Dia: Dilluns, 9 de setembre de 2013
Lloc: Aula S02, FME, UPC.
A càrrec de: Luigi Chierchia, Roma Tre
Títol: The general Nekhoroshev theorem
Resum: There exist many proofs of Nekhoroshev's theorem in the quasi-convex case together with sharp estimates on the stability exponents, however, no complete proofs of the general steep case exist besides the original one by Nekhoroshev himself (beautiful but not optimal). I will discuss a new fully constructive and explicit proof of Nekhoroshev's theorem in the general steep case.
This is joint work with Massimiliano Guzzo and Giancarlo Benettin.
Dia: Dilluns, 16 de setembre de 2013
Lloc: Aula S2 (soterrani), Facultat de Matemàtiques, UB.
A càrrec de: Mario Natiello, Centre for Mathematical Sciences, Lund University
Títol: Braids and periodic orbits of 3D dynamical systems with a Poincaré section
Resum: Periodic orbits of 3-d dynamical systems admitting a Poincaré section can be described as braids. This characterisation can be transported to the Poincaré section and Poincaré map, resulting in the braid type. Information from braid types allows to estimate bounds for the topological entropy of the map while revealing detailed orbit information from the original system, such as the orbits that are necessarily present along with the given one(s) and their organisation. The search for "present" and "absent" orbits has coined the ideas of "orbit forcing" and "orbit pruning". We review this characterisation with some examples, focusing on systems whose Poincaré section is homotopic to a disc.
Dia: Dimecres, 25 de setembre de 2013
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: John Butcher, prof. emerit, Dept. of Mathematics, University of Auckland, New Zealand
Títol: The order of numerical methods for differential equations
Resum: The order of a numerical method for initial value problems is a useful guide to its accuracy. To obtain accurate results with low computational cost, high order is usually preferable to low order.
It is interesting that the traditional theory of order, on which the numerical methods of Runge, Heun and Kutta were based, is incorrect for orders greater than 4 and a more rigorous theory will be presented. This modern order theory leads to an algebraic system based on the composition group for Runge-Kutta methods; its applications include the order of canonical methods and the order of so-called general linear methods.
Dia: Dimecres, 2 d'octubre de 2013
Lloc: Aula 105, FME, UPC.
A càrrec de: Sergey Gonchenko, Nizhny Novgorod State University
Títol: Richness of dynamics in nonholonomic models of a Celtic stone
Resum: We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show, that in the first model (the so-call BM-model of a Celtic stone) the chaotic dynamics onsets sharply, under a supcritical period doubling bifurcation of a stable limit cycle, and it goes certain stages of developing when varying a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them as discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Dia: Dimecres, 16 d'octubre de 2013
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Tere M-Seara, Departament de Matematica Aplicada I, UPC
Títol: Regularisation of non-smooth systems using singular perturbation theory
Resum: In this talk we study the dynamics of regularized Filipov systems using singular perturbation theory. The main goal is to understand how bifurcations which are typical for non-smooth systems evolve to classical well known bifurcations when the system is regularized.
This work does a detailed study of the so-called sliding bifurcations.
This is a join work with C.Bonet
Dia: Dimecres, 30 d'octubre de 2013
Lloc: Aula 105, FME, UPC.
A càrrec de: Josep Maria Mondelo, UAB.
Títol: Aplicacions a missions a asteroides
Resum: El setembre de 2007 va ser llançada la missió Dawn per la NASA, amb destinació els asteroides Vesta i Ceres. Es tracta de la primera missió que ha d'orbitar dos cossos celestes diferents, tots dos situats al cinturó d'asteroides. Pot fer això gràcies a l'elevat impuls específic del seu motor de propulsió iònica. Aquest tipus de motor exerceix una força molt petita però que es pot mantenir durant molt de temps (més d'un mes). Llevat d'uns jets de control d'orientació, la propulsió iònica constitueix l'únic mitjà de propulsió de Dawn. El fet de desconèixer amb precisió el camp gravitatori al voltant de Vesta i la incapacitat de Dawn de fer grans maniobres (en no disposar d'un motor potent de propulsió química) va produir una certa inquietud dins l'equip de disseny de missió envers de travessar la ressonància 1:1 en l'aproximació a Vesta, que va tenir lloc (amb èxit) l'estiu de 2011.
Aquesta ha estat la motivació de la feina d'aquesta xerrada, que va començar el setembre de 2009. Consisteix en el càlcul numèric de famílies d'òrbites periòdiques originades als dos punts fixos al voltant de Vesta en coordenades rotatòries (Vesta-estacionaris) que son de tipus centre x centre x sella, així com de les seves varietats invariants. L'estudi es completa amb el càlcul de la varietat centre-estable-inestable (forma normal parcial) d'aquests punts fixos. Tots els càlculs han estat efectuats amb una expansió del camp gravitatori de Vesta en harmònics esfèrics fins a ordre i grau 8.
La feina d'aquesta xerrada és conjunta amb Stephen B. Broschart (NASA-JPL), Àlex Haro (Univ. de Barcelona) i Benjamin F. Villac (Univ. of California, Irvine).
Dia: Dimecres, 20 de novembre de 2013
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Mike Jeffrey. University of Bristol
Títol: Hiding in the gaps: have we been looking at nonsmooth dynamics the wrong way?
Resum: Countless switches of countless kinds happen around us every moment, from electrical and biological circuits, to impacts and human decisions. Some are a means of control, some are violent transitions that cause damage and instability. Attempts to extend dynamical theory to such nonsmooth systems has been going on since dynamics was first conceived. But have we been looking at them the wrong way all that time? Filippov assumed nonsmooth systems were regular perturbations of something smooth. But by allowing for singular perturbations, a whole new world of dynamics is opened up inside the switch, quantifiable and observable, but only through global (i.e. not local) dynamics. In a sense, this means shifting from a local view to a "view from infinity". Once singular perturbative terms are included, the robustness of nonsmooth dynamics can at last be established, including all its most quirky phenomena, such as determinacy-breaking events, which might provide simple geometrical origins for seemingly noisy behaviour in diverse biological, mechanical, and electrical problems.
Dia: Dimecres, 27 de novembre de 2013
Lloc: Aula 105, FME, UPC.
A càrrec de: Heinz Hanssmann, Universiteit Utrecht
Títol: The 1:1 resonance in Hamiltonian systems
Resum: Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearization. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are known for most resonances k:l except the semi-simple cases 1:1 and 1:-1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable approximation. The normal form of an Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. Thus we study S1-symmetric systems. The question we wish to address is about the co-dimension of such a system in semi-simple 1:1 resonance with respect to left-right-equivalence, where the right action is S1-equivariant. The result is a co-dimension seven unfolding of the central singularity. Four of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system. This is joint work with Igor Hoveijn (Groningen)
A càrrec de: Marian Gidea, Yeshiva University
Títol: Global Diffusion on a Tight Three-Sphere
Resum: We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. On some fixed energy level, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a three-sphere, on which the Hamiltonian satisfies a strict convexity condition, and whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on the three-sphere is equivalent to the Reeb flow for the induced contact form. The strict convexity condition implies that the contact structure on the three-sphere is tight. When a small, generic coupling is added to the system, the normally hyperbolic invariant manifold is preserved, and its stable and unstable manifolds split, yielding transverse intersections. We show that there exist trajectories that sweep the whole range of frequencies of motion on the three-sphere. In this sense, the perturbed system exhibits global diffusion on the tight three-sphere.
Dia: Dimecres, 11 de desembre de 2013
Lloc: Aula 105, FME, UPC.
A càrrec de: Roberto Barrio (University of Zaragoza, Spain)
Títol: "Fast-slow chaos" in neurons: Hindmarsh-Rose model
Resum: In this talk we study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, evaluation of Lyapunov exponents in bi-parameter diagrams, ... We demonstrate how the organizing centers-points correspond to codimension-two homoclinic bifurcations along with fold and period-doubling bifurcation curves, that structure the biparametric plane. Besides, they form macro-chaotic regions resembling "onion bulb scales" and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis. These onion bulbs also generate topological changes in the structure of the chaotic attractors that is studied by obtaining their topological templates.
Dia: Dimecres, 18 de desembre de 2013
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael de la Llave, Georgia Institute of Technology
Títol: Patterns of equilibria in periodic and quasi-periodic media
Resum: Many problems in Physics and mechanics are described by an energy which the equilibrium configurations minimizes. Very often the problem also has symmetries (periodicity or quasi-periodidity). For example, one may consider deposition of materials in crystals or quasi-crystals.
The interaction between minimization and symmetries leads to very interesting compromises. The minimizers often have some repetitive structures which can be analyzed mathematically with methods of the calculus of variations or with KAM theory (which deals with smooth solutions). There is also a body of knowledge obtained by numerical exploration which is not completely rigorous yet.
NOTA: Como experiencia nueva en el seminario, se intentará transmitir la charla urbi et orbe usando software de webex.
Dia: Dimecres, 22 de gener de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Josep Sardanyés, PhD Complex Systems Lab (Universitat Pompeu Fabra, UPF) Institut de Biologia Evolutiva (CSIC-UPF)
Títol: Studying Coevolution with Dynamical Systems
Resum: Coevolution involves the mutual evolution of two (pairwise coevolution) or several (diffusive coevolution) interacting species. Coevolution has been identified at different biological scales, from molecular, microbial, ecological, to macroevolution. Due to the difficulties of characterizing coevolution in vivo, the theory of dynamical systems has played a key role toward the understanding of coevolutionary processes by means of mathematical models. In this seminar I will first introduce key concepts of biological coevolution, also explaining the main types of coevolution and several examples of coevolution in real ecosystems. Then, I will review some recent theoretical advances on the dynamics of antagonistic coevolution, e.g., predator-prey and host-parasite systems, within the framework of the Red Queen theory.
Dia: Dimecres, 12 de febrer de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez Ros, Departament de Matemàtica Aplicada I, UPC
Títol: Sobre la región de estabilidad en microtrones
Modelamos las oscilaciones en la fase de los electrones que se mueven en un microtrón mediante una cierta aplicación que preserva área con un punto fijo en el origen. El origen representa la llamada trayectoria síncrona de una partícula de referencia en el haz de electrones. Estudiamos la estabilidad no lineal del origen en función de la fase síncrona, que es un parámetro del sistema relacionado con el número de rotación del origen visto como punto fijo elíptico de la aplicación. Estimamos el tamaño y la forma del dominio de estabilidad que rodea al origen, cuya componente conexa principal está delimitada por la ultima curva invariante rotacional (UCIR, LRIC en inglés). Describimos la evolución de la UCIR cuando la fase síncrona varía y aproximamos esas UCIRs por niveles de energía adecuados de ciertos Hamiltonianos. Finalmente, clarificamos el papel que juegan las curvas invariantes estables e inestables de algunos puntos (fijos o periódicos) hiperbólicos.Pienso que será una charla especialmente útil para todos aquellos estudiantes que están siguiendo el curso de Carles y Amadeu. Muchos de los conceptos allí introducidos se necesitan a lo largo del trabajo. A saber, la forma normal de Birkhoff, el número de rotación, cálculo y análisis separatrices, cálculo y análisis de curvas invariantes, estudio de fenómenos de resonancia, etc.
Trabajo conjunto con Yuri Kubyshin, Osvaldo Larreal y Tere Seara
Dia: Dimecres, 19 de febrer de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Pere Gutiérrez, UPC
Títol: Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
Resum: We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector ω √ ε , with ω=(1,Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré-Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.Trabajo conjunto con Amadeu Delshams y Marina Gonchenko
Dia: Dimecres, 12 de març de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Albert Granados, INRIA Paris-Rocquencourt
Títol: Period-adding and firing-rate in hybrid spiking models
Resum: In this work we consider a general periodically driven hybrid system based on the integrate-and-fire model, widely used in neuroscience. Our unique assumption is that the system is monotonic, possesses an attracting subthreshold equilibrium point and it is forced by means of periodic pulsatile (square wave) function. In contrast to classical methods, in our approach we use the stroboscopic map instead of the so-called firing-map, and becomes a discontinuous map. By applying theory for piecewise-smooth systems, we avoid relying on particular computations and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions. We rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude and the duty cycle of the pulse. We show that it is covered by regions of existence of periodic orbits given by period adding structures. They completely describe all the possible spiking asymptotic dynamics and the behavior of the firing rate, which is a devil's staircase. Our results allow us to show that the firing-rate also follows a devil's staircase with non-monotonic steps when the frequency of the input is varied, and that there is an optimal response in the whole frequency domain.Dia: Dimecres, 9 d'abril de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Piotr Zgliczynski, Jagiellonian University, Krakow
Títol: Diffusion through non-transversal transition chains
Resum: The standard Arnold's mechanism of diffusion requires the existence of pseudo-orbits, provided by a transition chain, which is formed by a sequence of non-resonant invariant tori connected along transverse heteroclinic orbits. Then a standard shadowing argument shows the existence of nearby true trajectories of the system. We present a new shadowing argument for the case of non-transversal heteroclinic orbits, which can be applied to systems of big dimensions, like infinite-dimensional Hamiltonian systems.(joint work with Amadeu Delshams and Adrià Simon)
Dia: Dimecres, 30 d'abril de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Universitat de Barcelona
Títol: How much chaotic is a simple Hamiltonian system and why
Resum: We consider some very simple Hamiltonian systems, variants or generalizations of the Hénon-Heiles system in 2 and 3 degrees of freedom, resonant and non-resonant. For all of them we assume that there exists a positive definite totally elliptic fixed point. Around it the levels of energy are diffeomorphic to S3 or S5.A measure of the domain of chaoticity is estimated by looking at the frequency of positive Lyapunov exponents in a sample of initial conditions.
The question we study is how this depends on the model, on the level of energy and parameters. We identify the dynamical objects responsible for the observed behaviour.
Dia: Dimecres, 21 de maig de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Enrique Vigil Alvarez, Departamento de Matemáticas, Universidad de Oviedo
Títol: Modelos de dinámicas asociadas a bifurcaciones homoclínicas 3D: Expanding Baker Maps
Resum: Es conocido que dada una familia uniparamétrica de difeomorfismos bidimensionales que despliega una tangencia homoclínica, se puede construir una familia de aplicaciones límite retorno que guarda una estrecha relación con la aplicación cuadrática unidimensional, utilizada en numerosos resultados entre los que figura la demostración de la existencia de atractores extraños persistentes en entornos de órbitas homoclínicas 2D.
En el artículo [1] se construye una familia de aplicaciones límite retorno asociada al despliegue de tangencias homoclínicas en familias de difeomorfismos definidos en variedades de dimensión tres. En [2], los autores realizan un estudio numérico intensivo de dichas aplicaciones que deja entrever la existencia de atractores extraños con uno y dos exponentes de Lyapunov positivos. Con la intención de explicar analíticamente estos comportamientos se definen ciertas aplicaciones bidimensionales y lineales a trozos, denominadas Expanding Baker Maps (EBM), que presentan muchos de los atractores extraños mencionados pero en un escenario más sencillo. El objetivo de la sesión será mostrar cómo surgen dichas EBM y su relación con la familia de aplicaciones límite retorno, además de ver que, efectivamente, los atractores asociados a estas aplicaciones son atractores extraños. Todo esto se recoge en [3] y [4].
Referencias:
[1] J. C. Tatjer: Three-dimensional dissipative di.eomorphisms with homoclinic tangencies. Ergod. Theory Dyn. Syst., 21, 249-302 (2001).
[2] A. Pumariño, and J. C. Tatjer: Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms. Nonlinearity, 19, 2833-2852 (2006).
[3] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil: Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete and continuous dynamical systems series B, Volume 19, Number 2 (2014)
[4] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil: Chaotic dynamics for 2-D tent maps. (sometido para publicación) (2014)
Trabajo conjunto con A. Pumariño, J. A. Rodríguez, y J. C. Tatjer.
Dia: Dimecres, 11 de juny de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Arturo Olvera, IIMAS, Univ. Nacional Autonoma de Mexico
Títol: Parametric stability of the Levitron
Resum: The Levitron is a magnetic spinning top which levitate by the interaction of an external axial magnetic field. The dynamic of this device had been studied by M. Berry, Dulling & Meiss, etc. Using the Maxwell equations and the the equation of the rigid body we obtain an Hamiltonian system with six degrees of freedom. The Levitron has stable levitation if the spinning frequency belong to a specific interval of frequencies. Our first goal is to study the damping of the spinning top, we did numerical simulation of the complete set of equation and we compare horizontal oscillation with the linear approximation of the stable fixed point. The second goal is to add a parametric perturbation in order to compensate the losses of the mechanical system. A numerical study was done to obtain a bifurcation diagram of the parametric perturbation. A stable region was found, these points correspond to permanent levitation of the spinning top.
This is a joint work with Abraham de la Rosa.
Dia: Dimecres, 18 de juny de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Prof. dr H.W. Broer, University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science
Títol: Near-horizon celestial phenomena, inspired by Minnaert and Wegener, Bernoulli and Hamilton
Resum: In geometric optics light rays are being defined with the Fermat Principle, asserting that light rays follow paths of shortest time. Later this was turned into the Hamilton Principle. It turns out that quite a number of optical phenomena in the atmosphere can be explained with a medium only consisting of two homogeneous layers. We'll discuss blank strips or zones in the setting sun, but also fata morganas. For smooth refraction index profiles the theory gets differential geometric, where light rays are formed by geodesics. This immediately gives rise to a dynamical system. In the simplest approximation the systems are integrable, but also perturbations are possible where chaos can emerge. A large number of these models can be realized in terms of a surface of revolution.
Dia: Dimecres, 25 de juny de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marta Canadell, Universitat de Barcelona
Títol: Computation of normally hyperbolic invariant manifolds
Resum: In this talk we explain some methods for the computation of normally hyperbolic invariant manifolds (NHIM) in families of discrete dynamical systems based on the parameterization method. The application of the parameterization method leads to solving invariance equations for which we use a Newton-like method adapted to the dynamics and the geometry of the invariant manifold and its invariant bundles.
Particularly, we present two different kind of methods to compute normally hyperbolic invariant tori, NHIT. The first method is based on a KAM-like theorem in a-posteriori format for the existence of quasi-periodic invariant tori, which provides us an efficient algorithm for computing NHIT, by adjusting parameters of the family. The second method allows us to compute a NHIT and its internal dynamics, which is a priori unknown. We implement both methods to continue the invariant tori with respect to parameters, and to explore different mechanisms of breakdown.
This is a join work with Alex Haro.
Dia: Dimecres, 2 de juliol de 2014
Lloc: Aula 005, FME, UPC.
A càrrec de: Renato Calleja, Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM)
Títol: Construction of quasi-periodic response solutions for forced systems with strong damping
Resum: I will present a method for constructing quasi-periodic response solutions (i.e. quasi-periodic solutions with the same frequency as the forcing) for over-damped systems. Our method applies to non-linear wave equations subject to very strong damping and quasi-periodic external forcing and to the varactor equation in electronic engineering. The strong damping leads to very few small divisors which allows to prove the existence by using a contraction mapping argument requiring very weak non-resonance conditions on the frequency.
This is joint work with A. Celletti, L. Corsi, and R. de la Llave.
A càrrec de: Sonia Pinto de Carvalho, Departamento de Matematica - UFMG - Brasil
Títol: Billiards on Surfaces of Constant Curvature
Resum: The billiard problem consists in the free motion of a point
particle in a region enclosed by a closed curve (the billiard table),
being reflected elastically at the impacts with the boundary.
If the billiard table is contained on a surface of constant curvature,
then the motion occurs along a geodesic and the particle changes
direction at the impacts making equal angles with the tangent to the
boundary.
As in the plane case, this problems define a 2-dimensional discrete
and conservative dynamical system. In this joint work with Luciano
Coutinho dos Santos (CEFET-MG, Brasil) we present some dynamical
properties of such billiards when the boundary curve is convex.
Dia: Dimecres, 16 de juliol de 2014
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Barak Weiss, University of Tel Aviv
Títol: Badly approximable vectors on fractals and random walks
Resum: In joint work with David Simmons, we prove that certain natural fractal measures arising from iterated function systems of contracting similarities, give zero measure to badly approximable vectors. Previous results in this direction involved a reduction to the measure rigidity results of Lindenstrauss. Our approach involved a reduction to the analysis of stationary measures for certain semigroups acting on homogeneous spaces, extending work of Benoist and Quint.
No background in diophantine approximation will be assumed in the talk and all necessary prerequisites will be given.
Last updated: Thursday, 02-Oct-2014 08:23:40 CEST