fold singularity of Filippov systems
Resum: This will be an informal presentation about the study of the fold-fold singularity in Filippov Systems, its unfoldings and the dynamics of its regularization. We will show some unexpected bifurcations of periodic orbits arising in the regularized system which are not present in the original unfolding of the Filippov system. This is a joint work with C. Bonet and J. F. Larrosa
Dia: Dimecres, 20 d'abril de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alan Champneys, University of Bristol
Títol: Bistability and the theory of life, rainforests and (almost) everything
Resum: This talk describes work with current PhD students Bert Wuyts and Nicolas Verschuren. Reaction-diffusion models have been proposed since the work of Alan Turing as a model for spatial patterning. These the combined effects of natural phenomena that want to diffuse in space at the same time as undergoing local-level interactions. This talk considers two such problems for which the local interaction naturally produces bistability between a low-intensity and a high-intensity state. The first problem, with Bert, concerns data on rainforest/savana bistability. Here there have been recent studies in high profile journals that suggest a possible climate-induced tipping point leading to desertification of the rainforest. By looking at additional data on closeness to human impact, we show that the data in pristine regions instead displays spatially separated regions of rainforest and savanna, controlled by average rainfall levels. We using reaction-diffusion models that local bistability of tree density together with fire diffusion predicts sharp fronts (Maxwell points) that explain these data. The second problem, with Nicolas, concerns how cells develop polarity, the first step to them forming inhomogeneous structures that go to make up the rich diversity of cell types seen on earth. A fundamental mechanism involving small G-proteins in active and inactive forms is revisited and shown to lead to subcritical Turing patterns. The subcriticality leads to bistability between patterned and nonpatterned states which results in either sharp fronts or localised structures. A mixture of Hamiltonian normal-form and asymptotic methods can explain these results.
Dia: Dimecres, 27 d'abril de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Elena Bossolini, Technical University of Denmark
Títol: Slow-fast analysis of earthquake faulting
Resum: We consider the Burridge-Knopoff rate and state dependent friction law to study earthquake fault dynamics. We aim to give an analytical motivation for the periodicity of the earthquake episodes. We show that the system has an embedded slow-fast structure with time scale parameter ε. At the singular value ε=0 we find an attracting critical manifold on which there is a degenerate Hopf bifurcation, since the system has an underlying Hamiltonian structure. The Hamiltonian system has a special orbit separating bounded orbits from unbounded ones. A compact subset of the bounded orbits can be continued by Fenichel's theory and a Melnikov-type method into a family of attracting limit cycles for ε≪1. Numerical simulations for ε≪1 however, show that the periodic behaviour persists even outside the region of validity of the Melnikov result. To cover this part, we perform a compactification of the phase space and show that we can close the unbounded orbits at infinity. In particular we find a return mechanism which explains the formation of periodic orbits at infinity around the bifurcation value. Using Fenichel's theory and the blow-up method we argue on the persistence of the results in the non-singular case ε>0.
Dia: Dimecres, 4 de maig de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Àngel Jorba, Universitat de Barcelona
Títol: Some dynamical aspects of the augmented Hill problem
Resum: The augmented Hill problem is a 3DoF Hamiltonian model for the motion of a Solar sail near an asteroid, which is written as the classical Hill Hamiltonian plus an extra term that accounts for the effect of the Solar radiation pressure. In the talk we will describe the dynamics of this model focusing on the dynamics near the L2 point, which is always of the type saddle x centre x centre. It turns out that, for some values of the parameters, the two centres are in a 1:1 resonance. The main part of the presentation will be devoted to the discussion of the (rich) dynamics that appears near this resonance.
This is join work with Ariadna Farrés and Josep Maria Mondelo.
Dia: Dimecres, 11 de maig de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Narcís Miguel i Baños, IMUB
Títol: Escape rates through a Cantorus in Area Preserving Maps
Resum: The breakdown of an invariant curve of an APM can drastically change its dynamics, since it is no longer a barrier to transport. According to Aubry-Mather theory, after this breakdown, a remnant of the invariant curve persists: a Cantor set with the same rotation number as the just broken curve. Despite the holes in the Cantor set, created by manifolds of nearby periodic orbits, allow orbits to travel across it, the time to do so can be extremely large.
In this talk we will present the results of an extensive numerical evaluation of escape rates in the case of the Chirikov Standard Map, right after the destruction of the golden number rotational invariant curve. We will link them to the Greene-MacKay renormalisation theory and the MacKay-Meiss-Percival transport theory. Together, they allow to interpret our results by identifying the objects in the phase space that play a leading role in the strong increase of the time to cross the Cantorus. We will pay special attention to the effect of the tiny islands of stability on the vicinity of the Cantor set.
This is a joint work with Carles Simó and Arturo Vieiro.
Dia: Dimecres, 18 de maig de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rodrigo Schaefer, Universitat Politécnica de Catalunya
Títol: Global instability in Hamiltonian Systems and several Scattering maps
Resum: In this work we illustrate the Arnold diffusion in a concrete example - the a priori unstable Hamiltonian system of 2+½ degrees of freedom H(p,q,I,φ,s) = p²/2+cos q -1 +I²/2 + h(q,φ,s;ε) - proving that for any small periodic perturbation of the form h(q,φ,s;ε) = εcos q (a₀₀ + a₁₀cosφ + a₀₁cos s), (a₁₀a₀₁ ≠ 0 and ε≠0 small enough) there is global instability for the action, i.e., I(0)≤ -Iε <Iε≤ I(T) for some T and for any positive Iε≤ C log 1/ε; for some constant C. For this, we apply a geometrical mechanism based in the so-called Scattering map.
This work has the following structure: In a first stage, for a more restricted case we use only one Scattering map. Later, in the general case we combine a Scattering map and the inner map (inner dynamics) to prove the main result (the existence of the instability for any μ) and, we show some examples of "ways of diffusion" if we consider multiple combination of several scattering maps. Finally, we give an estimate for the "time of diffusion" for a special case.Dia: Dimecres, 25 de maig de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Alfonso Sorrentino, Università degli Studi di Roma "Tor Vergata"
Títol: Aubry-Mather theory for conformally symplectic systems
Resum: In this talk I shall present an analog of Aubry-Mather theory for a class of dissipative systems, e.g. conformally symplectic systems.
Dia: Dimecres, 1 de juny de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rodrigo Treviño, Brooklyn College, City University of New York
Títol: Quasicrystals, ergodic theory and cohomology
Resum: I will talk about some recent results on deviation of ergodic averages for systems coming from aperiodic tilings and aperiodic point sets which are self affine (the Penrose tiling is an example of these). These rely on fun interactions between ergodic theory and some cohomology theories associated with aperiodic point sets. Time permiting I will discuss applications to problems of diffraction, counting problems, and future directions. This is joint work with S. Schmieding.
Dia: Dimecres, 8 de juny de 2016
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Gabriella Pinzari, Università di Napoli
Títol: An integral for the outer perturbing function in the three-body problem.
Resum: Poincaré proved that the n-body problem has no other integrals than the energy, the linear momentum and the angular momentum.
This strong assertion implies that when the number of bodies is three or more, the problem is not integrable. And in fact most of the ongoing research is devoted to look for orbits exhibiting both stability and instability features. In the case of the planetary problem, namely, when one of the masses is much larger than the others, it makes sense to look at the so-called averaged problem. Again, this problem is not integrable, even though this does not follow from Poincaré's assertion, but from a recent work by Féjoz and Guardia.
The goal of my talk is to prove that a different ("outer") average of the perturbing function exhibits an integral which is indeed responsible of certain symmetries in the Hamiltonian, known in the literature. Other applications are foreseen and, if there is time, I shall talk about them.
Dia: Dimecres, 15 de juny de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Narcís Miguel i Baños, IMUB.
Títol: Stickiness in 2D and 3D conservative systems
Resum: Typically, conservative systems are neither integrable nor fully chaotic, but exhibit a mixed phase space. In this context, the dynamics of chaotic orbits near the boundary of regular zones is still not completely understood, mainly due to the rich geometry of the phase space around these regular components.
Chaotic orbits can spend a long time near regular zones, giving rise to the 'stickiness' phenomenon: the lengths of stays around regular components behaves as a negative power law. Note that the probability distribution of these lengths of stays may have some unbounded moments.
In this talk we will deal with this problem in two contexts. First, we will consider the stickiness effect of an island of stability in area preserving maps (APM). In our example, the dynamics in a compact set containing the island is close to the Hénon map. This is due to a bifurcation that gives rise to elliptic-hyperbolic periodic orbits and to the behaviour around islands which have been created. Second, the stickiness effect of a 'stability bubble' in a family of 3D volume preserving maps (VPM) will be addressed. In this case the dynamics of the bubble under study will be close to a discretization of the Michelson system of ODE. This is due to a Hopf-zero bifurcation that gives rise to a couple of saddle points with near coincident 2D manifolds close to a 2D sphere.
In both examples, the structures under study are embedded in the phase space of seemingly fully chaotic APM of the 2-torus and of VPM of the 3-torus, respectively. In some cases, in lifts of these maps to cylinders, the presence of regular components can give rise to anomalous diffusive properties.
This is a joint work with J.D. Meiss, C. Simó and A. Vieiro.
Dia: Dimecres, 22 de juny de 2016
Lloc: Aula Capella (Planta baixa, Pati de Lletres), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Departament de Matemàtiques i Informàtica, UB
Títol: Some questions looking for solutions in Dynamical Systems
Resum: Dynamical systems appear in many models in all sciences and technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms.
Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results.
In the talk some of these questions, for which the speaker is not aware of the existence of a good solution, will be presented.
Dia: Dimecres, 6 de juliol de 2016
Lloc: Aula S01, FME, UPC.
A càrrec de: Marina Gonchenko, Departament de Matemàtiques i Informàtica, UB
Títol: Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps
Resum: We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. The method we use is based on the construction of first return maps near a given nontransversal homoclic orbit. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps. We analyse bifurcation diagrams of the cubic Hénon maps paying special attention to the problem of 1:4 resonance. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. This is a joint work with Sergey Gonchenko and Ivan Ovsyannikov.
Last updated: Fri Nov 8 20:09:53 2024