Dia: Dimecres 6 de setembre de 2006.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Vadim Kaloshin, California Institute of Technology, Pasadena, USA .

Títol: Nonlocal instability of the planar 3 body problem .

Resum: The Restricted Planar Circular 3 Body Problem (RPC3BP) which is the simplest nonintegrable 3 body problem. Usually it is viewed as a model for planar either Sun-Jupiter-Asteriod or Sun-Earth-Moon system. Stability v.s. instability of such a system is one of long standing problems. Using Aubry-Mather theory, Mather variational method, and numerical analysis, we managed to prove existence of a rich variety of unstable motions. For example, an Asteroid could have a nearly elliptic orbit of say eccenticity 0.75 in the past and escape to infinity along nearly parabolic orbit of eccentricity more than 1 in the future. These motions could be interpreted as Arnold diffusion for this system. Instability results for RPC3BP imply instability for more general planar 3 body problems. This is a joint work with T. Nguyen and D. Pavlov

Dia: Dimecres 8 de novembre de 2006.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Henk Broer, Dept. Mathematics, U. Groningen .

Títol: On parametrized KAM theory .

• Hora: 17h30m

A càrrec de: H. Hanssmann, Dept. Mathematics, U. Utrecht .

Títol: On the destruction of resonant Lagrangean tori in Hamiltonian Systems .

Resum: Poincaré's fundamental problem of dynamics concerns the behaviour of an integrable Hamiltonian system under a (small) non-integrable perturbation. Under rather weak conditions K(olmogorov)A(rnol'd)M(oser) theory settles this question for the majority of initial values. The perturbed motion is (again) quasi-periodic, the number of frequencies equals the number of degrees of freedom. KAM theory proves such Lagrangean tori to persist provided that the frequencies are bounded away from resonances by means of Diophantine inequalities.

How do Lagrangean tori with resonant frequencies behave under perturbation? We concentrate on a single resonance, whence many n-parameter families of n-tori are expected to be generated by the perturbation ; here n+1 is the number of degrees of freedom. For quasi-convex systems we explain the pattern how these families of lower-dimensional tori come into existence, and then discuss what happens if the assumption of quasi-convexity is dropped.

Dia: Dimecres 29 de novembre de 2006.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Peter Veerman, Dept. of Mathematics and Statistics, Portland State University .

Títol: Avalanches and Granular Dynamics .

Resum: When you drop sand on top of a sand pile, initially the mound will become steeper and steeper. But finally, the slope will stop increasing and the falling particles will tend to roll far down the mound. The equations for any realistic model of such a process would be much too complicated to glean substantial analytic insight from. We therefore develop a paradigm for this kind of dynamics. We give a mathematical treatment of the dynamics of a particle falling down an inclined slope with a simple periodic profile (a "staircase"). The idea is to develop new insights about the dynamics of individual particles in an "avalanche" motion. Such motions happen in granular material when it is poured on a pile. This is a currently very active area of research with many applications. Our basic question is: What causes an avalanche, and what are the circumstances under which it stops?

Dia: Dimecres 20 de desembre de 2006.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Santiago Ibañez, Depto. Matematicas, Univ. de Oviedo .

Títol: Una panoramica sobre la singularidad Hopf-Cero .

Resum: Se denomina singularidad Hopf-Cero a todo punto de equilibrio de un campo de vectores tridimensional donde el sistema linealizado presenta una pareja de autovalores imaginarios puros y un autovalor cero. La clasificacion topologica de estas singularidades al nivel de la codimension dos fue obtenida por Takens en 1974. Desde entonces la resolucion de los correspondientes problemas de bifurcacion local ha sido, y todavia es, objeto de un gran numerode esfuerzos. En esta charla se discutiran los avances obtenidos, los problemas abiertos y las perspectivas futuras. Solo nos ocuparemos de uno de los tipos topologicos de los seis que se presentan con codimension dos. Se trata de aquel que, como se explicara, es susceptible de ser germen de comportamientos caoticos.

El recorrido por la literatura comenzara con los resultados de Guckenheimer y Holmes sobre el diagrama de bifurcacion en familias invariantes por rotaciones (cualquier despliegue de una singularidad Hopf-Cero admite una forma normal que a cualquier orden es invariante por rotaciones).Se discutiran despues los resultados de Broer y Vegter y de Gaspard sobre la existencia de conexiones homoclinicas de tipo Shilnikov (y por lo tanto de dinamica caotica) en los despliegues de la singularidad. Por ultimo, como recientes desarrollos de las anteriores perspectivas, se comentaran lostrabajos de Lamb,Teixeira Webster y de Champneys y Kirk.

El principal problema abierto que se discutira es el de la obtencion de condiciones analiticas que permitan responder (afirmativa o negativamente) a la pregunta de si un despliegue dado incluye conexiones homoclinicas de tipo Shilnikov. Para abordar esta cuestion uno ha de enfrentarse con problemas de escisiones exponencialmente pequenyas de separatrices donde los avances mas significativos se deben a resultados recientes de Baldoma y M-Seara. En este mismo contexto, la singularidad Hopf-Cero considerada se conectara con el estudio del despliegue de la singularidad nilpotente tridimensional de codimension tres y con el denominado sistema de Michelson. Como veremos se trata este ultimo de un despliegue muy especifico de la singularidad Hopf-Cero que ha sido ampliamente estudiado en la literatura. De nuevo haremos una breve presentacion de los resultados obtenidos para esta familia.

Para concluir se esbozaran los enfoques que pretendemos desarrollar en el trabajo que actualmente estamos elaborando.

• Hora: 17h30m

A càrrec de: Stefanella Boatto, Depto de Matematica Aplicada, Universidade Federal de Rio de Janeiro .

Títol: Periodic solutions of the Euler equation on surface with constant curvature .

Resum: Classes of steady and periodic solutions are investigated for the incompressible Euler equation. Of particular interest is the stability of "discrete solutions" of the type of point-vortices on surfaces with constant curvature, on domains without boundaries. The study makes use of an infinite dimensional Hamiltonian formulation of the vorticity equation when the rotation of a planet is taken into account [see T.G. Shepherd, Hamiltonian Dynamicsi, Encyclopedia of Atmospheric Sciences, Academic Press, pp 929--938, 2003; B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176 (2003), no. 1, 116--144.].

Dia: Dimecres 24 de gener de 2007.

## A Swinging Atwood's Machine and Non-Integrability day

Lloc: Aula B1 (planta baixa, costat Gran Via), Facultat de Matemàtiques, UB.

• Hora:   9h30m Swinging Atwood's Machine: Experimental Results (1). Jose-Philippe Perez and Olivier Pujol.
• Hora: 10h30m Coffee break
• Hora: 11h15m Galoisian Obstructions to Integrability from the First-Order Variational Equations (2). Juan J. Morales-Ruiz.
• Hora: 12h30m Galoisian Obstructions to Integrability from the Higher Variational Equations (3)(tentative title and abstract). Jean-Pierre Ramis.

• Hora: 16h       Symbolic Computation for the Variational Approaches to Integrability; Application to the Atwood Machine(4). Jacques-Arthur Weil and Sergi Simón.
• Hora: 17h       Coffee break
• Hora: 17h30m Numerical Tools to Detect Non-Integrability(5). Carles Simó.

Resums:

1. This presentation focuses on some experimental results relative to the Swinging Atwood's Machine (SAM) dynamics, which, to our knowledge, has not been yet investigated in this way. SAM is a nonlinear system with two degrees of freedom derived from the well-known simple Atwood machine. The latter was built in 1784 by George Atwood - a Physics teacher from London who constructed various apparatus in order to illustrate his Physics lectures - to experimentally demonstrate the uniformly accelerated motion of a falling system in the earth gravity field g with an acceleration lower than g. In Atwood's machine, two masses are mechanically linked by an inextensible thread and a pulley, whereas in SAM one of the mass (m) is allowed to swing in a plane while the other mass plays the role of a counterweight (M), so that SAM is a parametric pendulum with a variable length depending on mu = M/m.

For about twenty years, many theoretical and numerical studies have been performed about the mechanical behaviour of SAM. Particularly, it has been shown that SAM is a very rich dynamical system exhibiting enormous different trajectories depending on mu-values. For mu > 1 it has been qualitatively shown by using Poincare maps that SAM revealed regular and chaotic behaviours, the latter becoming prominent as mu increases. An interesting and surprising result is for mu = 3 for which SAM appears to be integrable, a conclusion also supported theoretically by Hamilton Jacobi's theory.

First, we will describe in detail the constructed apparatus by showing some photos of SAM and its different elements. A schematic representation will be then given in order to derive the equations of motion and videos on different experiments performed will be shown; in particular for the value mu = 3. Experimental results coming from analyses of the video sequences will be then presented and a re-analysis of the motion is proposed. Especially the contribution of the pulleys is no more neglected contrary to the previous theoretical studies since experiments show that they can rotate around their revolution axis and that their dimension has to be taken into account. Numerical simulations to solve equations of motion will be also presented and compared with previous studies. Finally, some perspectives for further theoretical studies will be proposed, in particular concerning the integrability of SAM.

2. Given an autonomous dynamical system with an integral curve Gamma, the variational equations of the system along Gamma are the linear homogeneous system whose principal fundamental matrix is the linear part of the flow of the system along Gamma. A distinguished example, though by far not the only one, is a Hamiltonian system such as the ones commonly studied in Astrophysics. Although such systems are considered real in a widespread number of cases, everything will be considered in the complex analytical setting here.

So far we have a (generally nonlinear) system and a linear system linked to the former. Heuristics of the non-integrability result central to this talk are firmly rooted in the following: if we assume the initial system "integrable" in some reasonable sense, then the corresponding variational system along any integral curve must be also integrable in the sense of linear Galois differential theory. Any attempt at ad-hoc formulations of this heuristic principle has only one possible drawback: the need for a notion of "integrability" of the original system; but such a specific notion is made available in the Hamiltonian case by the Liouville-Arnold Theorem. With this as a premise, the main result presented in this talk, proven by Juan J. Morales-Ruiz and Jean-Pierre Ramis, is the following: if an n-degree-of-freedom Hamiltonian H has n independent first integrals in pairwise involution defined on a neighborhood of an integral curve Gamma, then, the identity component of the Galois group of the variational equations of X_H along Gamma is a commutative group.

A general overview on the result will be done, as well as the assessment of some of the basic framework leading, including the preliminary lemmas, one of which is far more than just subservient to the final result. A comment will be made on how a special case, namely that of classical Hamiltonians, allows for a dramatic simplification of the conditions of the main theorem.

3. Inasmuch as in the previous talk, and although real Hamiltonians are considered for the most part in applications of differential Galois theory, a complex analytical setting is maintained for all Hamiltonian systems X_H considered in the present talk. In such a setting, a conjecture was made by Juan J. Morales-Ruiz calling for a natural extension of the previous Morales-Ramis criterion to higher variational equations; this conjecture was based on suggestion by Carles Simo, in turn stemming from numerical and analytical evidence provided by the study of the higher terms in the jet for a number of special cases. Since, while nonlinear in and of themselves, higher variational equations are easily proven to be equivalent to linear systems (after adjoining an adequate set of redundant variables), Galois differential theory finds no theoretic obstacle to its application in this extended problem.

With this fact in mind, and in joint work with Ramis, a proof was finally obtained by all three authors for the conjectured result: a necessary condition for the meromorphic complete integrability of X_H is the commutativity of the Galois group G_k of each variational equation of arbitrary order k>0 along any integral curve Gamma.

With this result rigorously proven, it has subsequently been possible to close important open problems of integrability which had eluded the criterion presented in the previous talk. An example will be exposed here as well as a general outlook on the algebraic framework surrounding the central result.

4. Abstract not yet available

5. A preliminary tool is a good integration routine for ODE. High order Taylor methods, with a truncation error small even compared to the computer epsilon, are good candidates. Main integration errors are due to round off errors and the propagation of these ones thourgh the dynamics of the system. With this tool one can face different alternatives to put in evidence the non-integrability:

1. Poincare maps P: computed on a given level of energy and, eventually, taking also into account other known integrals. First and higher order differentials of P can also be computed numerically.
2. Computation of Lyapunov exponents: either the maximal one or all of them. Several methods can be used to "filter" the wild behavior and speed up convergence.
3. Rotation number and, in general, frequency analysis. They are based on topological tools and in a collocation method for the fine tuning of frequencies.
4. Computation of homoclinic orbits and/or heteroclinic chains and checks of transversality. Using a) this is relatively easy in the case of hyperbolic periodic orbits. It becomes a more subtle problem when the problem is very close to integrable (possible exponentially small phenomena) and in the case of partially normally invariant tori (because of dimensionality).
5. Direct tests of the failure of the necessary integrability conditions coming from talks (2) and (3).

Dia: Dimecres 31 de gener de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 17h

A càrrec de: David Gomez-Ullate, U. Complutense de Madrid .

Títol: Dynamical systems on infinitely sheeted Riemann surfaces .

Resum: Our work is part of a program whose aim is to understand the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time τ. In this talk we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surfaces are known to be infinitely sheeted coverings of the complex time plane, ramified at an infinite set of points whose projection in the τ-plane is dense. The main novelty of this work is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on the Riemann surface that are projected to a circle on the complex plane τ. Due to the branching, the solutions may have different periods or may not be periodic at all. This is joint work with Yuri Fedorov (UPC).

Dia: Dimecres 14 de febrer de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Maciej Capiński, Math. Department, University of Science and Technology, Krakow .

Títol: Transition Chains in the Planar Restricted Elliptic Three Body Problem .

Resum: In the planar restricted circular three body problem, for the values $C$ of the Jacobi constant smaller but close to the value $C_2$ associated with the critical point $L_2$ , there exists a family of the Lyapunov periodic orbits around the equilibrium point. We will show that when the planar restricted elliptic three body problem is considered as a perturbation of the circular problem, most of the Lyapunov orbits persist and are perturbed into a Cantor set of invariant tori. We will also show that there exist transition chains between the tori, which arise from transversal intersections of the corresponding invariant manifolds. These intersections are not restricted to a constant energy manifold and each transition involves a change of energy.

• Hora: 17h30m

A càrrec de: Gonzalo Contreras, CIMAT Guanajuato, Mexico .

Títol: Geodesic flows with positive topological entropy, twist maps and dominated splittings .

Resum: We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a C^2 generic Riemannian metric has a non-trivial hyperbolic basic set in its geodesic flow.

Preprint available in http://www.cimat.mx/~gonzalo/papers/

Dia: Dimecres 21 de febrer de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Maciej Capiński, Math. Department, University of Science and Technology, Krakow .

Títol: Covering Relations and Non Autonomous Perturbations of ODEs .

Resum: Covering relations are a topological tool for detecting periodic orbits, symbolic dynamics and chaotic behavior for autonomous ODEs. We will first introduce the method of the covering relations and then extend it onto systems with a time dependent perturbation. As an example we will apply the method to non-autonomous perturbations of the Rössler equations to show that for small perturbation they possess symbolic dynamics.

Dia: Dimecres 28 de febrer de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Piotr Zgliczynski, Institute of Computer Science, Jagiellonian University, Krakow .

Títol: Some computer assisted proofs of bifurcations for ODE and maps .

Resum: I will discuss some aspects of computer assisted proofs of bifurcations. Local ones: period doubling (for Henon map) or symmetry breaking for ODEs and PDEs (Kuramoto-Sivashinski PDE) and the global - the cocoon bifurcation in the Michelson system (this part is a joint work with D. Wilczak and H. Kokubu)

Dia: Dimecres 2 de maig de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Carsten Petersen, Roskilde University, DK .

Títol: The Yoccoz Combinatorial-Analytic Invariant .

Resum: In a joint work with Pascale Roesch we prove that there is a natural dynamically defined bijection between the standard Mandelbrot set and the so called Parabolic Mandelbrot set. Conjecturally this map is a homeomorphism. An essential ingredient in this proof is to be able to identify a quadratic polynomial dynamically. In order to do so we have introduced the Yoccoz Combinatorial Analytic invariant which takes its name from Yoccoz puzzles. In the talk I shall discuss the bijection above and its prerequisites, leading onto a description of the Yoccoz Combinatorial-Analytic Invariant.

Dia: Dimecres 9 de maig de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Josep Sardanyes, UPF .

Títol: Hypercycle kinetics: from dynamical systems to evolution .

Resum: The hypercycle is a network of catalytically-coupled self-replicating macromolecules. This dynamical system has been suggested to be involved in key evolutionary steps in prebiotic evolution towards the first forms of life. Several evolutionary advantages are attributed to these kind of networks. For instance, the possibility of overcoming the error threshold found in error-prone self-replicating molecules. Moreover, the analysis of the bifurcation scenario towards the extinction of the hypercycle shows that this network is able to delay its extinction by means of a ghost i.e. saddle remnant, in phase space. The ghost involves a delayed transition towards the collapse of the hypercycle, causing a memory effect in concentration phase space. This dynamical property could have been relevant in a fluctuating environment.

Dia: Dimecres 16 de maig de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Marco Antonio Teixeira, Depto. Matematica, Univ. Campinas, Brazil .

Títol: Invariant varieties of relay systems .

Resum: We study the geometric qualitative behavior of a class of discontinuous vector fields in 4D around typical singularities. We are mainly interested in giving conditions under which there exist one-parameter families of periodic orbits, result which can be seen as an analogous to the Lyapunov Center Theorem. The focus is on certain discontinuous systems having some symmetric properties.

Dia: Dimecres 30 de maig de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Mario Ponce, Univ. d'Orsay .

Títol: Dinamica local de transformaciones holomorfas fibradas .

Resum: Consideramos una transformacion de la forma $(\theta,z)\mapsto(\theta+\alpha,f_{\theta}(z))$ a fibras holomorfas. Estudiaremos el comportamiento local de esta transformacion en una vecindad de una curva invariante. Veremos que esta curva juega el rol de un centro en torno del cual la dinamica se organiza, controlada por algunas datos infinitesimales (multiplicador, numero de rotacion fibrado) asociados a la curva. Si el tiempo lo permite, mostraremos que la existencia de una tal curva invariante esta ligada a propiedades aritmeticas de los numeros de rotacion.

Dia: Dimecres 13 de juny de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Dmitri Treschev, Moscow State Univ. i Russian Academy of Sciences .

Títol: Gibbs entropy and its physical meaning .

Resum: Let (M,μ) be a measure space and let ν be another measure on M: dν = ρ dμ, (the function ρ is said to be the Radon-Nikodim derivative dν/ dμ). We define the Gibbs entropy $s\left(\rho \right) = \int \rho log\rho d\mu$ and some of its modifications (the coarse-grained entropy).

We plan to discuss the question: how physical are these objects in the sense:

• do they grow if a μ-preserving dynamics appears?
• what properties of the dynamics are responsible for this growth?
• in what extent this "growth" can be independent of the arbitrariness of the construction?

Here one should keep in mind that if the dynamics is reversible, some strong restrictions are imposed on any entropy-like dynamical quantity.

• Hora: 17h30m

A càrrec de: Maria Eugenia Sansaturio i Oscar Arratia, Univ. de Valladolid .

Títol: A frenzied week with Apophis .

Resum: On Dec 20, 2004 the Minor Planet Center issued the MPEC 2004-Y25 announcing the discovery of a new NEA with designation 2004 MN4.

Only two days later, when the Christmas holidays were about to begin, it was already apparent that this asteroid, currently known as Apophis, would be notorious: our close-approach monitoring system, CLOMON2, was already showing a Virtual Impactor (VI) in 2029 reaching the level 2 in the Torino Scale... the first asteroid to reach that level since our monitoring system had been opperational. However, this was just the beginning of what it was to come in the subsequent days.

In this lecture we will give an overview of the NEODyS-CLOMON2 system and provide the details on how Apophis' collision scenario evolved, the way NEODyS' team handled it and the crazy Christmas holidays we had due to this unexpected guest.

Dia: Dimecres 4 de juliol de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 18h

A càrrec de: David Sauzin, Institut de Mécanique Céleste et de Calcul des éphémérides (IMCCE) .

Títol: On the quasianalyticity properties of spaces of monogenic functions suitable for one-dimensional small divisor problems .

Resum: We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems, in the spirit of M.Herman's work on the linearization of the diffeomorphisms of the circle. Our spaces are formed of functions which are monogenic in the sense of E.Borel: they are C1-holomorphic (in the Whitney sense) on certain compact sets Kj of the Riemann sphere, as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of Kj with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity; in the case of the circle diffeomorphisms, the multiplier is related to the rotation number via the exponential map). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the Kj's.

This is a joint work with Stefano Marmi that we are currently finishing.

Dia: Dilluns 9 de juliol de 2007.

Lloc: Aula B1 (planta baixa), Facultat de Matemàtiques, UB.

Hora: 11h

Lectura tesi doctoral de Sergi Simón Estrada

Títol:On the Meromorphic Non-integrability of Some Problems in Celestial Mechanics.

Dia: Dimecres 11 de juliol de 2007.

Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.

• Hora: 16h

A càrrec de: Zbigniew Hajto, Universitat Politecnica de Cracovia .

Títol: Partial differential fields and Jacobian conjecture .

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Last updated: Mon Aug 31 12:21:59 MEST 2007