Dia: Dimecres, 16 d'octubre de 2019

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Pablo Roldán, Yeshiva University

Títol: Topological Data Analysis of Financial Time Series

Resum: We introduce a methodology that combines topological data analysis with a machine learning technique k-means clustering in order to characterize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor. Then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions in the cryptocurrency markets.

Given the audience of this seminar, I will try to emphasize the connections of our methodology to dynamical systems, in particular to bifurcation theory.

This is joint work with M. Gidea (Yeshiva University) and Yuri Katz (Standard and Poors).

Reference: M. Gidea, D. Goldsmith, Y. Katz, P. Roldan, Y. Shmalo: Topological Recognition of Critical Transitions in Time Series of Cryptocurrencies, 2019, Physica A (accepted).

Dia: Dimecres, 16 d'octubre de 2019

Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Alexey Kazakov, National Research University Higher School of Economics, Nizhny Novgorod, Russia

Títol: On pseudohyperbolic attractors, quasiattractors and their examples

Resum: In this talk, we will discuss two different types of chaotic attractors: pseudohyperbolic attractors and quasiattractors. Each orbit on pseudohyperbolic attractor is unstable and this property persists for all small perturbations of the system. Quasiattractors either contain stable periodic orbits or such orbits appear for arbitrarily small perturbation. We demonstrate some examples of both pseudohyperbolic attractors and quasiattractors and explain how to verify whether a chaotic attractor belongs to a class of pseudohyperbolic ones or it is a quasiattractor.

Dia: Dimecres, 30 d'octubre de 2019

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Heinz Hansmann, Universiteit Utrecht

Títol: Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance

Resum: We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.

Dia: Dimecres, 6 de novembre de 2019

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Dmitry Turaev, Imperial College London

Títol: On triple instability

Resum: We show that bifurcations of a periodic orbit with three (or more) multipliers equal to 1 lead to chaotic dynamics of ultimate richness.

Dia: Dimecres, 13 de novembre de 2019

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Sergey Gonchenko, Lobachevsky University of Nizhny Novgorod

Títol: Mixed dynamics and reversible perturbations of conservative Hénon maps

Resum: In the introductory part of the talk, we give a brief overview of the basic concepts of the theory of mixed dynamics - the third type of dynamical chaos complementary to the two well-known its forms, conservative chaos and strange attractors. The main goal is to study the mechanisms of the emergence of mixed dynamics under (small) perturbations of conservative systems. In the most natural way, the mixed dynamics arises when the perturbations are reversible. We construct reversible perturbations of two-dimensional conservative Henon-like maps and study accompanying typical symmetry-breaking bifurcations. This is a joint work with M. Gonchenko and K. Safonov.

Dia: Dimecres, 27 de novembre de 2019

Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Gerard Farré, Royal Institute of Technology (KTH)

Títol: Instabilities of analytic quasi-periodic tori

Resum: We will present the results of a recent joint work with B.Fayad on the stability of quasi-periodic tori for Hamiltonian systems. In particular, we show the existence of real analytic Hamiltonians with a Lyapunov unstable quasi-periodic torus of arbitrary frequency. Furthermore, for Diophantine frequencies these tori can be chosen to be KAM stable, meaning that the original torus is accummulated by a set of invariant tori whose relative measure tends to one. We will also present some other similar interesting examples in the context of stability of invariant quasi-periodic tori.

Dia: Dimecres, 4 de desembre de 2019

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m
- Koon, W. S., Lo, M. W., Marsden, J. E., Ross, S. D.: ’Resonance and Capture of Jupiter Comets’, Celestial Mechanics and Dynamical Astronomy 81:27-38, 2001.
- Jorba, À.: ’Numerical Computation of the Normal Behaviour of Invariant Curves of n-Dimensional Maps’, Nonlinearity 14: 943-976, 2001.

A càrrec de: Gladston Duarte, Universitat de Barcelona / BGSMath

Títol: Invariant Manifolds near L1 and L2 in the Planar Elliptic Restricted Three-Body Problem

Resum: In this work we investigate the connections between the stable and unstable manifolds of tori around the points L1 and L2 of the Planar Elliptic Restricted Three-Body Problem (PERTBP). The study of connections between the invariant manifolds of the periodic orbits around these points, in the Planar Circular RTBP, and the creation of bridges between different types of orbits was already done in [1]. In the case of considering an elliptical movement, we investigate how the analysis of the orbit of comet 39P/Oterma can be improved in a more quantitative way. We compute the dynamical objects that interact with this comet (mixing the tools presented in [2] and the parallel shooting technique), and use some temporal+spatial sections in the phase space to better visualize these objects together with Oterma, when fitting its data into this model.

References:

Dia: Dimecres, 11 de desembre de 2019

Lloc: Aula T1 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m
- M. Moczurad, P. Zgliczynski, Central configurations in planar n-body problem for n=5,6,7 with equal masses, arXiv:1812.07279, Celestial Mechanics and Dynamical Astronomy, (2019) 131: 46.

A càrrec de: Piotr Zgliczynski, Jagiellonian University

Títol: Central configurations in planar n-body problem for n=5,6,7 with equal masses

Resum: We give a computer assisted proof of the full listing of central configuration for n-body problem for Newtonian potential on the plane for n=5,6,7 with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For n=8,9,10 we establish the existence of central configurations without any reflectional symmetry.

References:

Dia: Dimecres, 15 de gener de 2020

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Carles Simó, Universitat de Barcelona

Títol: A simple family of exceptional maps with chaotic behavior

Resum: We consider as chaotic a system which has sensitive dependence to initial conditions and topological transitivity. A simple family of maps in the 2D torus is considered for which the set of points that display chaotic behavior has full Lebesgue measure. However the maps have neither homoclinic nor heteroclinic orbits. The role of returning infinitely many times near the only fixed point (parabolic) is taken by quasi-periodicity. The presentation shall be completed by several generalizations and numerical examples. Some possible extensions will be mentioned.

Dia: Dimecres, 22 de gener de 2020

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB. Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Renato Calleja, Departamento de Matemáticas y Mecánica, IIMAS-UNAM

Títol: Whiskered KAM Tori of Conformally Symplectic Systems

Resum: Many physical problems are described by conformally symplectic
systems. We study the existence of whiskered tori in a family *f _{μ}* of
conformally symplectic maps depending on parameters

Our main result is formulated in an a-posteriori format. Given an
approximately invariant embedding of the torus for a parameter value
*μ _{0}* with an approximately invariant splitting, there is an invariant
embedding and invariant splittings for new parameters.

Using the results of formal expansions as the starting point for the a-posteriori method, we study the domains of analiticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement.

Joint work with A. Celletti and R. de la Llave.

Dia: Dimecres, 29 de gener de 2020

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m
- Hora: 16h45m Café i galetes
- Hora: 17h00m
- À. Jorba, M. Jorba-Cuscó and J.J. Rosales (2020) "The vicinity of the Earth-Moon L1 point in the Bicircular Problem". Cel. Mech. (to appear)
- J.J. Rosales, À. Jorba, M. Jorba-Cuscó (2020) "Families of Halo-like invariant tori around L2 in the Sun-Earth-Moon Bicircular Problem". Preprint.

A càrrec de: Begoña Nicolás, Departament de Matemàtiques i Informàtica, UB

Títol: Transport and invariant manifolds near *L _{3}* in the Earth-Moon Bicircular model

Resum: The talk focuses on the role of *L _{3}* to organize families
of trajectories going from Earth to Moon and viceversa, and entering
or leaving the Earth-Moon system. As a first model, we have consider
the planar Bicircular problem to account for the gravitational
effect of the Sun. The first step has been to compute a family of
quasi-periodic orbits near

Joint work with Àngel Jorba

A càrrec de: José J. Rosales, Departament de Matemàtiques i Informàtica, UB

Títol: Some results on the dynamics around the Earth-Moon L1 and L2 points in the Bicircular Problem

Resum: The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem (RTBP) that includes the direct gravitational effect of the Sun on an infinitesimal particle. In this talk we use the BCP model to study the dynamics around the Moon L1 and L2 regions. We use two techniques: reduction to the center manifold, and continuation of 2D invariant tori.

The reduction to the center manifold proves to be useful around L1, and it provides good vistual artifacts to understand the dynamics [1]. This technique is not useful around L2 because the radius of convergence is too small, and we use continuation of 2D invariant tori instead to get an insight on the dynamics.

For L1, it is showed that the existence of two families of quasi-periodic Lyapunov orbits, one planar and one vertical. The planar Lyapunov family undergoes a (quasi-periodic) pitchfork bifurcation giving rise to two families of quasi-periodic Halo orbits. Between them, there is a family of Lissajous quasi-periodic orbits, with three basic frequencies.

For L2, a total of six families of invariant 2D tori are found. Two of them are planar Lyapunov quasi-periodic orbits, and four of them are vertical. On the the vertical families comes from a direct continuation of the RTBP Halo orbits. Another one comes from resonant quasi-halo in the RTBP. The other two still need to be classified [2].

Joint work with Àngel Jorba and Marc Jorba Cuscó

Dia: Dimecres, 12 de febrer de 2020

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Ugo Locatelli, Università di Roma "Tor Vergata"

Títol: Invariant tori in exoplanetary systems: from theory to applications

Resum: As a preliminary introduction, KAM theory is briefly recalled by discussing in a unified way two algorithms, which construct the usual (maximal) invariant tori and the (lower-dimensional) elliptic tori. This is made by adapting to our purposes the approach developed by Poeschel [Math. Z. (1989)]. Therefore, we focus on the applications to exoplanetary systems, by describing a sort of KAM reverse method designed so to estimate the (unknown) values of the mutual inclinations. In particular, some results previously obtained [Volpi et al., CMDA (2018)] are discussed and the method to improve them is sketched. Actually, such a new approach is based on a careful combination of the constructive algorithms described in the first part of the talk. Finally, we show the first results produced by the new method.

This work is based on a research project made in collaboration with C. Caracciolo, M. Sansottera and M. Volpi.

Dia: Dimecres, 19 de febrer de 2020

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Dmitrii Todorov, Centre de Recerca Matemàtica

Títol: Some shadowing and inverse shadowing results

Resum: Among various types of stability properties for dynamical
systems, there is class of "per-iteration" stability properties called
"shadowing properties".
A typical property like that would say that behavior of individual
trajectories does not depend much (L^{∞}-distance-wise) on a constantly
applied small perturbation.
While it is not a surprise that such properties usually hold for simple
systems with simple attractors (at least to some extent), they also do hold
for some chaotic systems as well. It is well-known (and makes a part of
stability conjecture proof) that uniformly hyperbolic systems have various
nice shadowing properties. It would be desirable to be able to say the same
for less strongly chaotic systems (e.g. because they are the ones that
usually come from applications).
I will present some results somewhat destroying this hope.

This was a subject of my PhD and I have not worked on it for years by now. If time permits, I will say a couple of words about my more recent projects as well.

Dia: Dimecres, 26 de febrer de 2020

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.

- Hora: 16h00m Café i galetes
- Hora: 16h15m

A càrrec de: Otavio Gomide, Universidade Federal de Goiás

Títol: Small amplitude breathers for reversible Klein-Gordon equations

Resum: reathers are nontrivial time-periodic and spatially localized solutions of evolutionary Partial Differential Equations (PDE's). It is known that the sine-Gordon equation (a special case of Klein-Gordon equation) admits an explicit family of breathers. Nevertheless this kind of solution is expected to be rare in other Klein-Gordon equations. In this work, we discuss the non-existence of small amplitude breathers for reversible Klein-Gordon equations (RKG) through a rigorous analysis. Roughly speaking, we look at the RKG as an evolutionary PDE with respect to the spatial variable in such a way that the breathers becomes a homoclinic orbits to a critical point (origin). We obtain an asymptotic formula for the distance between the stable and unstable manifolds of such critical point which happens to be exponentially small with respect to the amplitude of the breather and therefore classical Melnikov Theory cannot be used. This is a joint work with M. Guardia, T. Seara and C. Zeng.

Dia: Dimecres, 4 de març de 2020

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.

- Hora: 16h00m
- Hora: 17h45m Café i galetes
- Hora: 18h00m

A càrrec de: Jason Mireles-James, Florida Atlantic University

Títol: Validated numerical methods for delay differential equations

Resum: Delay differential equations (DDEs) are often used to model system where there are communication lags between subsystems, or where there is a substantial pause between the time when a system receives a stimulus and when it reacts. Much like an ordinary differential equation (ODE), a DDE generates a dynamical system. However the analysis is complicated by the fact that the dynamics act on a suitable function space of past histories, hence the phase space is infinite dimensional.

Computer assisted proofs of existence for periodic orbits of DDEs have been in the literature for about ten years, since the Ph.D. thesis of J.P. Lessard. I will discuss some more recent work on validated numerical methods for stability analysis of equilibrium solutions and their attached unstable manifolds. If time permits I will discuss also a C^1 integration scheme which provides a rigorous enclosure of the solution of an initial value problem and its derivative.

A càrrec de: Maciej Capinski, AGH University of Science and Technology

Títol: Arnold Diffusion and Stochastic Behaviour

Resum: We will discuss a construction of a stochastic process on energy levels in perturbed Hamiltonian systems. The method follows from shadowing of dynamics of two coupled horseshoes. It leads to a family of stochastic processes, which converge to a Brownian motion with drift, as the perturbation parameter converges to zero. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion, for appropriate sets of initial conditions. The convergence is in the sense of the functional central limit theorem. We give an example of such construction in the PRE3BP.

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Last updated: Tuesday, 09-Mar-2021 16:48:07 CET