Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Luis Barreira, Instituto Superior Tecnico, U.Tec. Lisboa .
Títol: Poincare recurrence: from qualitative to quantitative .
Resum: Poincare's recurrence theorem is a fundamental result, but only of qualitative nature. In particular, it gives no information about the frequency of visits to a given set. This drawback was partly surpassed with the first versions of the ergodic theorem, which however only consider one aspect of the quantitative nature. More recently, there has been a growing interest in the area, starting with the seminal work of Boshernitzan, Ornstein and Weiss. Building on their results, I want to discuss a very sharp description of the quantitative nature of recurrence, and in particular a somewhat unexpected relation between the so-called recurrence rate and the pointwise dimension: essentially they are the same almost everywhere. The necessary material from hyperbolic dynamics and ergodic theory will be recalled along the way.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Piotr Zgliczynski, Jagiellonian University, Krakow .
Títol: Some results on multidimensional perturbations of 1-dim maps .
Resum: We consider continuous maps F:R x Rk → R x Rk which are close to F0(x,y)=(f(x),0), where f:R → R is continuous.
We address the following question, assume that f has an interesting dynamical property, will it continue to F if |F(z)-F0(z)| is small for z in some suitable compact sets. In this context as the 'interesting dynamical properties' we will consider the set of periods and the topological entropy. We will present some positive answers in this direction.
References
P. Zgliczynski, Sharkovskii's Theorem for multidimensional perturbations of one-dimensional maps. Ergodic Theory and Dynamical Systems 19 (1999), 1655-1684.
M. Misiurewicz and P. Zgliczynski, Topological entropy for multidimensional perturbations of one dimensional maps. Int. J. of Bifurcation and Chaos 11 (2001), 1443-1446.
Ming-Chia Li and P. Zgliczynski, On stability of forcing relations for multidimensional perturbations of interval maps, preprint, http://www.ii.uj.edu.pl/˜zgliczyn/papers/publ.htm
A càrrec de: Maciej Capinski, AGH Univ. Sci. and Technology, Krakow .
Títol: Cone Conditions and Covering Relations for Normally Hyperbolic Invariant Manifolds .
Resum: We present a new proof of the existence of normally hyperbolic invariant manifolds for maps. In contrast to the usual approach, the proof is performed in the phase space of the dynamical system, rather than on an infinitely dimensional functional space. This fact allows to formulate the assumptions in such a way that they are verifiable using rigorous numerics. The method gives explicit bounds on the region in which the manifold is contained and an explicit estimate on the size of the perturbation under which the manifold persists. The proof is based on the method of covering relations, Brouwer fixed point theory and cone conditions.
Lloc: Auditori, CRM.
A càrrec de: Alfonso Sorrentino, CEREMADE - Universite Paris-Dauphine .
Títol: Tutorial on Mather's theory for Lagrangian systems .
Resum: This talk is meant to be a brief (hopefully comprehensive) introduction to Mather's variational approach to the study of positive definite Lagrangian systems. Time permitting, I will discuss how this theory may be related to the study of weak (sub)solutions of Hamilton-equation (Weak KAM theory) and how these techniques can be used, for example, to deduce global uniqueness results for KAM tori, with a prescribed rotation vector or a prescribed average action.
A càrrec de: Pierre Berger, CRM .
Títol: Tutorial on Structural stability in dynamical systems and singularity theory .
Resum:
First we will recall all the classical statements of structural stability in both fields:
- Infinitesimal stability equivalent to stability
- Genericity and Characterizations
Then we will describe some of the geometry of structurally stable
maps, via the formalism of stratifications, for both fields. After
motivating the study of the structurally stable endomorphisms of
manifolds (structurally stable bundles in invertible dynamical
systems, physics, morphogenesis), we will state and sketch the proof
of a theorem on stability of attractor-repellor endomorphisms with
singularities.
Lloc: Auditori, CRM.
A càrrec de: Renato Calleja, University of Texas at Austin .
Títol: Fast numerical computation of quasi-periodic equilibrium states in statistical mechanics and twist maps .
Resum: We develop fast algorithms to compute quasi-periodic equilibrium states of one dimensional models in statistical mechanics. The models considered include as particular cases, Frenkel-Kontorova models, Heisenberg XY models, as well as problems from dynamical systems such as twist mappings and monotone recurrences. In the dynamical cases, the quasi-periodic solutions are KAM tori. We formulate a numerically accessible criterion for the computation of the analyticity breakdown of tori in systems described by a variational principle.
A càrrec de: Enrico Valdinoci, Università di Roma Tor Vergata .
Títol: Geometric properties of semilinear PDEs .
Resum: We present some symmetry results for semilinear PDEs of elliptic type arising from phase transition models, with possible application to degenerate and fractional operators, and to stratified and periodic media. We also discuss some connections with Aubry-Mather theory and with rigidity properties of overdetermined problems.
Lloc: Aula IMUB (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Wayne Hayes, Univ. of California at Irvine .
Títol: Solar System: Surfing the Edge of Chaos .
Resum: The stability of our Solar System has been debated since Newton devised the laws of gravitation to explain planetary motion. Newton himself doubted the long-term stability of the Solar System, and the question has remained unanswered despite centuries of intense study by generations of illustrious names such as Laplace, Langrange, Gauss, and Poincare. Finally, in the 1990s, with the advent of computers fast enough to accurately integrate the equations of motion of the planets for billions of years, the question has finally been settled: for the next 5 billion years, the shapes of the planetary orbits will remain roughly as they are now. This is called "practical stability": none of the known planets will collide with each other, fall into the Sun, or be ejected from the Solar System, for the next 5 billion years.
Although the Solar System is now known to be practically stable, it may still be "chaotic". This means that we might - or might not - be able to predict their precise positions within their orbits, for the next 5 billion years. The precise positions of the planets can affect the tilt of each planet's axis, and so can have a measurable effect on the climate. For the past 15 years, there has been debate about whether the Solar System exhibits chaos or not: when performing accurate integrations of the planetary motions, some astronomers observe chaos, and some do not. This is particularly disturbing because it is known that inaccurate integration can inject chaos into a numerical solution that would otherwise be stable.
My research is numerical analysis, and in this talk I will demonstrate how I closed the 15-year debate on chaos in the solar system by performing the most accurate, long-term integrations of the orbits of the planets that has ever been done. The answer surprised even the astronomical community, and was published in Nature Physics.
Lloc: Auditori CRM.
A càrrec de: Albert Fathi, Ecole Normal Superieure de Lyon .
Títol: Denjoy-Schwartz and Hamilton-Jacobi .
Resum: Given a C2 Hamiltonian H(x,p), C2-strictly convex in the moment variable, it has been shown by Patrick Bernard that one can always find C1 strict subsolutions with locally Lipschitz derivative of the Hamilton-Jacobi equation. After explaining the general background, the talk will concentrate on the constraints imposed on smoother critical subsolutions by the implications of the classical Denjoy-Schwartz theory of Dynamical Systems on surfaces.
A càrrec de: Laurent Niederman, Universite Paris Sud & Observatoire de Paris (IMCCE) .
Títol: Genericity of Lochak resonant estimates for partially convex quasi integrable Hamiltonian systems. Application to FPU chains .
Resum: In the 70's, Nekhoroshev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as "steepness".
In 1990, Lochak proved that in finite but exponentially large intervals of time the resonances contribute to the stability of motion provided that the unperturbed Hamiltonian is convex. More precisely, a rigorous local version Nekhoroshev theorem was proved in case of convexity which implies that the stability time increases under resonance conditions.
Lloc: Auditori CRM.
A càrrec de: Vadim Kaloshin, University of Maryland .
Títol: Arnold diffusion in a lattice .
Resum: We present several example of lattice Hamiltonian systems exhibiting Arnold diffusion. It includes nearest neighbors couples pendulums, rotors, coupled oscillators.
This is a joint work with M. Levi and M. Saprykina.
A càrrec de: Wei-Min Wang, Univ. Paris Sud .
Títol: A tutorial on multiscale analysis .
Resum: Many phenomena (e.g. localization and diffusion) look different depending on the scale at which they are observed. Understanding these phenomena requires understanding how behaviours in one scale affect the others.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Sergey Bolotin, University of Wisconsin, Madison and Moscow Steklov Mathematical Institute .
Títol: Skew products of nearly integrable symplectic maps .
Resum: We discuss dynamics of skew products of finite collections of symplectic maps, i.e., dynamics of compositions of these maps taken in random order. Such dynamical systems appear as scattering maps of normally hyperbolic invariant manifolds of Hamiltonian systems. To construct a solution of the Hamiltonian system shadowing an orbit of the scattering map, one needs this orbit to be hyperbolic. We construct hyperbolic orbits for skew products of nearly integrable symplectic maps. For a single nearly integrable map, finding hyperbolic orbits is a hard problem related to exponentially small splitting of separatrices. Fortunately the problem is much simpler for skew products. The motivation for this work comes from the study of almost collision orbits in the 3 body problem
A càrrec de: Víctor Muñoz, Universidad de Valladolid .
Títol: Neutral functional differential equations with infinite delay .
Resum: We present results on the structure of the omega-limit sets of a dynamical system arising in the study of monotone non-autonomous infinite delay functional differential equations in the framework of skew-product semiflows. More precisely, we establish the 1-covering property of omega-limit sets under convenient hypotheses. Besides, we extend these results to the neutral case and make a remark upon their applications to compartmental systems
Lloc: CRM, Auditori.
A càrrec de: Vadim Zharnitsky, Univ. of Illinois at Urbana-Champaign .
Títol: Quasilinear dynamics in nonlinear Schroedinger equation with periodic boundary conditions .
Resum: We describe a new scenario of an almost linear behavior in strongly nonlinear systems. This pheomenon is demonstrated for the cubic one dimensional nonlinear Schroedinger equation with periodic boundary conditions, where we have nearly complete understanding of the phenomenon. The nonlinearity gets "averaged out" by the high frequency solutions and this leads to an averaging type theorem for PDEs. This is joint work with M. Burak Erdogan.
A càrrec de: Michael Benedicks, KTH, Stockholm .
Títol: Non-linear evolution equations, Coupled Map Lattices and non-invertible dynamics in 2-d .
Resum: The talk is inspired by recent work of Pesin and Yurchenko who discovered nonlinear evolution equations which seem to have "strange attractors". More specifically, after discretisation one obtains a Coupled Map Lattice (CML), whose local maps have strange attractors. We will also try to relate this to Sinai-Ruelle-Bowen measures for CML:s (work by Bunimovich-Sinai and Kupiainen-Bricmont).
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Francesca Tovena, Universitá di Roma Tor Vergata .
Títol: Index theorems for holomorphic maps and foliations .
Resum: We describe a unified approach to index theorems for foliations
and for holomorphic endomorphisms of a complex manifold, showing how
to get the Camacho-Sad-like index theorems present in the literature
(and a couple of new ones) as particular instances of a single
construction, based on the existence of a universal partial
connection on the normal bundle of a submanifold of a complex
manifold.
(Joint work with M. Abate and F. Bracci)
A càrrec de: Marco Abate, Universitá di Pisa .
Títol: A Poincaré-Bendixson theorem for homogeneous vector fields and meromorphic connections .
Resum: In one complex dimension, a holomorphic germ tangent to the
identity is locally topologically conjugated to the time-1 map of a
homogeneous vector field. In particular, the study of the real flow
of (complex) homogeneous vector fields in (complex) dimension one
provides a large amount of informations on the local dynamics of
functions tangent to the identity.
This suggests that the study of the real flow of complex homogeneous
vector fields might also help to understand the local dynamics of
holomorphic map tangent to the identity in complex dimension two, at
least in generic cases. In this talk I shall describe how it is
possible to reduce the study of the real 1-dimensional flow of a
complex 2-dimensional homogenous vector field to the study of the
geodesic flow of a meromorphic connection on the complex projective
line, and how to use how to use this reduction and a new
Poincare-Bendixson theorem for meromorphic connections to describe
the recurrent behavior of such flows.
(Joint work with F. Tovena)
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jason Gallas, Instituto de Fisica, Universidade Federal do Rio Grande do Sul .
Títol: Infinite cascades of hubs and spirals (Numerical simulations in parameter space of simple flows) .
Resum: We discuss three examples of ODEs containing a remarkable "periodicity hub", like the one recently reported in Phys. Rev. Lett. 101, 054101 (2008). Hubs are focal points inside chaotic phases from where an infinite hierarchy of nested spirals emanates. By suitably tuning two parameters simultaneously, both waveform and periodicity may be increased continuously without bound and without ever crossing the surrounding chaotic phase. Familiar period-adding phenomena emerge as restricted one-parameter slices of an exceptionally intricate and very regular onion-like parameter surface centered at the focal hub which seems to "organize" all the dynamics around it. After describing the intricacies of isolated hubs, we discuss a flow displaying infinite hub cascades, aligned along simple curves in parameter space. The presentation consists essentially of colored bitmaps obtained numerically, illustrating complicated phase diagrams. Hubs do not seem to fit a Shilnikov homoclinic scenario. We present numerical evidence that they are not related to T-points.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Daniel Peralta-Salas, Dpto. de Matemáticas, Universidad Carlos III de Madrid .
Títol: Submanifolds which can be leaves of foliations in Euclidean space .
Resum: The theory of integrable embeddings recently introduced by G. Hector and the speaker consists in characterizing the submanifolds of open manifolds which are contained in (or are exactly given by) the zero-set of some submersion. In Euclidean space this allows to classify all submanifolds which can be leaves of foliations, thus giving rise to some striking corollaries: e.g. S3 cannot be leaf of a foliacion neither in R5 nor R6. The techniques that we use involve the Phillips-Gromov h-principle and the theories of immersions and obstructions. The goal of this talk is to give an overview of these results.
NOTA IMPORTANT: A les 5 pm, després de l'exposició, hi haurà una
reunió a la mateixa aula per a parlar de temes d'ORGANITZACIÓ del
seminari en el FUTUR. És IMPORTANT que hi hagi la MÀXIMA ASSISTÈNCIA.
La Tere Martínez-Seara i Alonso té algunes propostes de coses
interessants a fer, i jo també.
Ànim i que VINGUI TOTHOM!
Gràcies! Carles
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Non-integrability of Hamiltonian systems through high order variational equations: A soft introduction to results and examples .
Resum: In this talk we shall deal with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to study these problems. We shall display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations. This is a summary of joint work with Regina Martinez.
Lloc: Aula 101 (1er pis), FME, UPC.
A càrrec de: Rafael Ramírez Ros, Dept. Matemàtica Aplicada I, UPC .
Títol: Del cálculo de áreas en aplicaciones conservando área al cálculo de volúmenes en aplicaciones conservando volumen .
NOTA: A les 15h 30m hi haurà càfe, preliminar a la xerrada.
Nota prèvia al resum facilitada pel conferenciant:
Estaría bien que la audiencia repasara previamente algunos conceptos básicos de geometría diferencial. Concretamente, trabajaré con formas diferenciales: integrándolas (teorema de Stokes), derivándolas (derivada exterior, derivada interior, derivada de Lie) y empujándolas (pushforward, pullback). Si alguien no sabe que es una forma diferencial, se va a aburrir bastante durante la charla (a no ser que sea el propio "speaker", en cuyo caso la charla sería de todo menos aburrida).
Resum: Esta charla informal y divulgativa tendrá, si el tiempo y la autoridad competentes no lo impiden, cinco partes, cinco.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Chara Pantazi, Dept. Matemàtica Aplicada I, UPC .
Títol: Inverse problems for invariant algebraic curves: Explicit computations .
Resum: In this talk first we will present a short review of the Darboux theory of integrability. Poincare posed the following problem "determine all the invariant algebraic curves for a given polynomial vector field". Here, in this talk we will deal with the inverse problem: "Find all vector fields X having some fixed set of algebraic curves invariants". We will show that this inverse problem is quite well-understood and algorithmically accessible.
This is a joint work with C. Christopher, J. Llibre and S. Walcher.
References: C. Christopher, J. Llibre, Ch. Pantazi, S. Walcher: Inverse problems for invariant algebraic curves: Explicit computations. Proc. Roy. Soc. Edinburgh 139A, 1 16 (2009).
Lloc: Aula 101 (1er pis), FME, UPC.
A càrrec de: Konstantinos Efstathiou, Dept of Math., U. of Groningen .
Títol: The hydrogen atom in electric and magnetic fields .
Resum: The hydrogen atom in external electric and magnetic fields is one of the most classical and well-studied physical systems which has played a fundamental role in the development of quantum mechanics. It is thus remarkable that new things can still be said about it. In this talk we study the hydrogen atom using an integrable approximation that is constructed through two successive normalizations. Studying this integrable approximation we classify systems with different near orthogonal external field configurations in terms of their basic qualitative characteristics. Finally, we discuss the problem of relating the classification results obtained for the constructed integrable approximation to the dynamics of the original non-integrable 3 degree of freedom Hamiltonian system.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Marina Gonchenko, Dept. Matemàtica Aplicada I, UPC .
Títol: On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies .
Resum: We study bifurcations of two-dimensional symplectic diffeomorphisms with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings. This is a joint work with S. Gonchenko.
Lloc: Aula 101 (1r pis), FME, UPC.
A càrrec de: Ariadna Farrés, Dpt. MAiA, UB .
Títol: On the dynamics of a Solar Sail near L1 .
Resum: We are interested in understanding the natural dynamics of a Solar Sail in the Earth-Sun system. To model its motion we consider the Restricted Three Body Problem adding the Solar Radiation Pressure. This model has a 2D family of equilibria parametrised by the two angles that define the sail orientation. In this talk we will focus on the family of fixed points that are close to the classical collinear point L1. We will describe the periodic and quasiperiodic motion that there for different sail orientations.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Pau Martin, UPC .
Títol: Invariant manifolds of hyperbolic sets in weakly coupled lattice maps and their decay properties .
Resum: Weakly coupled map lattices are maps on lattices such that the dynamics on one site of the lattice has small effect on sites that are far away. This is vague concept and here we present a functional analysis framework which allows an efficient quantitative study of the decay properties of the interactions. We use this framework to prove stable manifold theorems and show that the manifolds are as smooth as the maps and share the same decay properties.
Lloc: Aula 101(1r pis), FME, UPC.
A càrrec de: Arturo Vieiro, Dept. de Mat. Aplicada i Anàlisi, UB .
Títol: Dynamical description and inner-outer splitting of resonances of APMs .
Resum: We are interested in the (semi-local) dynamics around an elliptic fixed point E0 of a one parameter family of real analytic area preserving maps (APMs). Generically, when changing the parameter, different chains of resonant islands bifurcate from E0. These islands have a pendulum-like phase space structure. First, we will derive (from Birkhoff normal form) a suitable Hamiltonian model useful to describe the dynamics within these resonant islands. Then, we will study the splittings of the separatrices bounding these pendulum-like islands using the Hamiltonian model obtained (properly modified for technical reasons) as a limit Hamiltonian. In a pendulum-like island there are two "main" splittings. They are associated to the separatrix connexions of the classical pendulum. For a resonant chain of islands we will refer to these "main" splittings as the inner and outer splittings according to their distance to E0. It turns out that both splittings are generically different, being the outer one the largest (at least if the island is located close enough to E0).
These results have been recently published in [1]. In the talk we will summarize the main ideas of [2] and we will discuss how they should be adapted to prove the difference on the splittings of separatrices, including the case of analytic non-entire maps. We will show the agreement of the theoretical results with the numerical observations using the well-known Hénon map as a paradigmatic example. (This is a joint work with C. Simó)
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Linda Keen, Lehman College, CUNY .
Títol: Boundaries of Bounded Fatou Components of Quadratic Maps .
Resum: In this talk, based on joint work with Ross Flek, we characterize those external rays that land on the bounded Fatou components of hyperbolic and parabolic quadratic maps. For those maps not in the main cardioid of the Mandelbrot set, we prove that these rays form a Cantor subset of the circle at infinity. Our techniques involve both the orbit portraits of Goldberg and Milnor that relate the dynamic and parameter planes and the Thurston theory of laminations for quadratic maps. This classification is important since it provides a way to characterize buried Julia sets of a class of degree two rational maps which are conjugate to self-matings of the above quadratics.
Lloc: Aula 101(1r pis), FME, UPC.
A càrrec de: Mercè Ollé, Dpt. MA1, UPC .
Títol: Horseshoe motion and homoclinic phenomena in the RTBP .
Resum: We consider the RTBP. On one hand, we deal with symmetric horseshoe periodic orbits (HPO), motivated by astronomical applications. The understanding of the families of HPO, when varying the mass parameter, relies on the μ=0 case and on the invariant manifolds of the collinear point L3. We explore the behaviour of such manifolds and show the cascade phenomenon of multi-round symmetric homoclinic orbits. Finally the homoclinic orbits to the Lyapunov families of Li, for i=1,2,3 are also analysed.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Dolors Puigjaner, Dept. Enginyeria Informàtica i Matemàtiques, U. Rovira i Virgili .
Títol: A dynamical systems approach to Lagrangian transport in a Rayleigh-Bénard problem .
Resum: We are interested in understanding the mixing and transport properties of fluid flow systems. In this talk we will focus on the application of dynamical systems tools to the analysis of the dynamics of fluid particle trajectories for the Rayleigh-Bénard problem in a cube. We will analyze the stability properties and bifurcation of fixed points and main periodic orbits. We will show that fixed points, including the ones at the boundaries, play a key role in the global dynamics of the flows. Suitable Poincare maps will be used to show that regions of chaotic motion and regions of regular motion coexist inside the cavity. We will present the methodology developed to identify the boundaries of these three-dimensional regions. The chaotic nature of the flows will be quantified by the maximal Lyapunov exponent and the metric entropy. In addition, we will present some topological relations that can be considered to check the results. (This is a joint work with C. Simó, J. Herrero and F. Giralt)
Lloc: Aula 101(1r pis), FME, UPC.
A càrrec de: Pablo Cincotta, Univ. Nacional de La Plata .
Títol: Alternative ways to measure chaos and diffusion in phase space .
Resum: In the present effort we provide results and discussion concerning alternative ways to measure chaos and diffusion in phase space. All these issues are thoroughly discussed herein by dealing with a multidimensional conservative map that would be representative of the dynamics of resonance interaction. Moreover, the computation of both the diffusion coefficient --defined through the variance of the unperturbed action variables-- as well as the Shannon or Arnol'd Entropy in order to measure the extent of chaotic diffusion processes is addressed.
APPENDIX
A brief discussion about the performance of different software packages or routines to integrate ODEs, as Taylor, DOPRI8, BS, will be presented.
Joint work with Claudia Giordano
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alejandro Luque, Dept. Matemàtica Aplicada I, UPC .
Títol: El método de la parametrización para toros normalmente elípticos en sistemas hamiltonianos .
Resum: Supongo que cualquier ciudadano de Barcelona que haya trabajado en sistemas dinámicos habrá leído u oído acerca del método de la parametrización o teoría KAM sin acción-ángulo en algún momento. La idea subyacente en estos métodos es que si tenemos una aproximación de un toro invariante, cumpliendo ciertas condiciones de no resonancia y no degeneración, entonces podemos construir un esquema cuadrático que converge a un toro invariante.
Este enfoque proporciona una serie de ventajas respecto a los métodos clásicos (no hay que recurrir al uso de transformaciones canónicas ni de coordenadas acción-ángulo, se pueden considerar problemas no perturbativos, etc) y se puede implementar para obtener métodos numéricos muy eficientes para calcular tales objetos.
En este trabajo adaptamos estas ideas en el contexto de toros de dimensión menor (a los grados de libertad del sistema) normalmente elípticos (¡las direcciones normales al toro oscilan!). Para realizar la construcción con éxito debemos considerar simultáneamente aspectos de reducibilidad de sistemas de ecuaciones casi-periódicos
Trabajo conjunto con Jordi Villanueva
Lloc: Aula 6.42 (A/C), ETSEIB, UPC. Planta 6, sortint ascensor a la dreta
A càrrec de: Marcel Guardia .
Títol: Exponentially small splitting of separatrices of the pendulum: two different examples .
Resum: In this talk we study the splitting of separatrices for the pendulum with certain fast periodic perturbations.
First we consider the classical case in which the perturbation only depends on time, and we see in which cases Melnikov integral predicts correctly the splitting. We study also, in that case, certain values of the parameters which are close to a codimension two bifurcation which can be understood as a very simple toy model of a second order simple resonance, and we conjecture how the size of the exponentially small splitting changes.
Finally, we consider the pendulum with a perturbation which is meromorphic and we see how the size of splitting depends on the size of the strip of analicity of the perturbation, leading to non exponentially small splitting when the strip is small enough.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Vassili Gelfreich, Math. Dept. Warwick University .
Títol: Universal richness of dynamics near a generic elliptic fixed point .
Resum: We will look into the dynamics of an area-preserving map near an elliptic fixed point and show that a non-transversal homoclinic point can be created by an arbitrarily small perturbation (both in smooth and analytic category). Using earlier results by Gonchenko, Turaev and Shilnikov we conclude that generically the local dynamics is universally rich, i.e., renormalised iterates of the map are dense in the set of all area-preserving maps defined on a disk. This talk is based on a current joint work with D.Turaev.
Lloc: Aula 6.42 (A/C), ETSEIB, UPC. Planta 6, sortint ascensor a la dreta
A càrrec de: V. Kaloshin .
Títol: Quasi-ergodic hypothesis and examples of a nearly integrable Hamiltonian systems with large transitive sets .
Resum: We shall construct a nearly integral system of 3 degree of freedom such that it has an orbit dense in a set of almost maximal Lebesgue measure on a fixed energy surface. The proof relies on Mather variational method and theory of normal forms. This is a joint work with Ke Zhang and Yong Zheng.
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