Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: A. Celletti, Dipto. di Matematica, U. di Roma "Tor Vergata" .
Títol: KAM stability and Celestial Mechanics .
Resum: KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to "physical systems" for "observable" values of the perturbation parameters. Here, we consider the Restricted, Circular, Planar, Three-Body Problem (RCPTBP),i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small the RCPTBP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCPTBP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points "close" to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCPTBP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Juan José Morales, Dept. Matemàtica Aplicada II, UPC .
Títol: Una nota sobre los transcendentes de Painlevé .
A càrrec de: Ernest Fontich, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Trencament de separatrius exponencialment petit en un cas feblement hiperbòlic. (Treball conjunt amb I. Baldomà) .
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez Ros, Dept. Matemàtica Aplicada I, UPC .
Títol: Escisión singular y bifurcaciones invisibles en algunos billares planos .
Resum: We present a numerical study of some billiard tables depending on a perturbative parameter $ε ≥ 0$ and a hyperbolicity parameter $h > 0$. These tables are ellipses for $ε=0$ and circumferences in the limit $h \to 0^+$. Elliptic billiard tables are integrable and have four separatrices connecting their hyperbolic two-periodic points.
These connections break up when $ε > 0$. As $h\to 0^+$, the area of the main lobes of the resulting turnstile (which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories) behaves like an exponential term $ε \e^{-π^2/h}$ times an asymptotic series $Σ_{j≥ 0} α^ε_j h^{2j}$ such that $α^ε_0 ≠ 0$.
This series is Gevrey-1 of type $ρ=1/2π^2$, so that its Borel transform is convergent on a disk of radius $2π^2$. In the limit $ε \to 0$, the series $Σ_{j≥ 0} α^0_j h^{2j}$ is an analytic function which can be explicitly computed with a discrete Melnikov method. The asymptotic series $Σ_{j≥0} ω^ε_j h^{2j}$ associated to the second exponential term $ε\e^{-2π^2/h}$ has the same properties.
Finally, we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space.
Our computations have been performed in multiple-precision arithmetic (namely, with several thousands decimal digits) and rely strongly on the expansion of the local invariant curves up to very high orders (namely, with several hundreds Taylor coefficients). Our programs have been written using the PARI/GP calculator.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de:F. Javier Muñoz Almaraz, Dept. MAiA, UB .
Títol: Relacion entre las integrales primeras y la continuacion de orbitas periodicas .
Resum:A un sistema con integrales primeras no triviales, no se le puede aplicar el metodo de continuacion de pseudo-longitud de arco de forma directa para encontrar familias de orbitas periodicas. Veremos como un desplegamiento del sistema permite justificar con el teorema de la funcion implicita que las orbitas periodicas del sistema original se organizan en familias. Estas ideas nos permiten desarrollar un metodo de continuacion para los sistemas con integrales primeras.
Tambien comentaremos como se aplican estas tecnicas para la continuacion de orbitas periodicas simetricas respecto de reversibilidades en sistemas con integrales primeras.
Dos ejemplos nos serviran de guia:
Si se desea mas informacion y articulos, puede consultarse la pagina web http://www.ma2.us.es/~javi/investigacion
Trabajo conjunto con J. Galan, E. Freire, A. Vandervauwhede y E. Doedel.
A càrrec de: Askold Perelomov, Inst. of Theor. and Exper. Physics, Moscow i Depto. de Fisica Teorica, U. Zaragoza .
Títol: The Kovalevskaya top: an elementary approach .
Resum: The goal of this talk is to give an elementary and very short solution to equations of motion for the Kovalevskaya top. Our results are based on the original papers by Kovalevskaya, Kotter and Weber.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de:Yuri Fedorov, Dept. Matemàtica Aplicada I, UPC .
Títol: Discrete Nonholonomic LL Systems on Lie Groups .
Resum: Recently the formalism of variational integrators (discrete Lagrangian systems) was extended to systems with nonholonomic constraints.
We apply this formalism to the case when the configuration space is a Lie group, whereas the Lagrangian and the constraints are left-invariant.
As an example, we construct a discrete version of a classical integrable problem of nonholonomic mechanics---the Suslov rigid body. It appears that, in contrast to discretizations of generic systems, the discrete Suslov system preserves a constrained energy integral and the resulting discrete dynamics is very similar to that of the continuous problem.
(Veure també http://arxiv.org/pdf/math.DS/0409415
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jacky Cresson, Equipe de Mathematiques de Besançon, Universite de Franche-Comte .
Títol: Hyperbolicity versus partial hyperbolicity and degenerate crossing .
Resum: A whole description of the transversality-torsion phenomenon, some conjectures about it and a discussion of degenerate (not transversal) crossing.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Joaquim Puig, Dept. de Matematica Aplicada I, UPC .
Títol: Una versio no pertorbativa del teorema de reductibilitat d'Eliasson .
Resum: En aquesta xerrada presentarem un resultat de reductibilitat de sistemes lineals discrets i quasi-periodics, provinents d'equacions discretes de Schrodinger o tipus Harper que exten un resultat d'Eliasson (1991). Aquest tipus d'equacions sorgeixen naturalment en diversos problemes de sistemes dinamics, especialment hamiltonians, aixi com en models de la fisica quantica. Veurem que es possible demostrar reductibilitat per gairebe tot (en sentit de la mesura de Lebesgue) valor de l'energia i per una frequencia diofantica si el potencial (o forc,ament) es mes petit que una certa constant que no depen de la condicio diofantica precisa, malgrat que el problema admet una formulacio KAM. Acabarem amb unes quantes aplicacions d'aquest resultat.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jorge Galan, Universidad de Sevilla .
Títol: A bifurcation approach to the Riemann Hypothesis .
Resum: In a remarkable paper in 1859 Riemann obtained an analytic formula for the number of primes under a preassigned value. He introduced the so called Riemann zeta function as a function of a complex variable defined in the half plane Re(s)> 1 as an absolutely convergent series and extended it to the whole complex plane by analytic continuation except for a single pole at s=1. Moreover he showed that it satisfies a functional equation that establishes a certain symmetry along the critical line Re(s)=1/2. The Riemann Hypothesis precisely affirms that the nontrivial zeros of the zeta function lie exactly on the critical line; i.e. they have a real part equal to one half. In the present work we approach the problem from a naïve point of view. We introduce an appropriate symmetric function of two real variables F(x,y) and establish a connection between the zeros of the zeta function and the branching bifurcation points of the solution of F(x,y)=0. Making use of the techniques of bifurcation theory, numerical continuation of solutions and a fluid dynamic interpretation we try to shed some light on this notoriously difficult problem.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Joseph Gerver, Dept. of Mathematics, Rutgers University .
Títol: Noncollision Singularities: Do Four Bodies Suffice? .
Resum: A heuristic model is presented for a solutions of the planar Newtonian four-body problem which has a noncollision singularity.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, Dpt. Matemàtica Aplicada i Anàlisi, UB .
Títol: Attractors in 3D diffeomorphisms .
Resum: We shall examine different families of 3D diffeomorphisms, like Henon maps driven by circle maps and their generalisations, and 3D Henon maps, a generalisation of the classical Henon maps. Different facts are proven, e.g. the existence of Henon-like attractors and of a discrete version of 3D Lorenz-like attractors. In contrast with the first type, last ones exist for open sets in the parameter space. Furthermore both families appear in a natural way when studying several unfoldings and return maps. A numerical scanning of the behaviour of these families, mainly using Lyapunov exponents, will be also presented. In particular we shall comment on the correct interpretation of the numerical results. Wrong interpretations can lead to believe that SNA exist. Finally we shall discuss several conjectures and open problems.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Martin Hemke, University of Kiel, .
Títol: Typical Orbits in the Julia Set .
Resum: We call a point or orbit typical, if the set of accumulation points of the forward orbit, namely its Ω-limit set, coincides with that of almost every point in the Julia set. It is an open question, whether there always exists a typical orbit, and related to the question whether the Julia set of a polynamial may have positive measure. We will study this question for the family of functions, whose set of singularities of the inverse is finite, counting multiplicity.
A càrrec de: Carles Simó, Dpt. Matemàtica Aplicada i Anàlisi, UB .
Títol: Attractors in 3D diffeomorphisms (2ond part) .
Resum: We shall examine different families of 3D diffeomorphisms, like Henon maps driven by circle maps and their generalisations, and 3D Henon maps, a generalisation of the classical Henon maps. Different facts are proven, e.g. the existence of Henon-like attractors and of a discrete version of 3D Lorenz-like attractors. In contrast with the first type, last ones exist for open sets in the parameter space. Furthermore both families appear in a natural way when studying several unfoldings and return maps. A numerical scanning of the behaviour of these families, mainly using Lyapunov exponents, will be also presented. In particular we shall comment on the correct interpretation of the numerical results. Wrong interpretations can lead to believe that SNA exist. Finally we shall discuss several conjectures and open problems.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Francesco Calogero, Dipartimento di Fisica, Universita di Roma "La Sapienza" .
Títol: Isochronous systems, and the transition from ordered to disordered motions .
Resum: A simple trick has been recently introduced that allows to modify a quite general evolution system, generating thereby a modified system that possesses an open domain of initial data, having full dimensionality in the space of initial data, such that all the motions emerging out of it are isochronous.
In this sense one can assert that "isochronous systems are not rare". Moreover, many of the isochronous systems obtained in this manner are quite interesting.
In this presentation these developments will be reviewed and several examples of classical many-body problems having this property will be discussed.
When the initial data are chosen outside the region of isochronicity, motions with higher periods might arise, or possibly aperiodic or chaotic motion. The mechanism that leads to these differents types of behaviour will also be discussed.
A càrrec de: David Gómez-Ullate, Departament de Matemàtica Aplicada I, UPC. .
Títol: Transition from simple to complicated motions seen as a travel on a Riemann surface .
Resum: One of the examples presented in the previous talk will be examined in greater detail: a system of three coupled first order ODEs. We have chosen this toy model to illustrate the mechanism of the transition from simple periodic orbits to motions of increasing complexity. The topological properties of the Riemann surfaces associated to the solutions of the system will be investigated since they are intimately related to the period and complexity of the orbit. This will enable us also to discuss the sensitive dependence of the system on initial conditions. The theoretical predictions for this model will be supplemented by the results of a numerical integration of the equations of motion. Animations of different motions will be shown.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Henk Broer, Dept. of Mathematics, Univ. of Groningen .
Títol: Unicity of KAM tori .
Resum: The classical KAM theorem establishes persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. It is a long standing question in how far the perturbed tori are unique. Using the fact that at the level of tori, there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine union of tori of full measure.
(joint work with Floris Takens)
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alexei Tsygvintsev, Ecole Normal Sup. de Lyon .
Títol: On the Stieltjes summability of Poincare-Sundman power series solutions of the 3-body problem .
Resum: The problem stated by Weierstrass in the 1880's asks for a method for the construction of power series solutions of the Newtonian 3-body problem converging on the entire t-axis. Karl Sundman observed in 1912 that it is always possible for solutions of non-zero angular momentum but the convergence was so slow that it is really useless in practice.
In this talk we discuss an alternative representation of global solutions of the 3-body problem within the framework of the analytic theory of continued fractions. Based primarily on rigidity properties of holomorphic maps, this approach gives a better capture of the global dynamics.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez Ros, Dept. Matemàtica Aplicada I, UPC .
Títol: Rotura de curvas invariantes resonantes en billares y billares duales asociados a circunferencias perturbadas .
Resum: Se pueden asociar dos aplicaciones twist preservando area a una curva convexa cerrada diferenciable: la aplicacion del billar clasico y la aplicacion del billar dual. Cuando la curva es una circunferencia, son integrables y sus espacios de fases estan foliados por curvas invariantes. Las curvas invariantes con un numero de rotacion racional son resonantes y no persisten bajo perturbaciones genericas.
Presentamos un criterio de tipo Melnikov obtenido mediante tecnicas variacionales para saber cuando un curva resonante concreta de esas aplicaciones del billar se rompen bajo una perturbacion concreta de la circunferencia.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Imma Baldoma, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Exponentially small splitting of heteroclinic orbits in a family in R3 .
Resum: We consider a field in R3 such that the origin is a fixed point with eigenvalues 0 and ±αi. After perturbing this system, we consider the second order normal form. By means of standard variable changes and scaling, we find ourselves in a singular perturbative context. The singular parameter is the angular velocity of the solutions.
The non perturbative terms are the ones coming from the normal form. When we consider the normal form, we get a heteroclinic curve associated with two saddle-type fixed points. Our aim is to measure the distance between the invariant manifolds coming from this heteroclinic connection of the full system. This distance will be exponentially small as a function of the angular velocity of the solutions.
A càrrec de: Stefano Luzzatto, Math. Dept., Imperial College, London .
Títol: Computable conditions for the verification of chaos in one-dimensional dynamics .
Resum: I will present some recent work concerning the verification of the occurrence of stochastic dynamics in families of dynamical systems. This problem is generally difficult because stochastic dynamics only occurs for topologically nowhere dense set of parameters. However, it is also interesting because this set often has positive probability.
I will discuss a combination of numerical, geometrical, and probabilistic techniques which make it possible to obtain explicit bounds for the measure of such a set. As an application of these results we obtain a first ever lower bound for the measure of the set of parameters corresponding to stochastic behaviour in the quadratic logistic family.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jordi Villanueva, Dept. Matemàtica Aplicada I, UPC .
Títol: Càlcul Numèric de Nombres de Rotació Diofàntics d'Aplicacions del Cercle .
Resum: Donat f:S->S un homeomorfisme del cercle preservant l'orientació, és ben conegut que podem definir el seu nombre de rotació, rot(f), com la mitjana asimptòtica de l'angle de rotació dels iterats per f de qualsevol punt del cercle, éssent independent del punt triat. Malauradament, el càlcul de rot(f) a partir de la seva definició convergeix molt lentament, i requereix molts iterats per obtenir una aproximació numèrica raonablement bona.
En aquesta xerrada presentem un mètode numèric que permet calcular aproximacions de rot(f), en alguns casos amb molta precisió, sota les hipòtesis de que f és analítica (o almenys prou diferenciable) i rot(f) diofàntic. En aquest contexte, usem fortament que l'aplicació és analíticament (o diferenciablement) conjugada a una rotació. El mètode es fonamenta en el càlcul de certes mitjanes dels iterats de f, de les quals, via la conjugació, en coneixem la seva expressió asimptòtica, seguit d'un procés d'extrapolació per obtenir rot(f).
A més de la formulació del mètode propiament dit, discutirem estimacions analítiques de l'error, algunes aplicacions i generalitzacions.
(treball conjunt amb T. Martínez-Seara)
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Frederic Gabern, Dept. Matemàtica Aplicada Anàlisi, UB .
Títol: Theory and Computation of Non-RRKM Lifetime Distributions and Rates in Chemical Systems with Three or More Degrees of Freedom .
Resum: The computation, starting from basic principles, of chemical reaction rates in realistic systems (with three or more degrees of freedom) has been a longstanding goal of the chemistry community. Our current work, which merges tube dynamics with Monte Carlo methods provides some key theoretical and computational tools for achieving this goal. We use basic tools of dynamical systems theory, merging the ideas of Koon et al. [Chaos 10, 427 (2000)], Gomez et al. [Nonlinearity 17, 1571 (2004)] and De Leon et al. [J. Chem. Phys. 94, 8310 (1991)], particularly the use of invariant manifold tubes that mediate the reaction, into the start of a comprehensive theory of lifetime distributions and rates of chemical reactions and scattering phenomena, even in systems that exhibit non-statistical behavior. Previously, the main problem with the application of tube dynamics has been with the analytical evaluation of volumes in phase spaces of arbitrary dimension. The present work overcomes this hurdle with some new ideas and implements them numerically. Specifically, an efficient algorithm that uses tube dynamics to provide the initial bounding box for a Monte Carlo volume determination is used. The combination of a fine scale method for understanding the phase space structure (invariant manifold theory) with statistical methods for practical computations (Monte Carlo) is the main novel contribution of this model problem, is not restricted by dimension, and is useful for higher degree of freedom systems as well.
(Joint work with W.S. Koon, J.E. Marsden and S.D. Ross)
A càrrec de: Yuri Fedorov, Dept. Matemàtica Aplicada I, UPC .
Títol: Algebraic closed geodesics on quadrics .
Resum: Generic geodesics on a two-dimensional ellipsoid $Q$ in the celebrated Jacobi problem are known to be quasiperiodic.
We show that they become periodic when the corresponding spectral curve is a hyperelliptic tangential cover of an elliptic one. In this case the geodesics themselves become elliptic curves in R^3. Each type of hyperelliptic tangential cover results in a one-parametric family of closed geodesic on $Q$.
Using the addition theorem for elliptic functions, we propose a simple approach to explicit description and classification of algebraic surfaces in $R^3$ that cut out closed geodesics on the ellipsoid. A gallery of such geodesics will be presented.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Anatoly Neishtadt, Space Research Institute Moscow, Russian Academy of Sciences .
Títol: On adiabatic perturbation theory for systems with elastic collisions .
Resum: Several problems with elastic collisions are considered: a ball between slowly moving walls, rays in slowly irregular waveguide with reflecting walls, an adiabatic piston. The justification of the applicability of a formal scheme of adiabatic perturbation theory to these problems is given.
(Joint work with I.V.Gorelyshev and A.I.Neishtadt)
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Lluis Benet, Centro de Ciencias Fisicas, UNAM .
Títol: Strands of narrow planetary rings: phase space considerations .
Resum: The aim of the talk is to present recent results based on a billiard system that explain the occurrence of strands in narrow planetary rings. I will first describe the dynamics of the billiard system and the connection to more realistic approaches. Then, I will show how small eccentricities yield strands.
Lloc: Aula T2 (antiga Aula 7, 2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alejandra González, Camerino, Italia .
Títol: Smoothing and geometric properties .
Resum: We consider $U\subset\RR^{d}$, an open and connected subset, endowed with an analytic exact symplectic (volume or contact) structure $\Omega$. We show that, under certain conditions, it is possible to define a linear operator $T_t$ taking functions in $C^{\ell}(U)$ which are exact symplectic (volume or contact) maps into analytic exact symplectic (volume or contact) maps and such that the sup norm of $T_t[f]-f$ goes to zero as $t$ goes to $\infty$. The operator $T_t$ is obtained by a suitable transformation of an analytic smoothing $S_t$ defined by the convolution operator with an analytic kernel.
A càrrec de: Oliver Diaz, University of Texas at Austin .
Títol: Sistemas dinámicos unidimensionales con ruido aleatorio débil .
Resum: Estudiamos el comportamiento asintótico de mapeos unidimensionales perturbados con ruido aleatorio débil. Obtenemos condiciones suficientes para la existencia de un Teorema de Límite Central para ciertas escalas de la magnitud del ruido.
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Last updated: Mon Jul 18 14:55:35 MEST 2005