Dia: Dimecres, 14 de setembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Jaume Paradela (UPC)
Títol: Unstable Motions in the Restricted Planar Elliptic Three Body Problem
Resum: Consider the restricted planar elliptic 3 body problem (RPE3BP), where a massless body moves on the gravitational field generated by two bodies, called primaries, which have masses μ∈(0,1/2] and 1-μ and revolve one around each other in Keplerian ellipses of eccentricity ε∈(0,1). We prove that the RPE3BP exhibits unstable dynamics: for any value μ∈ (0,1/2), we build orbits along which the angular momentum G of the massless body experiences an arbitrarily large variation provided the eccentricity of the primaries is small enough. This kind of unstable motion is also referred to as Arnold diffusion in the literature.
The proof relies on the existence of a transition chain of heteroclinic orbits to a Topological Normally Hyperbolic Invariant Cylinder (TNHIC). Two main sets of tools are developed to construct this diffusive mechanism in the RPE3BP. First, for the study of highly anisotropic splitting (involving exponentially and non exponentially small directions) between the invariant manifolds of pairs of partially hyperbolic invariant tori. Second, for the analysis of two distinct scattering maps associated to two different transversal intersections of the invariant manifolds of a (Topological) NHIC. In particular, for comparing them when they are exponentially close to each other. This is a joint work with Marcel Guardia and Tere Seara.
Dia: Dimecres, 28 de setembre de 2022
Lloc: Aula B1 (planta baixa), Facultat de Matemàtiques i Informàtica, UB.
També ONLINE https://ub-edu.zoom.us/j/99202881559?pwd=cDBlY3d4TS9Xam56bXVIU3c1TmtRUT09
A càrrec de: Pablo Roldán (Yeshiva University)
Títol: Continuation of relative equilibria in the n–body problem to spaces of constant curvature
Resum: The curved n–body problem is a natural extension of the planar Newtonian n–body problem to surfaces of non-zero constant curvature. We prove that all non-degenerate relative equilibria of the planar problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. For example, we extend Lagrange’s triangle configuration with different masses to both positive and negative curvature spaces.
Dia: Dimecres, 19 d'octubre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Alexey Kazakov (HSE University Nizhny Novgorod)
Títol: Four-winged and two-winged Lorenz attractors emerge due to bifurcations of a periodic orbit with multipliers (-1,i,-i)
Resum: We show that bifurcations of periodic orbits with multipliers (-1,i,-i) can lead to the birth of pseudohyperbolic Lorenz-like attractors of two different types: one is a discrete analogue of the classical Lorenz attractor and the other is new, a four-winged Simó angel. We also show that these two attractors exist for an orientation-reversing, quadratic, three-dimensional Hénon map, which implies the abundance of such attractors in a class of systems with homoclinic tangencies. The analysis is based on the study of a normal form for this bifurcation, a slow-fast three-dimensional system of differential equations with a Z4 symmetry. The existence of the continuous-time counterparts of the discrete 2- and 4-winged Lorenz-like attractors is established for the normal form as a part of an extensive numerical and theoretical analysis of its bifurcations. In particular, we establish that both 2- and 4-winged continuous-time Lorenz attractors are born out of a certain Z4-symmetric heteroclinic configurations with 3 saddles.
Dia: Dimecres, 26 d'octubre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
També ONLINE https://ub-edu.zoom.us/j/99202881559?pwd=cDBlY3d4TS9Xam56bXVIU3c1TmtRUT09
A càrrec de: Donato Scarcella, Paris-Dauphine University
Títol: Asymptotically quasiperiodic solutions for time-dependent Hamiltonians
Resum: In 1954 Kolmogorov laid the foundation for the so-called KAM theory. This theory shows the persistence of quasiperiodic solutions in nearly integrable Hamiltonian systems. It is motivated by classical problems in celestial mechanics, such as the n-body problem.
In this talk, we are interested in perturbations which depend on time non-quasiperiodically. We will analyze some properties of time-dependent Hamiltonians converging, when time tends to infinity, to autonomous Hamiltonians having an invariant torus supporting quasiperiodic solutions. We will look for solutions converging, asymptotically in time, to the quasiperiodic solutions of the unperturbed autonomous system.
Moreover, we will analyze the example in celestial mechanics of a planetary system perturbed by a given comet coming from and going back to infinity, asymptotically along a hyperbolic Keplerian orbit.
Dia: Dimecres, 2 de novembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Renato Calleja (IIMAS-UNAM, Universitat de Barcelona)
Títol: Quasi-periodic attractors up to the breakdown in the spin-orbit problem
Resum: We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and drift. This problem is an example of a conformally symplectic system, which is characterized by the property to transform the symplectic form into a multiple of itself. We construct and continue quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. The construction is based on a KAM theorem for conformally symplectic systems, which also provides estimates on the breakdown threshold of the invariant attractor. To construct the invariant attractor, we will use high precision numerical simulations to compute some of the required quantities. The algorithms are guaranteed to reach arbitrarily close to the border of existence, given enough computer resources. This talk refers to joint works with A. Celletti, J. Gimeno and R. de la Llave.
Dia: Dimecres, 9 de novembre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Andrew Clarke, Universitat de Barcelona
Títol: Why are inner planets not inclined?
Resum: Consider the 4-body problem with arbitrary masses in the regime where 3 bodies revolve around the other. We assume that the semimajor axes of the orbital ellipses are of different orders, and that there is non-negligible mutual inclination between the orbital planes of bodies 1 and 2. We prove that, given any finite itinerary of the angular momentum vector of body 2, there exist orbits of the 4-body problem shadowing this itinerary with arbitrary precision. From a geometric point of view, this implies that the eccentricity of the orbit of body 2 and the mutual inclination of the orbital planes of bodies 2 and 3 can be made to follow any finite itinerary. For example, the second planet can flip from prograde to retrograde nearly horizontal revolutions and back again arbitrarily many times.
Dia: Dimecres, 16 de novembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Makrina Agaoglou (ICMAT)
Títol: Transport processes and chemical reaction dynamics: The case of a Potential Energy Surface with four wells and an index-2 saddle
Resum: In this talk, we will explore the phase space structures governing isomerization dynamics on a potential energy surface with four wells and an index-2 saddle. For this model, we analyze the influence that coupling both degrees of freedom of the system and breaking the symmetry of the problem have on the geometrical template of phase space structures that characterizes reaction, using the method of Lagrangian Descriptors.
*Esta charla está parcialmente soportada con la Beca Leonardo a Investigadores y Creadores Culturales 2022 de la Fundación BBVA. La Fundación BBVA no se responsabiliza de las opiniones, comentarios y contenidos incluidos en este seminario los cuales son total y absoluta responsabilidad de sus autores.*
Dia: Dimecres, 23 de novembre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marina Vegué, Universitat Politècnica de Catalunya
Títol: Firing rate and synaptic weight distributions in plastic networks of spiking neurons
Resum: Networks of spiking neurons have been widely used as models to represent neuronal activity in the brain. These models are reasonably realistic but they are also difficult to treat analytically. Mean-field theory has nevertheless proven to be successful as a method for deriving some of their statistical properties at equilibrium such as the distribution of firing rates. However, these models assume a static connectivity, whereas the connection strengths in real brain networks evolve in time according to plasticity rules that depend on the neuronal activity.
In this talk I will present a way to extend the mean-field formalism to networks with synaptic weights that are prone to plastic, activity-dependent modulation. The plasticity in the model is mediated by the introduction of spike traces. A trace associated to one neuron represents a chemical signal that is released every time the neuron emits a spike and which is degraded over time. The temporal evolution of the trace is controlled by its degradation rate (i.e., a measure of how fast the spiking “memory” is lost) and by the neuron’s firing rate.
The mean-field formalism provides a set of equations whose solution specifies the firing rate and synaptic weight distributions at equilibrium. The equations are exact in the limit of traces with infinite memory but I will show that they already provide accurate results when the degradation rate lies within the physiological range. Overall, this work offers a way to explore and better understand the way in which plasticity shapes both activity and structure in neuronal networks.
Last updated: Wednesday, 30-Nov-2022 12:47:36 CET