Lloc: Aula 10, Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Rafael Obaya, Universidad de Valladolid .

  Títol: Algunos problemas dinamicos en una ecuacion escalar convexa .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Jean_Pierre Marco, Universite de Paris 6 .

  Títol: Uniform lower bounds of the splitting and optimality of Nekhoroshev exponents for multidimensional analytic systems .

  Resum:

  We construct a particular class of analytic Hamiltonian systems, perturbations of the completely integrable quasi-convex Hamiltonian $h(r)={1\over 2}(r_1^2+\cdots +r_{n-1}^2)+r_n$ on ${\bf T}^n\times{\bf R}^n$ ($n\geq4$), for which we can estimate from below the splitting of the invariant manifolds of a one-parameter family of hyperbolic tori. The lower bound we obtain is of the form $\Delta\geq C\exp(-c({1\over\epsilon})^a)$, with $a={1\over {2(n-3)}}$, $\epsilon$ being the analytic size of the perturbation of a fixed complex strip. A slight variation of that system is proved to have drifting orbits, whose speed have a lower bound of the same form.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Stefanella Boatto, Fields Institute and McMaster University .

  Títol: Billiards: diffusion approximation, boundary conditions and geometry .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Laurent Niederman, U. Paris-Sud (Orsay) .

  Títol: Generic sharp exponential stability among nearly integrable Hamiltonian systems .

  Resum: Results of exponential stability of the action variables under perturbation with a sharp exponent (1/2n) are valid for a generic set of integrable Hamiltonians.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Vassili Gelfreich, Math. Department, Warwick University .

  Títol: Discrete Bogdanov-Takens bifurcation .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Santiago Ibañez, Depto de Matematica Aplicada, U. Oviedo .

  Títol: Despliegue de la singularidad nilpotente de codimension tres en dimension tres .

  Resum: La familia cuadratica de campos de vectores en dimension tres dada por las ecuaciones

  x'=y     y'=z     z'=\lambda+\mu y+\nu z+x^2,

  con \lambda^2+\mu^2+\nu^2=1 juega un papel clave en el estudio del despliegue de la singularidad nilpotente de codimension tres en R^3. Concretamente, despues de oportunos reescalados, se convierte en familia limite de cualquier despliegue generico y por ello su estudio es fundamental para explicar el despliegue. En los ultimos a\~nos se han conseguido importantes avances en la comprension de las diferentes dinamicas que estan presentes en dicha familia. Comentaremos el estado actual del problema y detallaremos algunos de los resultados mas relevantes. En particular aquellos relativos a la existencia de bifurcaciones a orbitas homoclinicas de tipo Shil'nikov y tambien de las denominadas bifurcaciones ''cocoon''.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Anatoly Neishtadt, Space Research Institute, Moscu .

  Títol: Captures into resonances and scattering on resonances in volume-preserving systems .

  Resum: There exists a description of the phenomena of captures into resonances and scattering on resonances for systems with two fast rotating phases in the general context of systems close to integrable ones and in the context of slow-fast Hamiltonian systems.

  The talk will contain a description of these phenomena for volume-preserving systems with two rotating phases and two slow variables. Preservation of the phase volume allows to provide a rather detailed description of the dynamics similar to that available in the Hamiltonian case.

• Hora: 17h30m

  A càrrec de: Enrico Valdinoci, Dpt. di Matematica, U. di Roma Tor Vergata .

  Títol: Periodic and quasiperiodic orbits of the planetary problem of three bodies .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h30m

  A càrrec de: Robert D. Russell, Simon Fraser University .

  Títol: Numerical solution of PDEs using moving grids -- an overview .

  Resum: It is now well acknowledged that for the solution of most PDEs having nontrivial solutions it is advantageous if not essential to use adaptive methods. Unfortunately, given the wide array of adaptive techniques and their limited theoretical understanding, it is generally not obvious which should be preferred in any given circumstance.

  We shall discuss a class of moving mesh methods for solving time-dependent PDEs --- so-called MMPDE (moving mesh PDE) methods, which smoothly adapt the meshes in time during the problem integration. A key feature of the methods is that they can be interpreted and analysed using a continuous formulation, with the MMPDE solution being interpreted as a continuous change of variables from physical to computational space.

  For many PDEs with singular behaviour in the solution, these MMPDE methods can be particularly amenable to since one can often choose them such that the underlying qualitative solution structure is automatically preserved by the numerical solution in the limit. We shall give several examples of current interest and show how the interplay of analysis and numerics has led to a number of interesting discoveries. As well, we shall discuss some of the computational challenges in the field of adaptivity.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h30m

  A càrrec de: Nikola Petrov, Dept of Mathematics, University of Michigan .

  Títol: A simple introduction to wavelets and fractals (I) .

  Resum: In this series of two talks we will give a simple introduction to the main ideas of wavelet analysis and some of its applications. In the first talk we will give an overview of the general ideas of multiresolution analysis ideas behind the wavelet analysis, (try to) give a sketch of the construction of Daubechies wavelets, and discuss some connections with Fourier analysis.

  In the second talk, we will introduce some concepts related to global and pointwise regularity properties of functions and fractal properties of singular measures, and will explain briefly how wavelets help to analyze them.

  Nota: La segona part de la xerrada es fara el divendres a l'Aula de Sistemes Dinamics a la UPC

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h30m

  A càrrec de: Patrick Bonckaert, Limburgs Univ. at Diepenbeeck, Dept. Math. .

  Títol: Some equivalence results of near resonant stationary points .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Antonio Giorgilli, Dipto. Matematica e Applicazioni, Univ. Milano Bicocca .

  Títol: Metastable states in the Fermi-Pasta-Ulam system .

• Hora: 17h30m

  A càrrec de: Vladimir Gonchenko, Nizhny Novgorod Univ. .

  Títol: Homoclinic bifurcations and generalizing H\'enon map (GHM) .

  Resum: In many problems related to homoclinic bifurcations of codimension 1 the standard H\'enon map appears as the rescaled first-return map. The codimension 2 homoclinic problems lead to the so-called generalized H\'enon map (GHM)

  \bar x = y,

  \bar y = M_1 - M_2 x - y^2 + R xy + S y^3,

  where R and S are small. In the talk a short survey of problems related to GHM will be described and recent results about bifurcation structure of GHM will be presented.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Imma Baldoma, Dept. Matematica Aplicada i Analisi, UB .

  Títol: The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems .

  Resum: We study the existence and properties of solutions of Hamilton-Jacobi equations with a given behavior at infinity. The Hamilton-Jacobi equations we deal with can be interpreted as the inner equation of one and a half degrees of freedom rapidly forced Hamiltonian systems having a homoclinic connection.

• Hora: 17h30m

  A càrrec de: Vladimir Gonchenko, Nizhny Novgorod Univ. .

  Títol: Homoclinic bifurcations and generalizing H\'enon map (GHM) .

  Resum: Some recent results of the author (and coauthors: L.Shilnikov, D.Turaev, S.Gonchenko) on the study of bifurcations of multidimensional systems with homoclinic tangencies will be presented. The talk includes the theory of $\Omega$-moduli (continuous invariants of topological conjugacy on the set of nonwandering orbits), rescaled models of first return maps, the existence of mixed dynamics in Newhouse regions etc. Some open problems will be discussed.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Christian Henriksen, Dept. of Mathematics, Technical Univ. Denmark .

  Títol: Scaling Ratios and Triangles in Siegel disks .

  Resum: I'll present some previous joint work with X. Buff. Let $f(z)=e^(2i\pi\theta) z + z^2,$ where $\theta$ is a quadratic irrational. McMullen has proven that the Siegel disk of $f$ is self-similar about the critical point. We give a lower bound for the ratio of self-similarity. Then we show that for the golden mean $\theta = (sqrt{5}-1)/2$ you can place an Euclidian triangle in the Siegel disk with a vertex at the critical point.

• Hora: 17h30m

  A càrrec de: Sergi Simon, Dept. Matematica Aplicada i Analisi, UB .

  Títol: Algebraic proof of the non-integrability of Hill's Problem .

  Resum: Hill's lunar problem appears in Celestial Mechanics as a limit of the Restricted Three-Body Problem. Besides, information on the former shows light on several other three-body problems. It contains no parameters and is globally far from any simple well--known problem. Strong numerical evidences of its lack of integrability have been given in the past. Here an algebraic proof of non--integrability is presented. Beyond the result in itself, the paper can also be considered as an example of the application of differential Galois theory to a significant problem.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Bob Russell, Dept Math & Dept CS, SFU, Vancouver, BC .

  Títol: An Overview of Some Adaptive Methods for Solving Higher Dimensional PDEs .

  Resum: Supercomputers notwithstanding, the solution of time-dependent PDEs requires the use of adaptive techniques for which the underlying mesh is modified to capture special solution features. Given the vast array of techniques, it is difficult for a user to determine which ones will work well for their particular problems. In this talk, we shall consider two basic approaches for moving mesh points in time. The first relies on minimizing a suitable variational form, and the second involves computing the mesh velocities directly. Both essentially involve finding a coordinate transformation from physical coordinates to computational coordinates, and each faces a number of difficulties for higher dimensional PDEs. We discuss recent theoretical developments, related to classical mathematical problems, which help to both explain why these traditional difficulties occur and how they may be overcome. We also relate these adaptive mesh problems to some other general problems in science and engineering. Finally, some numerical examples are given to demonstrate the efficacy of these new implementations.

• Hora: 17h30m

  A càrrec de: Henk Broer, University of Groningen, Department of Mathematics .

  Títol: Geometry of resonance tongues .

  Resum: Resonance roughly means that two or more oscillations interact in one dynamical system. This may result in another oscillatory motion, or in some form of quasi-periodicity, or even in chaos. The organisation of the dynamics in parameter space often has a quite universal geometry, where resonance tongues can play a role. In the talk we discuss two contexts of resonance and indicate what is this geometry in each case. One case concerns the Hopf-Naimark-Sacker bifurcation and the other the Hill-Schroedinger equation with (quasi-periodic) forcing.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Nuria Fagella, Dept. de Mat. Aplicada i Analisi, UB .

  Títol: Herman rings and Arnold disks .

  Resum: We consider the family of rational maps, analogue of the Arnold family (1 complex dimensional), and we study the set (in C^2) of parameter values for which the maps possess an invariant Herman ring of a given rotation number (say alpha, a Brjuno number). We call this set D, the Arnold disk of rotation number alpha.

  We give an explicit parametrization of the Arnold disk that shows that D is isomorphic to a disk. More precisely we give a holomorphic and bijective map F: D(0,1) ---> D and compute explicitely the derivative of F at 0 to be F'(0)=(0 , r), where r is the conformal radius of the Siegel disk of a quadratic polynomial (or, in the case of the Arnold family, of the semistandard map).

  As a consequence, we give an estimate of the asymptotic size of the Herman rings as the nonlinear parameter tends to 0. This estimate improves a previous result in [Fagella, Martinez-Seara and Villanueva 2004, to appear in ETDS].

• Hora: 17h30m

  A càrrec de: Xavier Jarque, Dept. de Mat. Aplicada i Analisi, UB .

  Títol: On the Julia set of the exponential family .

  Resum: Let $E_{\lambda}$ the complex exponential family where $\lambda$ is such that the Julia set is the whole plane. I will start by presenting some well known results on the invariant sets sharing itineraries with respect the classical symbolic dynamics (the hairs). Then I will state our results for specific value of the parameter (namely, the Misiurewicz $2\pi 1$) and discuss the existence of diferent kind of Indecompossable Continuum for specific (non regular) itineraries.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h 30m

  A càrrec de: Carles Simo, Dept. de Mat. Aplicada i Analisi, UB .

  Títol: Dynamical systems, numerical experiments and super-computing .

  Resum: Dynamical systems study the evolution models of natural phenomena and the simplified models which help to understand them. They can be given in deterministic form, either by means of ordinary differential equations, partial differential equations or discrete maps. They are useful in all domains of science and technology.

  In their study tools from all areas of mathematics are used. But for systems with some degree of complexity it is impossible to produce a fairly complete description of the evolution in the space of states, and its dependence with respect to parameters, without using numerical techniques. They are essential for concrete applications and very useful even for theoretical studies. They can be seen as an experimental part of mathematics.

  In the last years it has become possible to achieve a generalisation in the systematic use of numerical experiments, due to the availability of large arrays of processors working in parallel with a reduced cost. But the impact of new algorithms has been even larger. This makes feasible to face problems of larger and larger complexity.

  In this talk some aspects of the general methodology and several examples will be presented.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Carsten Lunde Petersen, RUC, Denmark .

  Títol: Rebirth of Receeding limits .

  Resum: Let $\L_{\lamda,p/q}$ denote the set (limb of the connectedness locus $M_\lambda$) of quadratic rational maps with an attracting fixed point of multiplier $\lamda$ and a repelling or parabolic fixed point of combinatorial rotation number $p/q\in ]-\frac 1 2, \frac 1 2]\cap\Q$. I proved that when $\lambda converveges subhorocyclically in $\D$ to the conjugate root $\exp(-i2\pi p/q)$, the limb $\L_{\lamda,p/q}$ receeds to infinity in the modulispace of quadratic rational maps, that is maps in the limb diverge uniformly to infinity as ${|1-\lambda\e^{i2\pi p/q}|}^2/2\Re(\lambda\e^{i2\pi p/q})$ converge to $0$. Adam Epstein studied the fine structure of this degeneration and proved that the period $q$ renormalizations appropriately normalized converge to parabolic quadratic rational maps. In a joint work we prove that the renormalizations converge to maps in the parabolic Mandelbrot set $M_1$ of quadratic rational maps with a parabolic fixed point and connected Julia set. Moreover we study the different limits. (joint work with Adam Epstein)

• Hora: 17h30m

  A càrrec de: David Sauzin, IMCCE, Paris .

  Títol: Skew-products over the Bernoulli shift with infinite ergodic measure .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h30m

  A càrrec de: Joan Herrero, Dept. Enginyeria Quimica, URV, Tarragona .

  Títol: A finite-volume / finite-difference solver for Navier-Stokes equations .

  Resum: The numerical analysis of fluid dynamics is a field of great interest for many engineers, physicists and mathematicians. The rapid development of computers in the last decades makes it possible today to calculate complex three-dimensional flows that were out of reach twenty years ago. Navier-Stokes solvers are traditionally based on finite-difference (or, alternatively, finite-element) spatial discretizations. While spectral (and pseudo-spectral) methods for fluid dynamics have developed spectacularly in the last years, finite-difference (and finite-element) methods are still attractive nowadays because of their easiness to apply to complex flow problems.

  The most relevant features of a Navier-Stokes solver, based on a finite-volume / finite-difference approach, are presented. The conservation equations are first introduced in both continuous and discrete form. Aspects of the numerical implementation that are conceptually relevant and/or decisive for the practical efficiency of the code are then considered. These include the control-volume approach on a staggered grid, the enforcement of the mass conservation (i.e., dealing with pressure), the calculation of the finite-difference approximation coefficients, and the time-integration schemes used.

  Applications of the code to a variety of problems of both practical and fundamental interest are presented. These examples are also used to comment some practical aspects of the calculations such as are grid deployment, selection of the time step, CPU time and physical memory requirements, or visualization of results.

• Hora: 18h

  A càrrec de: Alexander B. Vladimirsky, Dept. of Mathematics Cornell University .

  Títol: A survey of methods for computing global (un)stable manifolds of vector fields .

  Resum: We will compare a number of alternative approaches to approximating k-dimensional invariant manifolds of vector fields in R^n, concentrating on the case k = 2.

  The discussed methods will include:

  • time-rescaled motion along the trajectories [Johnson, Jolly, & Kevrekidis];
  • geodesic level sets propagation [Guckenheimer & Worfolk] and [Krauskopf & Osinga];
  • BVP continuation of trajectories [Doedel];
  • computation of "fattened" trajectories [Henderson];
  • PDE approach [Guckenheimer & Vladimirsky];
  • box covering [Dellnitz & Junge].
  All methods will be illustrated by approximating a two-dimensional stable manifold of the origin in the Lorenz system. The discussion will concentrate on the key algorithmic ideas, accuracy, computational complexity, and the issues related to implementations for the case k > 2. This presentation is based on the upcoming survey paper by Krauskopf, Osinga, Doedel, Henderson, Guckenheimer, Vladimirsky, Dellnitz, and Junge (submitted to Int. J. Bifurcation Chaos).

Dia: Divendres 21 de maig de 2004.

Lloc: Aula 3 , Facultat de Matemàtiques, UB.

• Hora: 18h

  A càrrec de: Jean-Pierre Ramis, Univ. Paul Sabatier, Toulouse .

  Títol: About recent works on non-linear differential Galois theory .

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Mike Shub, Dept. of Mathematics, Univ. of Toronto .

  Títol: Stable Ergodicity .

  Resum: (see joint paper with C. Pugh on the January issue of the Bulletin of the AMS)

• Hora: 17h30m

  A càrrec de: Carles Simó, Dept. Mat. Apl. i Analisi, U. Barcelona .

  Títol: Resonance tongues, instability pockets and spectrum in quasi-periodic Hill--Schrödinger equations .

  Resum: Consider Hill's equation or the 1D Schrödinger equation or more general linear Hamiltonian systems, with a (parametric) quasi-periodic forcing. The problem under consideration is to study the resonance tongues, the regularity of its boundaries, the creation of instability pockets and to derive, from this, relevant properties for the spectrum in the case of Schrödinger equations. This is mainly based on joint works with H. Broer and J. Puig.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

  A càrrec de: Marcelo Viana, IMPA .

  Títol: Simple Lyapunov spectra .

  Resum: The Lyapunov spectrum of an system is the set of its Lyapunov exponents (relative to some especially relevant invariant measure). The spectrum is said to be simple if the number of distinct exponents coincides with the dimension of the underlying space. Equivalently, simplicity means that all Lyapunov exponents have multiplicity 1, where the multiplicity is the dimension of the corresponding Oseledets subspace.

  I shall present an explicit sufficient condition for simplicity, which is satisfied "generically". Moreover, I shall discuss some possible applications.

• Hora: 17h30m

  A càrrec de: Esther Barrabés, Univ. de Girona .

  Títol: Families of periodic horseshoe orbits in the RTBP .

  Resum: In the framework of the Restricted Three Body Problem, families of periodic horseshoe orbits are computed. First, we deal with planar orbits in order to find bifurcations from which families of 3D periodic horseshoe orbits are born. We compute the stability parameters of the spatial orbits and their inclination as well. We see how these parameters vary along the families with respect to the Jacobi constant.

Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h30m

  A càrrec de: Dmitry Treschev, Dept. Mathematics and Mechanics, Moscow State University (Lomonosov University) .

  Títol: Dynamics on the space of quantum observables .

  Resum: We introduce the algebra of quantum observables as the space of ``converging series'' in $\hat x_j$ (the operators of multiplication by $x_j$) and $\hat p_j = -i\hbar\partial / \partial x_j$. In these terms the analogy between quantum and classical mechanics becomes clear and straightforward. In particular, quantum analogs of many classical concepts (Darboux theorem, integrability, action-angle variables, normal forms) naturally appear.

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  Dia: Dimecres 7 de juliol de 2004.

  Lloc: Aula 7 (2on pis), Facultat de Matemàtiques, UB.

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  • Hora: 16h30m

    A càrrec de: Yuri Kifer, Institute of Mathematics, Hebrew University, Jerusalem .

    Títol: Recent Progress in Fully Coupled Averaging .

    Resum: In systems where both slow and fast motions coexist a direct studyi becomes complicated and the averaging principle prescribes to approximate the slow motion by averaging it in fast variables. This setup arises in the study of physical systems which can be viewed as a perturbation of an ideal (for instance, Hamiltonian) system with integrals of motion which start slowly changing after perturbation. When the fast motion does not depend on the slow one the averaging principle works under quite general conditions and this case is well understood by now. The more realistic situation when both motions depend on each other is more complicated and the averaging principle works usually (if at all) not poinwise (in initial conditions) but only in the $L^1$ sense. I shall discuss recent results in this direction, as well, as results (for families of Axiom A systems) about the asymptotic behaviour of the error in averaging, in particular, diffusion approximations and large deviations bounds for it, the latter leading to the stochastic resonance type behavior.

  • Hora: 18h

    A càrrec de: Luis Benet, CCF UNAM (Mexic) .

    Títol: Occurrence of narrow rings by sheperd moons .

    Resum: I discuss, using a billiard as example, a generic scenario for the occurrence of narrow rings which are sheperded, like the F Saturn ring and the $\epsilon$ ring of Uranus. The main assumption is to consider an ensemble of non-interacting particles belonging to the ring. The connection to $1/r$ potentials as well as some extensions will be also considered.

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  Last updated: Tue Jul 6 16:44:22 MEST 2004