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.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Kuntal Banerjee, Lab. Paul Sabatier, Université de Toulouse.
Títol: On the order of contact of the boundaries of the rational tongues in the standard family.
Resum:
It is a well known fact that the order of contact of the
boundaries of the rational tongue Tp/q is exactly q
in the standard family. It is less known that this phenomenon
is related to the fact that the map
z → e2π i p/q z eπ z has only one cycle of petals at
the parabolic point 0. We generalize this result for admissible
and guided families, and we show that the order of contact of
the boundaries Tp/q in such families is a multiple of q.
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Alejandra González, Dpt. MAiA, UB
Títol: A method to study degenerate KAM tori
Resum:
We present a method to study bifurcations of KAM tori, with fixed
Diophantine frequency.
This is based on the construction of a real-valued function, that we call
the potential, in such a way that
invariant tori correspond to critical points of the potential. The method
uses symplectic geometry and
a parametric KAM theorem based on the automatic reducibility of Lagrangian
tori.
This is a joint work with Alex Haro i Rafael de la Llave.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Eva Miranda (UPC)
Títol: Geometria dels sistemes integrables en varietats de Poisson
Resum: Les equacions de Hamilton corresponen a les equacions del flux del
camp Hamiltonià per una 2-forma no degenerada (simplèctica) que
localment escrivim en coordenades Darboux. A vegades aquesta 2-forma presenta
degeneracions (fluxs geodèsics magnetics, transformacions d'Appell de
sistemes de Newton... ) i la 2-forma associada al sistema ja no és una
forma simplèctica. El marc geomètric natural per escriure les
equacions Hamiltonianes associades és el d'estructures de Poisson.
L'objectiu d'aquesta xerrada és presentar alguns resultats de formes
normals locals i semiglobals per sistemes integrables en varietats de Poisson
(que poden tenir dimensió parella o senar). Aquests resultats de formes
normals permeten expressar (al menys parcialment) l'estructura
geomètrica (Poisson) en coordenades de Darboux i, en el cas regular,
l'aplicació moment en coordenades "acció" en un entorn d'una
varietat invariant compacta i per tant simplificar els càlculs.
Presentarem un teorema de coordenades acció-angle per sistemes
integrables (en sentit commutatiu i no commutatiu) en varietats de Poisson
qualsevol en un entorn d'una òrbita compacta.
També presentarem resultats més generals en el cas concret de
varietats de b-Poisson per sistemes Hamiltonians integrables amb singularitats
no-degenerades.
Els teorema de coordenades acció-angle per sistemes integrables en
varietats de Poisson és treball conjunt amb Camille Laurent-Gengoux i
Pol Vanhaecke de l'Université de Poitiers. Els resultats sobre formes
normals en varietats de b-Poisson són treball conjunt amb Victor
Guillemin i Ana Rita Pires del MIT.
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Carles Simó, UB.
Títol: On the role of Dynamical Systems in Celestial Mechanics.
Resum:
Dynamical Systems, DS, can be considered as partly initiated by
the approach taken by Poincaré to study some problems of Celestial
Mechanics, CM, mainly related to the Restricted Three-Body Problem.
During last century and, specially, in the last decades DS had a
tremendous development. The ideas and tools of DS are applied to all
domains of Science and Technology. A key point is the idea that DS aim
to understand the geometry of the phase space, the invariant objects
sitting on it, how they are related and how they change as a function of
parameters. In the talk we shall review the impact of the results of DS
when specialized to CM. More concretely we shall consider some aspects
like: central configurations, topological implications, collision and
regularisation, some kinds of periodic solutions, escape/capture
boundaries, invariant manifolds and its role in confining the dynamics,
regularity and Gevrey phenomena, exponentially small phenomena and
exponentially large diffusion times, practical stability and codimension
1 manifolds. Beyond some purely theoretical aspects, applications will
be made to dynamics in the solar system and to space mission design.
Still CM is one of the most challenging sources of problems for DS.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Mercè Romero, UB.
Títol: How can invariant manifolds can be used to model the spiral arms and rings in galaxies
Resum: Invariant manifolds have been used in different fields in applied
mathematics, celestial mechanics, chemistry. Recently they have been
applied in Galactic Dynamics to model the spiral arms and rings in
galaxies. Here I will expose how to set the equations of motion and study
the dynamics in the case of a galactic model. I will show the different
techniques used and the main results when comparing to observations.
Finally, I will briefly introduce the Gaia mission and how we can interact
to take benefit from each other.
Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Joaquim Font, UB.
Títol: Estudi numèric d'òrbites de segona espècie del RTBP pla i circular
Resum: Fixat el valor de la constant de Jacobi i un cert interval de temps
es descriu el conjunt d'òrbites del RTBP pla i circular que
tenen consecutives trobades amb el primari petit (òrbites de segona
espècie). Considerem la superfície de secció, S,
determinada pel pas d'una òrbita pel pericentre. El conjunt
d'òrbites de segona espècie es trobarà com a
intersecció de les varietats estable i inestable,
Ws,u, de la singularitat de col.lisió amb la
superfície de secció anterior. Usarem la solució del
problema de dos cossos per a calcular un "esquelet" de les corbes
Ws∩ S i Wu∩ S. En aquest cas el
conjunt de punts obtinguts Ws∩ Wu∩ S es
correspon amb la classificació donada per M. Hénon en termes
dels arcs S i T, que es poden usar per a construir un alfabet que descrigui
les òrbites de segona espècie. En el RTBP pla i circular es
calculen, numèricament, òrbites que combinen arcs
quasi-homoclínics de tipus S i T.
Lloc: Aula 101(1r pis), FME, UPC.
A càrrec de: Sergey V. Gonchenko, Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod, Rússia
Títol: Smale horseshoes of new types and Generalized Hénon maps
Resum:
We study hyperbolic dynamics and bifurcations for generalized
Hénon maps in the form
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.
Dia: Dimecres, 10 de febrer de 2010.
Dia: Dimecres, 17 de febrer de 2010.
Dia: Dimecres, 24 de febrer de 2010.
Dia: Dimecres, 3 de març de 2010.
Dia: Dimecres, 10 de març de 2010.
Dia: Dimecres, 17 de març de 2010.
Dia: Dimecres, 24 de març de 2010.
Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez Inostroza, URV.
Títol: Cartesian approach for constrained mechanical systems with three degrees of freedom.
Resum: In the history of mechanics, there have been two points of view for studying mechanical systems: Newtonian and Cartesian.
According the Descartes point of view, the motion of mechanical systems is described by the first-order differential equations in the N dimensional configuration space Q.
In this paper we develop the Cartesian approach for mechanical systems with three degrees of freedom and with constraint which is linear with respect to velocity. The obtained results we apply to discuss the integrability of the geodesic flows on the surface in the three dimensional Euclidian space and to analyze the integrability of a heavy rigid body in the Suslov and the Veselova cases.
In the modern scientific literature the study of the Descarte ideas we can find in the monographic of V.V. Kozlov Dynamical system X, General theory of vortices, Spriger, 2003, in which the author claims "solving dynamics problem is possible inside the configuration space".
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Elena Fantino (IEEC & UPC)
Títol: On invariant manifolds, weak stability boundaries, low-energy transfers, capture and escape in the Sun-Earth-Moon system
Resum: The invariant manifolds of periodic orbits around the L1 and L2 libration points play a role in several dynamical effects that take place in the planar circular restricted three-body problem (PCR3BP) and in the system composed by two PCR3BPs: regions of weak stability boundary, low-energy transfers, temporary captures in the vicinity of the libration points, escape from the system, collisions with the primaries. The implications are of relevance for space mission design and to understand the natural transport in the Solar System. The investigation of all the above issues in the system composed by the Sun, the Earth and the Moon is presented.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Jordi-Lluís Figueras, UB.
Títol: Rigorous analytic validation of invariant tori on the verge of a hyperbolicity breakdown
Resum: We implement computer assisted proofs for the existence of (normally hyperbolic) invariant tori in quasi-periodically forced systems. The proofs are based on a version of the Newton-Kantorovich theorem. We design a Fourier model for managing truncated Fourier series, providing rigorous estimates of the remainders. With these tools, we obtain rigorous and sharp bounds of the errors in the approximations of invariant tori, proving that those invariant tori do exist. We apply these techniques in different scenarios, with special emphasis to validate invariant tori close to their breakdown.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Stefanella Boatto, Instituto de Matematica, Universidade Federal de Rio de Janeiro, Brasil
Títol: Once upon a time some rings of vortices on spheres, plans and arbitrary closed surfaces
Resum: Vortex modeling has a long history. Descartes (1644) used it as a model for the solar systems. J. J. Thomsom (1883) used it as a model for the atom. We consider point-vortex systems, which can be regarded as "discrete" solutions of the Euler equation. Their dynamics is described by a Hamiltonian system of equations. We are interested in polygonal configurations and how their stability depends upon various dynamical variables. In the plane a polygon with seven vortices has been shown to be a special boundary case: polygons with N<7 vortices are (linearly and nonlinearly) stable while polygons with N>7 vortices are unstable. Why should N=7 be any special? What about more than a ring of vortices? What is the effect of the presence of polar vortices on the sphere? Does a given relative equilibrium persist? Others are formed or destroyed? What happens to the vortex dynamics when we deform a sphere ? A bit of Celestial Mechanics' techniques helped us to simplify a problem that has been studied during over a century. Still some open questions remain: in particular how far does it go the analogy between the N-body problem and the N-vortex problem? Is it possible that for some aspects one problem is included in the other?
At the end I shall discuss about another topic of applied mathematics:
Títol: A brief summary about some aspect of mathematics applied to elections.
Resum: In the USA April 2008 was the month of Mathematics of Voting. Various mathematicians considered the various mathematical issues related to Elections. Among others Don Saari, see http://math.uci.edu/~dsaari/Mathematics%20of%20voting.mov
I shall breifly illustrate some results of Mexican and Italian data analysis
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Alejandro Luque (UPC)
Títol: Algunas propiedades de subfamilias críticas de aplicaciones del círculo
Resum: Dada una familia dos-paramétrica de difeos del círculo, el conjunto de parámetros que da lugar a un número de rotación prefijado recibe el nombre de lengua de Arnold. La teoría KAM nos dice que para familias analíticas de difeos del círculo (que cumplen ciertas condiciones de no degeneración) la lengua de Arnold es una curva analítica cuando el número de rotación es Diofántico (ejem, en este caso hay quien hablaría de "pelo" de Arnold). Este resultadao no nos proporciona ninguna información si la família incluye una subfamilia de aplicaciones analíticas con un punto crítico.
El objetivo de esta charla es discutir cálculos numéricos sobre la diferenciabilidad de dichas lenguas (pelos) en el punto crítico. Es bastante curioso observar que las curvas mantienen cierta regularidad, teniendo en cuenta que la conjugación en el punto crítico es Hölder únicamente. Por medio del grupo de renormalización daremos una justificación de los fenómenos observados. En particular, la regularidad de las lenguas (pelos) en el punto crítico se puede controlar en términos del espectro de cierto operador. Por supuesto, en la charla haremos una introducción muy elemental del grupo de renormalización en este contexto.
Respecto a la parte numérica, a nadie le sorprenderá que explotemos un método introducido por T.M Seara y J. Villanueva basado en promediar adecuadamente los iterados y extrapolar. Con miras de evitar una sensación general de "déjà vu", no vamos a dar detalles sobre este procedimiento. A cambio, presentaremos un método de la parametrización para calcular eficientemente pelos (lenguas), obteniendo también información sobre las normas de Sobolev de los objetos implicados.
Esto es un trabajo conjunto con R. de la Llave.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Pau Rabassa, UB
Títol: Bifurcations and renormalization theory for quasi-periodically forced one-dimensional maps
Resum: Consider the Forced Logistic Map in the cylinder (θ, x ) = (θ+ω,α x(1-x) ( 1+ ε cos(θ))), where x is in the real line and θ is in the unit circle, α and ε are parameters and ω is a Diophantine value. This map is known to have very rich dynamics. Concretely the invariant curves of the map undergo period doubling bifurcations and reducibility losses.
Firstly, we will present a numerical study of the interaction of these two phenomena for some regions of the parameter space. We will also present analytic results and additional computations to give an explanation to the first computations on the parameter space. After this,we will present a numerical study of the reducibility loss of the periodic invariant curves for different periods and values of ω which evidence self-renormalizable properties of the family. We will show that these properties can be explained as a consequence of an universal behavior in a suitable class of maps.
For the one dimensional maps, it is well known that the universality can be explained using the renormalization operator. To give an explanation to the universal behavior observed in the q.p. case we will propose an extension of the renormalization operator to the case of one dimensional q.p. forced maps. Several properties of the operator will be presented. We will also analyze the consequences of the renormalization theory for a two parametric family of maps like the FLM. Finally, we will see that the universality of the q.p. can be reduced to the study of the dynamics of the q.p. renormalization operator, giving a desirable explanation of what causes the universality evidenced before.
This is a joint work with A. Jorba and J. C. Tatjer.
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Primitivo Acosta (IMA - Universidad Sergio Arboleda, Colombia)
Títol:
Resum: En mi tesis de doctorado Galoisian Approach to Supersymmetric Quantum Mechanics, disponible en http://ima.usergioarboleda.edu.co/primi/phdthesis.pdf, se presentó un método para transformar ecuaciones diferenciales con coeficientes funciones no racionales en ecuaciones diferenciales con coeficientes funciones racionales. El ingrediente principal es el cambio de variable hamiltoniano, el cual da origen a lo que se denominó "algebrización Hamiltoniana". Se plantea entonces una versión algebrizada de la mecánica cuántica supersimétrica. Se presentarán ejemplos para motivar a los asistentes.
Lloc: Aula B2 (Planta baixa), Facultat de Matemàtiques, UB.
A càrrec de: Marko Lindner, Chemnitz University of Technology, Faculty of Mathematics.
Títol: Spectra of Jacobi operators: Analysis and approximation
Resum: We look at non-selfadjoint bounded linear operators on vector-valued lp spaces and study their spectrum (in particular the essential spectrum) and pseudospectra. Our operators are given by infinite matrices with finitely many diagonals. We give a formula for the essential spectrum which will be discussed for concrete examples of matrices with almost periodic, slowly oscillating or random diagonals. For the case of tridiagonal matrices, we moreover give upper bounds on spectrum and pseudospectrum of the infinite matrix A in terms of pseudospectra of certain finite matrices of order n that are connected to submatrices of A. The latter sets approximate the (pseudo-)spectrum of A in the Haus- dorff metric as n goes to infinity.
Lloc: Aula 102(1r pis), FME, UPC.
A càrrec de: Marian Gidea, Department of Mathematics, Northeastern Illinois University.
Títol: Shadowing and Diffusion in Hamiltonian Systems
Resum: We will discuss some topological methods to prove the existence of shadowing orbits, i.e. orbits with prescribed itineraries, in monotone twist mappings of the annulus. We will also apply topological methods to show the existence of diffusing orbits, i.e., orbits that travel 'far' and 'chaotically' with respect to the action variable, in certain Hamiltonian systems close to integrable
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.
A càrrec de: Maria Aguareles, Universitat de Girona
Títol: Algunes propietats de les solucions de tipus vórtex de l'equació de Ginzburg-Landau
Resum: L'equació de Ginzburg-Landau (GL) és una equació en derivades parcials que modelitza fenòmens de superconductivitat i de superfluidesa. No obstant, les solucions de l'equació GL de tipus vórtex, que són solucions amb un grau de Brower (o índex) no nul, expressades en coordenades polars adopten una expressió especialment simple i esdevenen la solució d'una equació diferencial ordinària de segon ordre no autónoma amb certes condicions a l'origen i a l'infinit. En aquest seminari parlarem doncs d'aquesta darrera equació i veurem que té una única solució monótona creixent que connecta l'origen amb l'1 a l'infinit i que és asimptòtica Gevrey a una única série formal per valors del radi prou grans. A més, veurem que aquesta solució és localment a l'origen com αrn, essent n el grau de la solució, i la constant α resulta ser una funció del grau que és exponencialment petita en n, la qual en cosa dificulta notablement el seu cà;lcul numèric. Veurem també breument el mètode numèric emprat per calcular el valor de α.
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