Dia: Dimecres, 2 d'octubre de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.A càrrec de: Amadeu Delshams, UPC
Títol: Breakdown of homoclinic orbits to L1 of the hydrogen atom in a circularly polarized microwave field
Resum: We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian, which is a perturbation of size K > 0 of the standard rotating Kepler problem. In a rotating frame, the largest chaotic region of this system lies around a center-saddle equilibrium point L1 and its associated invariant manifolds. We compute the distance between stable and unstable manifolds of L1 by means of a semi-analytical method, which consists of combining normal form, Melnikov, and averaging methods with numerical methods.
Also, we introduce a new family of Hamiltonians, which we call Toy CP systems, to be able to compare our numerical results with the existing theoretical results in the literature. It should be noted that the distance between these stable and unstable manifolds is exponentially small in the perturbation parameter K (in analogy with the L3 libration point of the R3BP).
This is a UPC joint work with Mercè Ollé, Juan Ramon Pacha and Óscar Rodríguez.
Dia: Dimecres, 16 d'octubre de 2024
Lloc: Aula S03, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.A càrrec de: Maria Saprykina (KTH, Suècia)
Títol: KAM-rigidity for parabolic affine abelian actions on the torus Resum: Two famous instances of local rigidity for ℤ2-actions are the classical KAM rigidity of Diophantine toral translations and smooth rigidity of hyperbolic or partially hyperbolic higher rank actions proved by Damjanovic and Katok. To complete the study of local rigidity of affine ℤ2-actions on the torus, we address the case of parabolic affine actions. Consider an affine ℤ²-action (a,b) on 𝕋d generated by two commuting parabolic affine maps of the form a(x)=A(x)+α, b(x)=B(x)+β, where A,B ∈ SL(d,ℤ). We say that a linear map A∈ SL(d,ℤ) is parabolic of step 2 if (A−Id)²=0. An affine map a(x)=A(x)+α is of step 2 if A is of step 2. We say that the action (a,b) is KAM-rigid under volume-preserving perturbations if there exists σ ∈ ℕ, r0≤ σ and ε>0 satisfying the following. If r≤ r0 and (F,G)=(a+f,b+g ) is a smooth λ-preserving ℤ²-action such that
∥f∥r≤ ε, ∥g∥r≤ ε, ^ f ≔ ∫𝕋² f dλ=0, ^ g ≔ ∫𝕋² g dλ=0,
then there exists H=Id+h ∈ Diff∞λ(𝕋d) such that ∥h∥r-σ≤ ε and H∘(a+f)∘H-1 = a, H∘(b+g)∘H-1 = b.
Let 𝒯(A,B) denote the set of possible translation parts (α,β) in the affine actions with linear part (A, B), that is 𝒯(A,B)≔{α, β∈ ℝd ∣ (A-Id)β=(B-Id)α}. We present the following dichotomy for a commuting pair (A,B) of parabolic matrices, where A is step-2 (i.e., (A-Id)²=0):This is the result of a joint work with D. Damjanovic and B. Fayad.
A càrrec de: Kristian Bjerklöv (KTH, Suècia)
Títol: Monotone families of circle diffeomorphisms driven by expanding circle maps.
Resum: We consider monotone families of circle diffeomorphisms forced by strongly expanding circle maps. We obtain estimates of the fibered Lyapunov exponents for such systems and show that in the limit as the expansion tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on the expansion.
This is joint work with Raphaël Krikorian.
Dia: Dijous, 31 d'octubre de 2024
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.A càrrec de: Yanick Ricard
Títol: Convection in the mantle of the Earth and that of large planets
Resum: I will discuss the characteristics of solid
convection in the mantle of Earth and then address more specifically the effets of compressibility. The radial density of planets increases with depth due to compressibility, leading to impacts on their convective dynamics. To account for these effects, including the presence of a quasi-adiabatic temperature profile and entropy sources due to dissipation, the compressibility is expressed through a dissipation number proportional to the planet’s radius and gravity. In Earth’s mantle, compressibility effects are moderate, but in large rocky or liquid exoplanets (super-earths), the dissipation number can become very large. We explore the properties of compressible convection when the dissipation number is significant. We start by selecting a simple Murnaghan equation of state that embodies the fundamental properties of condensed matter at planetary conditions. Next, we analyze the characteristics of adiabatic profiles and demonstrate that the ratio between the bottom and top adiabatic temperatures is relatively small and probably less than 2. We examine the marginal stability of compressible mantles and reveal that they can undergo convection with either positive or negative superadiabatic Rayleigh numbers. Lastly, we delve into simulations of convection in 2D Cartesian geometry performed using the exact equations of mechanics, neglecting inertia (infinite Prandtl number case), and examine their consequences for super-earth dynamics.
Dia: Dimecres, 6 de novembre de 2024
Lloc: Aula S03, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.A càrrec de: Kristian Uldall Kristiansen (Technical University of Denmark)
Títol: The simple turning point: A new approach based upon GSPT, blowup and normal forms
Resum: In this talk, I will present some recent results (based upon a joint paper with Peter Szmolyan, JDE 2024) on the simple turning point problem, defined by ε² x″(t) + μ(t)x(t)=0, μ(0)=0, μ'(0)=1 and 0<ε≪ 1. The objective here is to follow the unstable manifold (line bundle in the (x,εẋ,t)-space) for t=-c<0 across the "turning point" t=0 to t=c>0 with c>0 fixed for all 0<ε≪ 1. The problem is well-known but our approach is new and it has some advantages. In particular, we use GSPT, blowup and normal forms and only rely on smoothness (not analyticitity) of the function μ. The normal form procedure uses summability results of center-like invariant manifolds (associated with certain Taylor-jets of the vector-field). Towards the end of my presentation, I will present an outline (idea) for how this procedure could potentially be used in different contexts with fast oscillations (future work).
Dia: Dimecres, 13 de novembre de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.A càrrec de: José Lamas (UPC)
Títol: Final motions and ejection-collision orbits in the 3 Body
Resum: We consider the Planar Circular Restricted 3 Body Problem (PCRTBP), which describes the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass located at the origin. In rotating coordinates, this system is Hamiltonian with two degrees of freedom. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions.
By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics.
In particular, we show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to the massive primary. Furthermore, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to the massive primary. Additionally, we also establish oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position or velocity is infinity while the inferior limit remains a real number.
Dia: Dimecres 29 de gener de 2025
Lloc: Aula S03, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Sabyasachi Mukherjee (Tata Institute for Fundamental Research, India)
Títol: Topology and geometry of quadrature domains via holomorphic dynamics
Resum: A domain in the complex plane is called a quadrature domain if it admits a Schwarz reflection map; i.e., a meromorphic map that extends continuously to the boundary as the identity map. Quadrature domains have important connections with statistical physics, fluid dynamics, and diverse areas of analysis. We will discuss how classical Riemann surface theory and dynamics of Schwarz reflection maps can be exploited to study the topology and singularities of quadrature domains. We will review earlier works of Gustafsson, Lee, and Makarov in this direction, and introduce classical ideas from holomorphic dynamics to provide sharp upper bounds on the connectivity and number of double points of quadrature domains. Time permitting, we will mention connections between Schwarz reflection dynamics and combination theorems for antiholomorphic polynomials and reflection groups.
A càrrec de: Maciej Capiński (AGH University of Kraków)
Títol: Arnold Diffusion in the Full Three Body Problem
Resum: We will present a geometric mechanism which leads to Arnold Diffusion in the three body problem. As an application of our method we will consider the Neptune-Triton-asteroid system, with the mass of the asteroid playing the role of the perturbation parameter. The proof is computer assisted.
This is joint work with Marian Gidea.
Dia: Dimecres 12 de febrer de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Douglas Duarte Novaes, Instituto de Matemática, Estatística e Computação Científica, (IMECC/UNICAMP), Brasil.
Títol: Limit tori in vector fields: detecting and counting
Resum: Invariant compact manifolds, such as equilibria, periodic orbits, and invariant tori, provide important information about the dynamics of differential systems. This knowledge is significantly increased when we can describe the behavior of nearby trajectories. In this talk, we present conditions that ensure the existence of invariant tori in perturbed differential systems, along with results on their regularity, stability, and dynamics. These conditions are based on higher-order averaged equations and extend classical theorems by Krylov, Bogoliubov, Mitropolsky, and Hale. We also explore a three-dimensional version of Hilbert’s 16th Problem, focusing on the number of isolated invariant tori in 3D polynomial vector fields. For a given degree m, we define N(m) as the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree m and provide a result that relates the number of normally hyperbolic invariant tori in 3D polynomial systems to the number of hyperbolic limit cycles in planar systems. This leads to a lower bound for N(m), showing that it grows as fast as m³/128.
Dia: Dimecres 19 de febrer de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Chiara Caracciolo, Uppsala University / Univ. di Padova
Títol: 3D Orbital Architecture of Exoplanetary Systems: KAM-Stability Analysis
Resum: We study the KAM-stability of several single star two-planet nonresonant extra-solar systems. It is likely that the observed exoplanets are the most massive of the system considered. Therefore, their robust stability is a crucial and necessary condition for the long- term survival of the system when considering potential additional exoplanets yet to be seen. Our study is based on the construction of a combination of lower-dimensional elliptic and KAM tori, so as to better approximate the dynamics in the framework of accurate secular models. For each extrasolar system, we explore the parameter space of both inclinations: the one with respect to the line of sight and the mutual inclination between the planets. Our approach shows that remarkable inclinations, resulting in three-dimensional architectures that are far from being coplanar, can be compatible with the KAM stability of the system. We find that the highest values of the mutual inclinations are comparable to those of the few systems for which the said inclinations are determined by the observations.
Dia: Dimecres 26 de febrer de 2025
Lloc: < span class=black>Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Santiago Barbieri (UPC)
Títol: Existence and nonexistence of invariant curves of coin billiards (joint work with A. Clarke)
Resum: In this talk I will consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map T of the annulus 𝔸 = 𝕋 × (0,π).
Together with A. Clarke, we proved the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary ∂𝔸; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of ∂𝔸 through which there passes no invariant essential curve; and the only coin billiard for which the phase space 𝔸 is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, I will describe the results of some numerical experiments on the elliptical coin billiard.
Last updated: Tue Mar 4 11:24:47 2025