Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Tere M-Seara, Dpt Matemàtica Aplicada I, UPC .
Títol: Resurgence of inner solutions for analytical perturbations of the McMillan map .
Resum: In the study of the exponentially small splitting that occurs in certain perturbations of the McMillan map a sequence of "inner equations" has to be considered. An essential step in the measure of the splitting is to know some special solutions of these equations and to be able to give an asymptotic value of their difference.
In this talk we give basic ideas from resurgence theory: we obtain the desired solutions as Borel-Laplace sums of the formal solutions, studying the analyticity of their Borel transforms. Moreover, using 'Ecalle's alien derivations we are able to measure the discrepancy between different Borel-Laplace sums.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Marco Antonio Teixeira, UNICAM, Brasil
. Títol: Stability of Discontinuous Systems
. Resum: In this talk we discuss the qualitative behavior of non-smooth
systems around a typical singularity. We present a general procedure to
classify such singularities. We deal with discontinuous vector fields in
R3 where the discontinuities are concentrated in a codimension-one
surface.
References:
[ST] Sotomayor J. and Teixeira M.A. - Vector fields near the boundary
of a
3-manifold , Lect. Notes in Math., 331, Springer Verlag, 1988, 169-195.
[T1] Teixeira M.A. Stability conditions for discontinuous vector fields, J. of
Di. Eq., 88, 1990, 15-29.
[T2] Teixeira M.A. Perturbation Theory for Non-smooth Systems. In Meyers, Robert (Ed.) -a aparecer em- Encyclopedia of Complexity and Systems
Science, Vol x, pp xx-xxx. Springer New York, to appear in Spring-2009.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Núria Fagella, UB
. Títol: Entire transcendental maps with two singular values and a
persistent Siegel disk
. Resum: We study the class of entire transcendental maps of finite order
with one critical point and one asymptotic value, which has exactly
one finite pre-image, and having a persistent Siegel disk. After
normalization this is a one parameter family
fa with a∈C* which includes the semi-standard map
λzez at a=1, approaches the exponential map when a→0
and a quadratic polynomial when a→∞. We investigate the
stable components of the parameter plane (capture components and
semi-hyperbolic components) and also some topological properties of
the Siegel disk in terms of the parameter
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Carsten Lunde Petersen, IMFUFA, NSM at Roskilde University
. Títol: Hyperbolic components in the space of cubic polynomials
. Resum: The first object to study in a family of dynamical systems are the
loci or components with hyperbolic dynamics or stable dynamics, in short
hyperbolic components. In the space of cubic polynomials there is one
unbounded hyperbolic component and countably many bounded hyperbolic
components. It turns out that there are four types of bounded hyperbolic
components. In this talk I will discuss how we can coordinatize such
components in a dynamically natural way with seemingly good extension
properties to the boundary of such components
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez-Ros (UPC)
. Títol: Jugando con las condiciones de Cayley
. Resum:
Se sabe que la dinamica del billar dentro de un elipsoide es
completamente integrable y, por tanto, facil de entender.
Casi todas las trayectorias se mueven sobre toros y en cada
toro la dinamica es una traslacion paralela con una frecuencia
que depende del toro.
Las condiciones de Cayley sirven para detectar los toros
cuyas trayectorias son periodicas. El resultado original de
Cayley (siglo XIX) trataba el caso 2D: las ellipses.
Dragovich & Radnovich lo generalizaron a dimension
arbitraria en 1998.
Estas condiciones son algebraicas, pero se complican
cuando el periodo es grande. El objetivo de la charla es
reformularlas de forma que permitan obtener algunos
resultados con poco (ejem!) esfuerzo. Por ejemplo,
explicitare cuales son los toros con trayectorias
periodicas de periodos "minimos" en los casos 2D y 3D.
Observaciones:
Sera una charla de pizarra, esbozando incluso alguna
demostracion. El contenido sera esencialmente algebraico;
a saber, operaciones astutas con polinomios.
La parte geometrica, dinamica y visual (ergo, mas divertida)
la explicara Pablo S. Casas en diciembre.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Yuri Fedorov (UPC)
. Títol: The Poisson equations in the nonholonomic Suslov problem:
integrability, meromorphic and hypergeometric solutions
. Resum:
One of the most known simple mechanical nonholonomic system is the
Suslov problem
describing the motion of a rigid body under a certain constraint on its
angular velocity. The problem was reduced
to a simple system on the Lie algebra so(3) and integrated by Suslov in
1903.
However, description of the unreduced motion in space turned out to be
a more complicated task.
In this talk we consider the linear Poisson equations describing this
motion and obtain necessary conditions for their solutions to be
meromorphic.
It appears that, under some extra minor restrictions, these conditions
are also sufficient and lead to
a family of explicit meromorphic solutions, which correspond to rather
special motions of the body in space.
We also give explicit extra polynomial integrals in this case.
In the general case the Poisson equations are transformed into a
generalized
third order hypergeometric equation. A study of its monodromy group
allowed us to solve the long standing
problem on calculation of the "scattering" angle: the angle between the
axes of limit permanent rotations of the body in space.
The talk is based on the results of a recent collaboration with
A Maciejewski and Maria Przybylska (Torun Centre for Astronomy, Poland).
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Pablo Sánchez Casas (UPC)
. Títol: La aplicación frecuencia para billares en elipsoides
. Resum:
e sabe que la dinámica del billar dentro de un elipsoide es
completamente integrable. Casi todas las trayectorias se mueven sobre
toros de Liouville. En cada toro la dinámica es una traslación
paralela con una frecuencia que depende del toro y éste se encuentra
identificado por dos parámetros cáusticos. La aplicación frecuencia es
la que asocia una frecuencia a cada pareja de parámetros cáusticos.
Presentamos algunas conjeturas sobre la aplicación frecuencia, basadas
en experimentos numéricos. Asimismo, describimos su significado
geométrico, dominio y rango, y observamos que se puede extender de
forma continua sobre valores singulares de los parámetros cáusticos,
aunque resulta ser "exponencialmente puntiaguda" en algunos casos.
En cuanto a las trayectorias periódicas, verificamos que son más
abundantes en elipsoides achatados que en los cercanos a esféricos.
Además, en el espacio de parámetros, calculamos las curvas de
bifurcación que marcan la desaparición de los toros con una frecuencia
fijada. Finalmente, mostramos diversas trayectorias de periodos 4, 5 y
6, como ejemplos de periodo mínimo según los distintos tipos de
cáusticas.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Maciej Capiński
. Títol: Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds
. Resum: We present a proof of existence of normally hyperbolic invariant manifolds for
maps. The proof is based on local estimates on derivatives of maps and allows
for rigourous-computer-assisted implementation. We give an example of a driven
logistic map in which even though standard (non-rigorous) computer simulation
gives misleading results, our method can still be applied.
A càrrec de: David Blázquez Sanz (Niigata University, Japan)
. Títol: Integrable non-autonomous linear Hamiltonian systems and their
canonical forms
. Resum: Non-autonomous linear Hamiltonian systems are, in general,
far from integrable. A simple reason we can show is that
its associated extended autonomous Hamiltonian system (that
we obtain by adding the dissipation as an additional variable)
is not linear anymore. In this talk we explore the relation
between the integrability of a non-autonomous Hamiltonian
system and its associated extended autonomous system.
Then, we introduce a suitable notion of integrability, that we
explore mainly in the linear case. For Hamiltonians of
2 and a half degrees of freedom we prove that:
It is done by direct application of Morales-Ramis theorems
on integrability and Kolchin-Kovacic theorems on reduction
of linear differential equations.
We also point out a geometric intepretation of the
differential Galois group as Liouville torus.
This is a Joint work with Sergio Carrillo, and part of his
Ms.C. thesis in Universidad Nacional de Colombia.
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael de la Llave, UT
. Títol: Convergence on differentiable functions in closed sets
. Resum: Many problems in dynamics consider functions defined in nested sets
which converge on a Cantor set, which is the intersection of the nested
sets. This occurs in all the problems when 'exclusion of parameters' takes
place.
The most natural definition of differentiability in closed sets is
Whitney differentiability, whereas in the open approximations one can use
the classical definition.
We show by examples that to conclude Whitney differentiability in the
limit set, it is not enough to estimate the convergence of the
derivatives. One
has to take into account geometric properties of the sets and speed of
convergence. We provide a theorem that shows that, if all this is taken
into account, one can indeed conclude that the limit function is
Whitney differentiable.
This is joint work with Prof. Xuemei Li.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Joaquim Puig, UPC.
Títol: Hi ha vida més enllà de l'Almost Mathieu?
Resum:
L'objectiu d'aquesta xerrada és presentar alguns avenços recents
en la dinàmica de cocicles lineals quasi-periòdics a SL(2,R), en
especial de tipus Schrödinger, i en operadors de Schrödinger amb
potencials quasi-periòdics. Després d'introduir les notacions i
resultats bàsics i de parlar un xic del que se sap per a l'Almost
Mathieu (el model més estudiat) veurem què es pot dir sobre operadors
més generals. Part del material que presentarem es basa en treballs amb
col·laboració amb Àlex Haro i Carles Simó.
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Ester Barrabés, Dept. d'Informàtica i Matemàtica Aplicada, Universitat de Girona.
Títol: A limit case of the "Ring Problem"
Resum: We study the dynamics of an extremely idealized model of a planetary
ring. In particular, we study the motion of an infinitesimal particle moving
under the gravitational influence of a large central body and a regular
n-gon of smaller bodies as n tends to infinity. Our goal is to
gain insight into the structure of thin, isolated rings.
Last updated: Fri Feb 5 15:16:14 2010
Dia: Dimecres, 28 d'octubre de 2009.
Dia: Dimecres, 28 d'octubre de 2009.
Dia: Dimecres, 18 de novembre de 2009.
Dia: Dimecres, 25 de novembre de 2009.
Dia: Dimecres, 2 de desembre de 2009.
Dia: Dimecres, 16 de desembre de 2009.
We also classify integrable linear Hamiltonian systems of
two-plus-one-half degrees of freedom and give their canonical
forms.
Dia: Dimecres, 13 de gener de 2010.
Dia: Dimecres, 20 de gener de 2010.
Dia: Dimecres, 3 de febrer de 2010.
Sessió actual.