Dia: Dimecres, 22 de setembre de 2021
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Maciej Capinski, AGH University of Science and Technology
Títol: Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions
Resum: We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system preserves the properties of topological expansion and contraction, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold
Dia: Dimecres, 29 de setembre de 2021
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Gemma Huguet, UPC
Títol: Oscillatory dynamics and neuronal communication
Resum: The goal of this talk is to illustrate how mathematics, and in particular the theory of dynamical systems, can contribute to the understanding of the fundamental mechanisms responsible for the activity of the nervous system.
In this talk we will focus on the study of neuronal oscillations, both regular and irregular. Oscillations are ubiquitous in the brain, but their role is not completely understood. We will present some neural models and show how dynamic systems theory tools can be used to provide a comprehensive analysis of dynamics. We will then focus on the role of oscillations in neuronal communication.
The Communication Through Coherence (CTC) theory (Fries, 2005, 2015) proposes that oscillations regulate the information flow. Thus, neural communication is established if the underlying oscillatory activity of the emitting and receiving populations is properly phase locked, so that inputs arrive at the peaks of excitability of the receiving population. The oscillators must be therefore phase-locked to accomplish strong communication.
We study the emerging phase-locking patterns of a neuronal Excitatory - Inhibitory (E-I) network under external periodic forcing, simulating the input from other oscillating neural groups. We use mean-field models, which provide an exact description of the macroscopic activity of a network and are amenable for mathematical analysis. We locate numerically the phase-locked states, arising from the study of the stroboscopic map. Finally, we discuss the implications of the computed phase-locked states on neuronal communication.
This is joint work with Alberto Pérez-Cervera, David Reyner-Parra and Tere M. Seara.
Dia: Dimecres, 10 de novembre de 2021
Lloc:
Aula 103, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Renato Calleja, IIMAS-UNAM
Títol: Some choreographies of the n-body problem: connections between polygons and the figure eight
Resum: N-body choreographies are periodic solutions to the N-body equations in which equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for continuing and proving the existence of choreographies in the gravitational body problem with the help of the digital computer. These arise from the polygonal system of bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. For odd numbers of bodies between n = 3 and n = 15 we find numerically that the figure eight choreography can be reached starting from the regular n-gon. Based on these calculations we extend conjectures claiming the n-gon and the eight are in the same continuation class for all odd numbers of bodies.
This is joint work with Carlos García-Azpeitia, Jason Mireles James, Jean-Philippe Lessard and is a continuation of work with Eusebius Doedel and C. García-Azpeitia.
Dia: Dimecres, 24 de novembre de 2021 Lloc: IMTech Colloquium ONLINE
https://meet.google.com/nvt-pgxd-tgn
A càrrec de: Lai-Sang Young (NYU)
Títol: Chaotic and random dynamical systems
Resum: In this talk I will compare and contrast chaotic (deterministic) dynamical systems with their stochastic counterparts, i.e. when small random perturbations are added to model uncontrolled fluctuations. Three groups of results, a mixture of old and new, will be discussed. The first has to do with how deterministic systems that are sufficiently chaotic produce signals that resemble (genuinely random) stochastic processes. Next I will compare the ergodic theories of chaotic systems and of random maps (such as stochastic flows generated by SDEs), my theme being that many results are nicer in the random category. That leads to my final point, which is that in some ways existing theory of chaotic systems requires too detailed information for it to be readily applicable, while a little bit of random noise can go a long way.
Dia: Dimecres, 1 de desembre de 2021 Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Oscar Rodríguez del Río (Università di Pisa)
Títol: Determinació orbital d'asteroides i brossa espacial. El problema del linkage
Resum: En aquesta xerrada veurem en què consisteix la determinació orbital, fent especial èmfasi en els mètodes de les integrals keplerianes i en la seva importància en l'aplicació a asteroides i brossa espacial. Així, presentarem la generalització d'un mètode del s. XIX introduït per Ottaviano Mossotti el qual permet el càlcul del moment angular d'un asteroide d'una forma molt eficient. A continuació, ens centrarem en un problema específic de determinació orbital, el problema del linkage. El problema del linkage consisteix a intentar unir dos o més conjunts d'observacions per tal de determinar una òrbita preliminar. És important notar que aquests conjunts d'observacions no contenen prou informació per si sols per produir una òrbita preliminar de l'objecte que volem estudiar. Finalment veurem una estratègia per lidiar amb aquests conjunts d'observacions combinant diferents mètodes.
Dia: Dimecres, 15 de desembre de 2021 Lloc:
Aula 103, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Santiago Barbieri (Université Paris-Saclay and Università degli Studi Roma Tre)
Títol: On the genericity of effectively stable integrable hamiltonian systems and on their algebraic properties
Resum: Hamiltonian systems constitute an important class of dynamical systems. Those hamiltonian systems which are integrable in the sense of Arnold-Liouville possess an important property: their solutions can be witten explicitly and the phase space is foliated by invariant tori carrying global quasi-periodic orbits. This kind of systems are exceptional but in applications it is not rare to see systems which are perturbations of integrable ones. A natural question is then to determine whether the stability of solutions is preserved for this latter type of systems. Kolmogorov-Arnold-Moser theory assures that, under generic hypotheses, a Cantor set of positive Lebesgue measure of invariant tori carrying quasi-periodic motions persists under a sufficiently small perturbation. On the other hand, instabilities may appear in the complementary of this set (Arnold diffusion). Moreover, a Theorem due to Nekhoroshev (1971-1977) shows that the solutions of a sufficiently regular integrable system verifying a transversality property known as "steepness" are stable over a long time under the effect of a suitably small perturbation. Nekhoroshev also showed (1973) that the steepness property is generic, both in measure and topologic sense, in the space of jets (Taylor polynomials) of sufficiently smooth functions. However, the proof of this result kept being poorly understood up to now and, surprisingly, the paper in which it is contained is hardly known, whereas the rest of the theory has been widely studied over the decades. Moreover, the definition of steepness is not constructive and no general rule to establish whether a given function is steep or not existed up to now, thus entailing a major problem in applications.In this seminar, I will start by explaining the main ideas behind Nekhoroshev's proof of the genericity of steepness by making use of a more modern language. Indeed, the proof strongly relies on arguments complex analysis and real algebraic geometry: the latter was much less developed than nowadays at the time that Nekhoroshev was writing, so that many passages appear to be quite obscure in the original article. Moreover, an important result of real algebraic geometry was buried in the proof and seems to have
been proved again by Roytwarf and Yomdin in 1997 by making use of different arguments (and generalized in many directions by subsequent works of many authors). Finally, I will show how a deep understanding of the genericity of steepness allows to determine explicit algebraic criteria in the space of jets which make it possible to establish whether a given function is steep or not.
(Joint work with L. Niederman)
Reference: N. N. Nekhoroshev, "Stable lower estimates for smooth mappings and
for gradients of smooth function",
Mathematics of the USSR-Sbornik, 1973, vol. 90 (132), no. 3, pp.432-478.
Dia: Dimecres, 23 de març de 2022 Lloc:
Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Adrián Ponce-Alvarez, Universitat Pompeu Fabra, Computational Neuroscience Group, Center for Brain and Cognition (CBC)
Títol: Neural networks at different scales and in different states
Resum: I will present my lines of research that include the study of neural
networks, from microcircuit to whole-brain activity, in different
cortical/brain states. My approach combines simple neural network models,
statistical model inference, information theory, and data analysis. In a
series of works, I have studied the emergence of patterned activity from
the interactions between network connectivity and neural dynamics, under
resting-state activity, task conditions, neuro-modulation, and different
cortical/brain states — e.g. synchronized/desynchronized cortical states,
sleep, and anesthesia.
Dia: Dimecres, 30 de març de 2022 Lloc:
Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Lu Bing-ying, Bremen University
Títol: Soliton resolution and asymptotic stability for the sine-Gordon equation
Resum: In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation ftt−fxx+sinf=0. Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.
Joint with Jiaqi Liu (Univ. Chinese Academy of Sciences), Gong Chen (Univ. Toronto)
Dia: Dimecres, 6 d'abril de 2022 Lloc:
Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Jessica Elisa Massetti (Università Roma Tre)
Títol: On the persistence of periodic tori for symplectic twist maps
Resum: Invariant tori that are foliated by periodic points are at the core of the fragility of integrable systems since they are somehow extremely easy to break, in counterposition to the generic robustness of the quasi-periodic ones considered by KAM theory. On the other hand, the investigation of rigidity of integrable twist maps, i.e. to understand to which extent it is possible to deform a map in a non-trivial way preserving some (or all) of its features, is related to important questions and conjectures in dynamics. In this talk I shall discuss the persistence of Lagrangian periodic tori for symplectic twist maps of the 2d-dimensional annulus and a rigidity property of completely integrable ones.
This is based on a joint work with Marie-Claude Arnaud and Alfonso Sorrentino.
Dia: Dimecres, 27 d'abril de 2022 Lloc:
Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Dimitri Pelinovski (McMaster University)
Títol: Exponentially small splitting for heteroclinic and homoclinic orbits in lattice equations
Resum: In many wave systems, propagation of traveling solitons or kinks is
prohibited because of
resonances with linear excitations. We show that wave systems with resonances
may admit an infinite
number of traveling solitons or kinks if the closest to the real axis
singularities of a limiting asymptotic
solution in the complex upper half plane appear in quartets. This quite
general statement
is illustrated by many examples, the latest of which is the lattice model with
saturable nonlinearity.
This is based on a joint work with Marie-Claude Arnaud and Alfonso Sorrentino.
Dia: Dimecres, 18 de maig de 2022 Lloc:
Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Dario Bambusi (Università degli studi di Milano)
Títol: Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori (towards a quantum Nekhoroshev theorem)
Resum: I will present a study of the time dependent Schrödinger equation
-𝐢∂tΨ=-ΔΨ+v(t,x,-𝐢∇)Ψ
on a flat d dimensional torus. Here V is a time dependent pseudodifferential operator of order strictly smaller than 2. The main result I will give is an estimate ensuring that the Sobolev norms of the solutions are bounded by tǫ. The proof is a quantization of the proof of the Nekhoroshev theorem, both analytic and geometric parts. Previous results of this kind were limited either to the case of bounded perturbations of the Laplacian or to quantization of systems with a trivial geometry of the resonances, lik harmonic oscillators or 1-d systems. In this seminar I will present the result and the main ideas of the proof.
A càrrec de: Beatrice Langella (SISSA, Trieste)
Títol: Growth of Sobolev norms in quasi integrable quantum systems
Resum: In this talk I will analyze an abstract linear time dependent Schrödinger equation of the form
-𝐢∂tΨ=(H0+V(t))Ψ (1)
with H0 a pseudo-differential operator of order d > 1 and V(t) a time dependent family of pseudo-differential operators of order strictly less than d. I will introduce abstract assumptions on H0, namely steepness and global quantum integrability, under which we can prove a |t|ε upper bound on the growth of Sobolev norms of all the solutions of (1). The result I will present applies to several models, as perturbations of the quantum anharmonic oscillator in dimension 2, and perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular: flat tori, Zoll manifolds, rotation invariant surfaces and Lie groups. The case of several particles on a Zoll manifold, a torus or a Lie group is also covered. The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem.
This is a joint work with Dario Bambusi.
Dia: Dimecres, 25 de maig de 2022 Lloc:
Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Pietro Baldi (Università di Napoli)
Títol: Normal form and existence time for the Kirchhoff equation
Resum: We consider the Kirchhoff equation
∂ttu - Δu (1 + ∫𝕋d |∇u|2dx) = 0
on the d-dimensional torus 𝕋d. This is a quasi-linear PDE with the structure of an infinite-dimensional Hamiltonian system, originally proposed as a nonlinear model for the oscillations of elastic strings and membranes.
In the talk we present two recent results, obtained in collaboration with Emanuele Haus, about the Cauchy problem with initial data of size ε in Sobolev class.
A càrrec de: Alberto Pérez-Cervera (Universidad Complutense de Madrid)
Títol: Isostables for Stochastic Oscillators
Resum: Phase-Amplitude variables are indispensable tools to characterize oscillatory dynamics. However, achieving an extension of these tools to stochastic oscillators, i.e. noisy excitable systems, has remained an open question until very recently. In this talk, we will present a framework (inspired on the parameterisation method) for the ‘phase-amplitude’ description of stochastic oscillators [1,2], and discuss possible applications.
This is a joint work with B.Lindner, P.Thomas, P.Houzelstein and B.Gutkin
Dia: Dimecres, 1 de juny de 2022
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Evelyn Sander, George Mason University
Títol: Rotation vectors for torus maps using the weighted Birkhoff average
Resum: In this talk, I will discuss numerical methods for studying one- and two-dimensional invariant tori based on the weighted Birkhoff average. These methods do not rely on symmetries, such as time-reversal symmetry, nor on approximating tori by periodic orbits. These methods are combined with computational number theoretic methods for calculating whether a number (resp. vector) is rational (resp. nonresonant) up to a certain tolerance. Together, these methods make it possible to distinguish between chaotic regions, islands, and invariant tori, while simultaneously giving a highly accurate estimate of the frequency vector of each torus. We demonstrate these methods for Arnold's circle map, Chirikov's standard map and some of its variants, and to the three-dimensional standard volume-preserving map.
This work is in collaboration with James Meiss.
Dia: Dimecres, 13 de juliol de 2022
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Jason Mireles-James, Florida Atlantic University
Títol: Computer assisted proof of transverse homoclinics at L4 in the CRTBP
Resum: In a recent paper with Shane Kepley and Maciej Capinski we prove the existence of several Ejection/Collision orbits, as well as near collision periodic and homoclinic orbits, in the Planar Circular Restricted Three Body Problem (https://arxiv.org/abs/2205.03922). The proofs are computer assisted and are especially useful in non-perturbative parameter regimes. We apply our arguments to questions where there is a long history of numerical investigation.
I will discuss a simple but general framework for studying two-point boundary value problems in systems defined in multiple coordinate systems where there are conserved quantities. The approach facilitates the proof of many different kinds of orbits using a similar template. I'll focus on the case of L4 homoclinics, where we need validated numerical integration schemes for the differential equation and it's variational equations, as well as validated error bounds on the L4 stable/unstable manifolds and their derivatives.Last updated: Thursday, 28-Mar-2024 12:33:35 CET