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 ofsolidconvection 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).
Last updated: Fri Nov 8 20:08:05 2024