Dia: Dimecres 14 de gener de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Gustaf Söderlind, Lund University, Sweden
Títol: Logarithmic norms with applications to nonlinear dynamics
Resum: Originally defined for matrices in 1958, the logarithmic norm bounds solutions to initial value problems of the form
u̇ = Au + r, u(0)=u0 ,
where r is a forcing function or perturbation and A is a bounded linear operator.
Using differential inequalities one obtains
∥ u(t) ∥ ≤ etM[A] ∥ u0 ∥ + ∫0t e(t-τ)M[A] ∥ r(τ) ∥ dτ ,
where the logarithmic norm M[A] is the best possible constant such that this bound holds for all t≥ 0.
The logarithmic norm has since been extended to cover nonlinear maps and differential operators, unifying and quantifying notions such as definiteness for matrices; monotonicity for nonlinear maps; and ellipticity for differential operators. It plays an essential role in stability analysis, with broad and versatile applications in initial value problems, elliptic boundary value problems and initial-boundary value partial differential equations, as well as their discretizations.
Based on the new book [Sode24], the talk gives a brief introduction to the essential background material, and proceeds to develop the theory for nonlinear maps. Some classical nonlinear systems (such as van der Pol, Lotka--Volterra, the Kermack--McKendrick SIR model, the Oregonator equation, and the Lorenz system) are analyzed with respect to stability, stiffness and splitting. Alternative but related techniques involving the divergence of the vector field are also discussed, illustrating phase volume compression as a necessary condition for stability.
[Sode24] G. Söderlind, Logarithmic Norms, Springer Series in Computational Mathematics 63, (2024)
A càrrec de: Christos Efthymiopoulos, University of Padova
Títol: Manifold models of spiral structure in galaxies
Resum: The unstable invariant manifolds associated with unstable periodic orbits or low-dimensional tori in the corotation region at the end of galactic bars have been used extensively in the modelling of galactic spiral structure in the last 20 years. After a short review of main results on the subject, the seminar will discuss how the conventional wisdom of dynamical systems' theory, including the invariance and stability properties of the manifolds, and the passage from manifolds to Lagrangian coherent structures, can lead to observational consequences as regards the morphology and the mean velocity flow through galactic spirals. We will finally discuss the comparison between manifold models and self-consistent N-body simulations of galaxies with chaotic spirals.
Dia: Dimecres 3 de setembre de 2025
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Kathrin Padberg-Gehle, Leuphana Universitaet Lueneburg
Títol: From trajectories to transport: network-based approaches for coherent sets and scalar mixing in turbulent flows
Resum: Understanding, quantifying and controlling transport and mixing processes are central in the study of fluid flows. Many different Lagrangian approaches have been proposed for detecting organizing flow structures, including recent data-based methods that aim to identify such coherent objects directly from simulated or measured tracer trajectories. Among these approaches are Lagrangian flow networks, where trajectories serve as network nodes and the links are weighted according to spatio-temporal distances between trajectories. Spectral clustering as well as simple network measures such as node degrees or clustering coefficients can be used to identify flow regions of different dynamical behavior. In this talk, we propose some extensions to the network-based framework that allow us
Dia: Dimecres 10 de setembre de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Dongchen Li, Imperial College
Títol: Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics
Resum: A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some central subspace has a non-empty interior and thus has a higher topological dimension than the set itself. We show that, for any symplectic Cr-diffeomorphism (where r is sufficiently large and finite, or r=∞,ω) of a 2N-dimensional (N>1) symplectic manifold, symplectic blenders can be obtained by an arbitrarily small symplectic perturbation near any one-dimensional whiskered KAM-torus that has a homoclinic orbit. Using this result, we prove that non-transverse homoclinic intersections between invariant manifolds of a saddle-center periodic point (i.e., it has exactly one pair of complex multipliers on the unit circle) are persistent in the following sense: the original map is in the Cr closure of a C1 open set in the space of symplectic Cr diffeomorphisms, where maps having such saddle-center homoclinic intersections are dense. These results also hold for Hamiltonian flows in the corresponding settings.
Dia: Dimecres 17 de setembre de 2025
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Przemysław Berk, Nikolaus Copernicus University in Toruń
Títol: Disjointness of rescalings of surface flows.
Resum: We consider certain classes of flows on compact surfaces such as locally Hamiltonian or translation flows, and we are investigating the relation of these flows with their rescalings, that is, their linear accelerations. I will show that in certain well parametrized classes of such flows, a typical one is disjoint (spectrally or in the sense of Furstenberg) with its rescalings (rational or real, depending on the class), i.e. it is non-isomorphic to an extreme degree. This talk is based on an article with Adam Kanigowski and an ongoing project with Corinna Ulcigrai.
Dia: Dimecres 1 d'octubre de 2025
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Robert Cardona, Universitat de Barcelona
Títol: Beyond helicity: dynamical invariants of 3D volume-preserving flows and a conjecture of Arnold and Khesin
Resum: Volume-preserving flows on 3-manifolds arise naturally in fluid dynamics and plasma physics, where the vorticity or magnetic field is transported by the motion of the fluid. A central role is played by the helicity, a classical invariant introduced by Woltjer and Moffatt that measures the average linking of flow lines. Helicity is the best-known “coadjoint invariant” of a volume-preserving flow, which implies that it is preserved under the natural symmetries of the fluid equations, and it enjoys important uniqueness properties among such invariants. In this talk, which is meant to be accessible, I will first explain and motivate the relevant notions, and then focus on the question of how far the uniqueness of helicity really goes. In particular, I will show that the Ruelle invariant is, in a precise sense, everywhere independent of helicity in the 𝒞 ¹-topology. This gives a strong negative answer to the 𝒞 ¹-case of a conjecture of Arnold and Khesin.
This is joint work with Julian Chaidez (University of Southern California) and Francisco Torres de Lizaur (Universidad de Sevilla).
Dia: Dimecres 15 d'octubre de 2025
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Otto Vaughn Osterman, University of Maryland
Títol: Length Spectrum Rigidity in Dispersing Billiard Systems
Resum: The problem of spectral rigidity in dispersing billiard systems is that of determining whether the perimeters of periodic orbits uniquely determine the billiard table up to isometry. We consider this problem for the class of dispersing billiard systems consisting of three convex scatterers in the plane satisfying the non-eclipse condition. We show that the perimeters of a particular set of periodic orbits for two such systems are identical if and only if their collision maps are locally analytically conjugated to each other near a particular homoclinic orbit.
Dia: Dimecres 22 d'octubre de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Dídac Gil Rams, UPC
Títol: Splitting of separatrices in generalized standard maps
Resum: We study transversal intersections between the invariant manifolds (stable and unstable) associated to an hyperbolic fixed point for a class of maps named generalized standard maps (defined in [1]). This generalization includes the Standard map, first studied by Lazutkin in [5], the perturbed McMillan map, studied in [6], or the Hénon map among others.
We obtain an asymptotic formula for the Lazutkin invariant, a value related to angle of intersection between the manifolds, which is exponentially small and its first term depends on a Stokes constant. To do so, one of the techniques that we use is the inner equation related to our generalized standard maps.
The Stokes constant is an involved quantity that comes from the study of the inner equation. The second part of our work consists in giving a general algorithm, useful for the studied maps, to compute an interval containing the value of the named constant by means of a computer assisted proof in CAPD (similarly as in [2]). Finally, we apply this algorithm to the Standard and Hénon maps in order to see that the intervals contain the values given by Gelfreich in [4] and [3], respectively.
Joint work with Inmaculada Baldomà Barraca (UPC), Maciej Capiński (AGH) and Pau Martín de la Torre (UPC).
Referències:
Dia: Dimecres 12 de novembre de 2025
Lloc: Aula S03, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Guowei Yu, Nankai University
Títol: (Bi)-hyperbolic and (bi)-parabolic motions in the restricted (N+1)-body problem
Resum: According to Chazy, the final motion of the restricted (N+1)-body problem has four possibilities: bounded, hyperbolic, parabolic and oscillatory. When the motion is hyperbolic or parabolic, the massless body will go to infinity along a definite asymptotic direction with a finite limiting energy. Then there are two basic questions: First given any initial time and position, as well as the asymptotic direction and limiting energy when time goes to positive infinity, is there a corresponding hyperbolic or parabolic motion realizing it; Second given any asymptotic directions and limiting energy, when time goes to both negative and positive infinity, it there a corresponding bi-hyperbolic or bi-parabolic motion realizing it? In this talk, we will report some of our progress in these two questions.
A càrrec de: Sergi Burniol Clotet, Universidad de la República, Uruguai
Títol: Rigidity of the Unstable Foliation
Resum: Anosov flows are one of the central examples of hyperbolic and chaotic dynamical systems. Associated to them are the stable and unstable foliations, whose leaves contract and expand exponentially along the flow. In this talk, we present a rigidity result for these foliations in the case of transitive Anosov flows on 3-manifolds: if the unstable foliations of two such flows are equivalent (that is, if there exists a homeomorphism mapping one foliation to the other), then the flows are topologically conjugate up to a constant change of time.
The classical examples of Anosov flows are the geodesic flows of compact negatively curved surfaces. In this setting, there are previous rigidity results due to Ratner, Marcus, and Abe, among others. The latter showed that if the unstable foliations of two such flows are equivalent, then the underlying surfaces are homothetic.
Dia: Dimecres 26 de novembre de 2025
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Anna Gierzkiewicz (Jagiellonian University)
Títol: Detecting oscillating orbits in the Sitnikov 3BP by computer assistance
Resum: In a joint work with M. Capiński and P. Martín, we prove the existence of oscillatory orbits (i.e., orbits that go closer and closer to infinity but always return to a fixed bounded region) for the Sitnikov 3BP in the case of three equal masses (m₁ = m).
The proof relies on analyzing the stable and unstable invariant manifolds of infinity and their intersections. We construct orbits shadowing these invariant manifolds by the method of correctly aligned windows. The proof is computer assisted with the use of CAPD library for C++.
Dia: Dimecres 3 de desembre de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Frank Trujillo, CRM
Títol: Hausdorff Dimension of Invariant Measures of AIETs
Resum: In this talk, we will discuss the Hausdorff dimension of invariant measures of affine interval exchange transformations (AIETs). AIETs are piecewise affine bijections of a bounded interval with a finite number of discontinuities. In general, these maps are uniquely ergodic, and their unique invariant probability measure is singular with respect to Lebesgue measure. We will provide explicit values for the Hausdorff dimension of this unique invariant probability measure for a particular class of AIETs, called self-similar AIETs.
This is joint work with P. Berk, K. Frączek, and Ł. Kotlewski.
Dia: Dimecres 10 de desembre de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Illya Koval, IST Austria
Títol: Fractal-like meanders appear in KAM theory
Resum: In KAM theory, a certain nondegeneracy of the Hessian of the unperturbed Hamiltonian is crucial to show the existence of invariant tori. These invariant tori have to be the graphs of Lipschitz functions. However, when such a type of nondegeneracy is violated, very little is known. Inspired by the works of Simó, we will show the existence of non-graph meandering invariant tori and that they are generic. We discuss meanders of the first order, second order, and even fractal-like meanders.
This is a joint work in progress with Vadim Kaloshin and Yi Pan.
Dia: Dimecres 17 de desembre de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Domènec Ruiz i Balet, UB
Títol: Connections between dynamical systems, deep learning and large language models
Resum: This introductory talk aims to introduce some of the most relevant machine learning architectures and understand them as dynamical systems. We will navegate through formulations, different problems and perspective and some of the current results.
Dia: Dimecres 28 de gener de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Renato Calleja (UNAM)
Títol: Invariant Tori in Hamiltonian and conformal-Hamiltonian Systems: Constructive and Geometric Perspectives
Resum: This talk presents recent and ongoing work on invariant tori in Hamiltonian systems with quasi-periodic time dependence, combining geometric insight and computational techniques. In collaboration with Pedro Porras and Alex Haro, we developed an a-posteriori KAM theorem for Lagrangian tori in flows, solved via a Newton-like scheme using symplectic frames and a new torsion matrix formulation. The method yields efficient, quadratically convergent algorithms under classical Diophantine and non-degeneracy conditions.
We also discuss results for conformally symplectic (dissipative) systems, including the spin–orbit problem and the breakdown of quasi-periodic attractors. Current joint work with Alex Haro and Arturo Vieiro explores secondary tori near elliptic points and resonance capture via Birkhoff normal forms.
The parameterization method serves as a unifying framework throughout, bridging conservative and dissipative regimes and linking rigorous theory with effective computation.
Dia: Dimecres 4 de febrer de 2026
Lloc: Aula 004, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Marc Jorba, UPC
Títol: Global dynamics in a simple predator-prey model
Resum: Dynamical systems provide a fundamental mathematical framework for understanding how complex and unpredictable biological behavior can emerge from simple nonlinear interactions. In this talk, we consider a map modeling the interaction between two species: a predator and a prey. The map is not invertible but an endomorphism. This feature is key to understanding the emergence of interesting phenomena. We will study how the standard invariant manifolds interact with the so-called 'Critical Lines' and the effect this interaction has on global dynamics. Numerical simulations suggest the existence of strange attractors, and infinitely many periodic orbits.
This is a join work with Lluís Alsedà and Tomás Lázaro.
Dia: Dimecres 11 de febrer de 2026
Lloc: Aula 004, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Alejandro Cabrera (UPC)
Títol: Relating dynamical systems and certain singular spaces
Resum: The aim of this talk is to give an overview of joint work with M. del Hoyo and E. Pujals (2020) in which we relate the study of (discrete) dynamical systems to that of its -typically singular- space of orbits, seen as a so-called differentiable stack. In the first part, I will motivate this observation and give a brief introduction to differentiable stacks. I will then explain the main results stating that, for simply connected manifolds, the stack of orbits allows one to recover a dynamics up to conjugation and inversion. I finally discuss some applications, including a geometric version of a celebrated result of M.Rieffel in the context of C* algebras for distinguishing irrational rotations of the circle.
Dia: Dimecres 18 de febrer de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Irene de Blasi, University of Turin, Italy
Títol: On the Birkhoff conjecture for Kepler billiards
Resum: We investigate the integrability of Kepler billiards, that is, mechanical billiard systems in which a particle moves under the influence of a gravitational center and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture. Our approach is based on the construction of symbolic dynamics at high energies, whose existence is ensured provided suitable conditions on the boundary are satisfied, and implies positive topological entropy.
Joint work with S. Baranzini, V, Barutello and S. Terracini
Dia: Dimecres 18 de febrer de 2026
Lloc: Aula 004, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: José M. Vega, Universidad Politécnica de Madrid
Títol: Higher order dynamic mode decomposition-like methods
Resum: Higher order dynamic mode decomposition (HODMD) is a recent postprocessing method, able to efficiently uncover the structure of dynamical systems outcomes exhibiting exponential growth/decay and/or periodic/quasiperiodic oscillations. It is an improvement/extension of a former method developed in fluid dynamics, called here standard dynamic mode decomposition (DMD), which does not always give the correct results for reasons that will be clarified in this talk.
An extension of HODMD will also be briefly described that is called spatiotemporal Koopman decomposition (STKD), obtained by applying HODMD in both space and time, for extended dynamical systems governed by PDEs. Such method uncovers the structure of pure and modulated traveling waves by analysing the involved frequencies and wavenumbers, which are computed with great accuracy.
Ideas and concepts in HODMD and STKD will be illustrated in toy models and tested using both, numerical and experimental data, coming from systems of scientific and technological interest.
Dia: Dimecres 11 de març de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Bernhard Reinke, Aix-Marseille University
Títol: Dynamics of higher-dimensional root finding methods
Resum: I will give an overview about the dynamics of higher-dimensional iterative root finding methods: given a complex polynomial in one variable of some degree $d$, these methods are based on iteration in $d$ complex variables and should converge to a vector consisting of the $d$ roots. I will focus on the "Weierstrass--(Durand--Kerner)" and the "Ehrlich--Aberth" methods. A fundamental question is whether these methods are generally convergent (so whether they "always work" as observed in practice), an obstruction for this is the existence of attracting cycles. Together with Dierk Schleicher and Micheal Stoll we established the existence of attracting cycles for the Weierstrass method. While the question of general convergence for the Ehrlich--Aberth method is still open, I would like to present some connections of the search for attracting cycles with the enumerative geometry of moduli spaces of rational curves, based on joint work with Rob Silversmith.
Dia: Dimecres 18 de març de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Mikhail Hlushchanka, University of Amsterdam
Títol: Quasi-visual approximations, tile graphs, and semi-hyperbolic maps
Resum: The topology and geometry of some (fractal) metric spaces can often be described by a sequence of increasingly finer covers. Each such cover provides a coarse picture of the space at increasing resolution. A quasi-visual approximation is a special sequence of covers of the space that satisfies certain natural conditions (concerning relative distances and diameters of the elements of the covers).
Quasi-visual approximations provide a convenient framework for addressing questions in quasiconformal geometry. In particular, quasi-visual approximations can be used to detect whether a given homeomorphism between two bounded metric spaces is a quasisymmetry. They are also naturally connected to the theory of Gromov hyperbolic spaces via the tile graph construction. In a dynamical setting, a sequence of covers of the underlying space may be produced naturally by starting with an initial cover and then pulling it back by the iterates of the map. We show that for rational maps such a dynamical sequence of covers of the Julia set is quasi-visual if and only if the map is semi-hyperbolic. (Based on joint work with Mario Bonk and Daniel Meyer)Dia: Dimecres 25 de març de 2026
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Marcel Guardia, UB.
Títol: Diffusive behavior along mean motion resonances in the restricted 3 body problem
Resum: Consider the Restricted Planar Elliptic Three Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. For eccentricity of Jupiter $e_0$ small enough, we show that there exists a family of probability measures supported at the $3 : 1$ mean motion resonance such that the push forward under the associated Hamiltonian flow has the following property. At the timescale $te_0^{-2}$, the distribution of the Jacobi constant of the Asteroid weakly converges to an Ito diffusion process on the line as $e_0\to 0$. This resonance corresponds to the biggest of the Kirkwood gap on the Asteroid belt in the Solar System. This is a joint work with V. Kaloshin, P. Martin and P. Roldan.
Dia: Dimecres 15 d'abril de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Rodrigo Treviño, University of Maryland
Títol: Can you hear the shape of a Cantor set?
Resum: While it is not immediately clear how to do analysis on Cantor sets, there are various notions of regularity on such spaces and (non-local) operators that embrace the role of a Laplacian. I will present some recent results on the intersection of the analysis and topology of the type of Cantor sets which show up in symbolic dynamics. Time permitting I will describe a Hodge-like theorem for Cantor sets, analogous to the classical Hodge theorem of differential geometry.
Dia: Dimecres 29 d'abril de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Xavier Buff, Institut de Mathématiques de Toulouse
Títol: Spiralling domains for germs tangent to the identity (joint work with Jasmin Raissy)
Resum: We prove that if a is a non zero real parameter, the polynomial F:C^2 → C^2 defined by F(x,y) = (x + y^2 + ax(x-y),y+x^2+ay(x-y)) has infinitely many Fatou components in which orbits converge to the origin. We also prove that the projection of those orbits to CP^1 spiral along geodesics of a meromorphic affine structure. This study relates to the study of the real-time trajectories of a homogeneous vector field on C^2, to the study of an affine structure obtained by gluing the sides of a polygon via complex affine maps, and to the study of the Fuchsian group generated by the order three map z→1/(1-z) and by the translation z→z+T where T is a real number greater than 3.
Dia: Dimecres 6 de maig de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Leon Staresinic (Universität Zürich)
Títol: The Topological Boshernitzan-Kornfeld Conjecture
Resum: Interval Translations Maps (ITMs) are a natural generalisation of the well-known Interval Exchange Transformations (IETs). They are obtained by dropping the bijectivity assumption for IETs. As such, they are exactly the finite piecewise isometries of the interval. There are two types of ITMs, finite-type and infinite-type ones. They are classified by their non-wandering sets: it is a finite union of intervals for finite-type maps, and contains a Cantor set for infinite-type maps.
One of the basic questions in the field is: How prevalent is each type of map in the parameter space? In this work, we show that the set of finite-type maps contains an open and dense subset of the parameter space of ITMs with a fixed number of intervals, which resolves in positive the topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.
This is a joint work with Kostiantyn Drach and Sebastian van Strien.
Dia: Dimecres 13 de maig de 2026
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Pau Martín, UPC
Títol: Universality in celestial mechanics
Resum: In this work we prove that, given any analytic area preserving map of the plane, there is a problem of celestial mechanics that approximates the map with an error arbitrarily small. More concretely, we prove that, given any compact subset K of the analytic area preserving maps of the plane and any ε>0, there exists a restricted planar N-body problem such its Poincaré map associated to a suitable section approximates any map in K with an error smaller than ε. The restricted planar N-body problem is associated to a suitable central configuration of N bodies, where the masses of the bodies play the role of parameters.
Dia: Dimecres 13 de maig de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Amadeu Delshams, UPC
Títol: A method of reduction for invariant curves of quasiperiodically forced maps
Resum: The existence of translated curves for quasiperiodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among the translated curves, the invariant curves are characterized as the solutions of a scalar bifurcation equation, from which their existence, stability as well as bifurcation can be easily described. This is a joint paper with Rafael Ortega: arXiv preprint
Dia: Dimecres 27 de maig de 2026
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Pablo Roldan, UPC
Títol: Efficient shadowing of pseudo-orbits generated by the transition map
Resum: In a recent paper [DGR26], we considered the spatial R3BP around the equilibrium point L1, and constructed diffusion
pseudo-orbits with optimal diffusion time. The pseudo-orbits are obtained by iterating the so-called transition map on a
suitable NHIM. They are chains of (finite-time) heteroclinic orbit segments, so they are not true orbits of the R3BP system.
In this talk, we will explain how to construct true diffusion orbits that shadow the pseudo-orbits. We will use:
- the transition map;
- the Birkhoff normal form around L1; and
- a (numerical) multiple shooting scheme.
In contrast to common shadowing lemmas used in theoretical papers, we construct shadowing orbits of similar length than the
pseudo-orbits.
This is joint work with A. Delshams and M. Gidea.
Dia: Dimecres 10 de juny de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Miriam Lorenzo Laguno (Universitat de Maryland)
Títol: Mixing properties of locally Hamiltonian flows on surfaces
Resum: I will introduce locally Hamiltonian flows on surfaces and discuss their representation as suspension flows over suitable Poincaré sections. I will review classical and recent results on the ergodic and mixing behavior of these flows, and present recent results on rates of mixing for a subclass of them.
Dia: Dimecres 17 de juny de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Tere M-Seara (UPC)
Títol: Chaos in generic analytic convex billiards
Resum: In this talk we will show the abundance of Chaos in analytic convex billiards.
It is well known that, for a convex billiard table, the corresponding billiard map is a twist map on the cylinder.
We will proof that such map has hyperbolic periodic orbits whose stable and unstable manifolds intersect transversally, giving rise to the existence of horseshoes and, therefore, to hyperbolic invariant sets whose dynamics is conjugated to the Bernoulli shift.
The abundance of transverse homoclinic orbits has been only proved for generic smooth billiard tables.
The novelty of our work is that, going back to some ideas of Zehnder of 1972, we prove that the set of analytic billiards such that "given any rational rotation number, there is a unique periodic orbit in the associated Aubry-Mather set, whose stable and unstable manifolds intersect and whose intersections are all transversal” form a Baire generic set with the usual analytic topology.
The result is in collaboration with I. Baldomá (U.Politecnica de Catalunya) A. Florio (U. Paris Dauphine) and M. Leguil (E. Polytechnique Paris).
Dia: Dimecres 8 de juliol de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Yunzhe Li (ISTA)
Títol: Deformations of the Standard Map Preserving Countably Many Spectral Data.
Resum: We construct nontrivial deformations of the standard map that preserve the symplectic actions of infinitely many periodic orbits accumulating on an invariant curve with a Liouvillean rotation number. This result can be viewed as an analogue of a spectral nonrigidity phenomenon for convex billiards. After a brief review of the background and related results, I will discuss key components of the proof, including a resonant normal-form construction that captures the dependence of periodic orbits on the resonant Fourier coefficients of the dynamics.
Dia: Dimecres 15 de juliol de 2026
Lloc: Aula S02, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Álvaro Fernández-Mora (Monash University)
Títol: Computer-assisted approach for the Infinitesimal Hilbert 16th problem
Resum: The infinitesimal Hilbert's 16th problem asks for a uniform upper bound on the number of limit cycles in polynomial Hamiltonian vector fields under polynomial perturbations. Let Z(n) denote this maximum number of limit cycles, where n is the degree of the perturbation and n+1 is the degree of the Hamiltonian. Upper bounds for Z(n) remain largely unknown even for low degrees. In this work, we present novel computer-assisted techniques to approach this classical problem. We apply our framework to construct an alternative proof of the celebrated result Z(2)=2. While the quadratic case is one of the few cases that has been fully solved, our primary objective is to build a scalable toolkit capable of overcoming the algebraic complexity that currently limits the study of Z(n) for large n.
Last updated: Mon Sep 1 12:18:35 2025