Dia: Dimecres, 16 d'octubre de 2019
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 fμ of
conformally symplectic maps depending on parameters μ. Whiskered tori
are tori on which the motion is a rotation but having as many
contracting/expanding directions as allowed by the preservation of the
geometric structure.
Our main result is formulated in an a-posteriori format. Given an
approximately invariant embedding of the torus for a parameter value
μ0 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.
A càrrec de: Begoña Nicolás, Departament de Matemàtiques i Informàtica, UB
Títol: Transport and invariant manifolds near L3 in the Earth-Moon Bicircular model
Resum: The talk focuses on the role of L3 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 L3 of saddle type. Then, the computation
of their stable and unstable manifolds provides connections between
Earth and Moon, and also gives rise to trajectories that enter and
leave the Earth-Moon system. Finally, by means of numerical
simulations based on the JPL ephemeris we show that these
connections seem to subsist in the real system.
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.
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.
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.
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.
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.
Last updated: Tuesday, 09-Mar-2021 16:48:07 CET
Sessions anteriors