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Dia: Dimecres, 23 de setembre de 2020
Lloc: ONLINE https://meet.google.com/xgx-wysa-wea
A càrrec de: Jezabel Curbelo, Universitat Politècnica de Catalunya
Títol: Characterizing the spatial distribution of transport and mixing in geophysical contexts by lagrangian tools
Resum: Transport and mixing are important processes for the spatial distribution of heat, momentum and trace gases in the atmosphere and oceans. Finding order in the apparent chaos that seems to govern transport processes is a challenge. To quantify the spatial distribution of eddy mixing in the atmosphere we are following a dynamical system approach in the Lagrangian framework. This talk presents several examples of our methodology revealing beautiful time dependent geometries that comprise an efficient underlying transport network. The results to be presented show that our approach can provide valuable insight into the phase space structures that govern and mediate the transport in geophysical contexts.
Dia: Dimecres, 28 d'octubre de 2020
Lloc: ONLINE https://eu.bbcollab.com/guest/e72c23cf69754e1ca7555595a9a0f2bf
A càrrec de: Clara Cufí Cabrè, Universitat Autònoma de Barcelona
Títol: Differentiable invariant manifolds of nilpotent parabolic points
Resum: We consider a map F of class 𝒞r with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of F, there exist invariant curves of class 𝒞r away from the fixed point, and that they are analytic when F is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of F on them.
This is a joint work with Ernest Fontich (Universitat de Barcelona).
Dia: Dimecres, 9 de desembre de 2020
Lloc: ONLINE https://eu.bbcollab.com/guest/e72c23cf69754e1ca7555595a9a0f2bf
A càrrec de: Josep Maria Mondelo, Universitat Autònoma de Barcelona
Títol: Flow map parameterization methods for invariant tori in Hamiltonian systems
Resum: The goal of this talk is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincare maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant tori and their invariant bundles around librational equilibrium points of the Restricted Three Body Problem. The invariant bundles are the first order approximations of the corresponding invariant manifolds, commonly known as the whiskers, which are very important in the dynamical organization and have important applications in space mission design.
Joint work with Àlex Haro.
Dia: Dimecres, 10 de març de 2021
Lloc: ONLINE https://meet.google.com/rbq-vszv-uvx
A càrrec de: Ezequiel Maderna, Universidad de la República
Títol: Soluciones globales de viscosidad para la ecuación de Hamilton-Jacobi en el problema clásico de N cuerpos.
Resum: Una situación muy probable en la evolución de un sistema de masas puntuales, cuyas posiciones evolucionan bajo el efecto de las atracciones gravitacionales mutuas, es la de divergencia de todas las distancias entre los cuerpos cuando el tiempo t → +∞. Esto sólo puede ocurrir cuando la energía total de la órbita h es no negativa. A estas soluciones del problema de N cuerpos las llamamos expansivas. Cuando la constante h es nula, es sabido que las distancias mutuas divergen como t2/3, y si bien el efecto de la gravedad se desvanece, su efecto fuerza de todos modos a que la configuración de los cuerpos en el espacio tenga una forma que debe aproximarse al conjunto de formas de las configuraciones centrales: un conjunto muy pequeño de formas, cuya finitud (que es sabida para el caso N≤4) se conjetura que vale para cualquier cantidad de cuerpos.
Vamos a ver que que cuando h>0 todas las formas sin colisiones se realizan como figuras límite de expansiones, aún eligiendo arbitrariamente las posiciones iniciales de los cuerpos. Para probar esto, estudiaremos la dinámica de los niveles de energía positivos de forma geométrica, construiremos las funciones de Busemann asociadas a la forma de expansión elegida, veremos que estas funciones son soluciones globales de viscosidad de la ecuación de Hamilton-Jacobi H(x,d_xu)=h, y sus curvas de gradiente son rayos geodésicos de la métrica de Jacobi-Maupertuis y asintóticos a la dirección elegida (trabajo en colaboración con A.Venturelli).
Si el tiempo lo permite revisaremos una lista importante de problemas abiertos relacionados, para los cuales pensamos que este enfoque puede resultar adecuado.
Dia: Dimecres, 17 de març de 2021
Lloc: ONLINE https://meet.google.com/hdk-gckv-ork
A càrrec de: Mar Giralt, Universitat Politècnica de Catalunya
Títol: Beyond all orders breakdown of the homoclinic connection to L3 in the Restricted Planar Circular 3-Body Problem
Resum: We consider the Restricted Planar Circular 3-Body Problem (RPC3BP) with primaries mass ratio μ small. This configuration has a saddle-center equilibrium point called L3 (collinear with the primaries and beyond the largest one) with a 1-dimensional stable and unstable manifold. Moreover, the modulus of the hyperbolic eigenvalues are smaller than the elliptic ones by a factor of √μ. In this talk, we present an asymptotic formula for the distance between the stable and unstable manifolds of L3. This distance is exponentially small with respect to √μ and therefore, classical perturbative methods do not apply.
One of the main challenges of the proof is to obtain a good first order with an homoclinic connection and to analyze the complex singularities of its time parametrization. To obtain the the leading term of the difference, the perturbed manifolds have to be analyzed close to these singularities.
This is a joint work with Inma Baldomá and Marcel Guardia.
Dia: Dimecres, 24 de març de 2021
Lloc: ONLINE https://meet.google.com/hdk-gckv-ork
A càrrec de: Marina Gonchenko, Universitat Politècnica de Catalunya
Títol: Reversible perturbations of Hénon-like maps
Resum: We consider area-preserving Hénon-like maps and their compositions and study smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. To construct these perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families which contain conservative orientable and non-orientable Hénon maps as well as the product of two Hénon maps whose Jacobians are mutually inverse. In particular, we study how reversible non-conservative perturbations affect to the structure of the 1:3 resonance, i.e. bifurcations of fixed points with eigenvalues e± i 2π/3, in the conservative cubic Hénon maps.
These are joint works with S. Gonchenko, K. Safonov, A. Kazakov, E. Samylina and A. Shykhmamedov.
Dia: Dimecres, 7 d'abril de 2021
Lloc: ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Filippo Giuliani, Universitat Politècnica de Catalunya
Títol: Time quasi-periodic traveling gravity water waves in infinite depth
Resum: I will present a result of existence and stability of quasi-periodic in time, small amplitude solutions of the pure gravity water waves bi-dimensional system with infinite depth. In this case the rest surface is a completely resonant elliptic fixed point of the phase space, in the sense that the linearized problem at the origin possesses an infinite dimensional subspace on which the dynamics is periodic. Then the search for quasi-periodic solutions requires a refined bifurcation nonlinear analysis, especially to deal with lower order non-trivial resonances, such as the well-known Benjamin-Feir resonances. The proof is based on a combination of a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. This is a joint work with R. Feola.
Dia: Dimecres, 21 d'abril de 2021
Lloc: ONLINE https://meet.google.com/ixe-qezq-uvp
A càrrec de: Marcel Guàrdia, Universitat Politècnica de Catalunya
Títol: Breakdown of small amplitude breathers for the nonlinear Klein-Gordon equation
Resum: Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions). We also construct generalized breathers, these are solutions which are periodic in time and in space are localized up to exponentially small (with respect to the amplitude) tails. This is a joint work with O. Gomide and T. Seara.
Dia: Dimecres, 5 de maig de 2021
Lloc: ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Eva Miranda (UPC-CRM-Observatoire de Paris) i Daniel Peralta Salas (ICMAT)
Títol: Looking at Euler flows through a contact mirror
Resum: The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, In the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago. Another application of this contact mirror concerns the study of singular periodic orbits (including escape orbits in Celestial mechanics) that we briefly discuss.
Time permitting, we end up this talk addressing an apparently different question: What kind of physics might be non-computational? The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere, But, can this result be improved?
This talk is based on several joint works with Cédric Oms and with Robert Cardona and Fran Presas.
Dia: Dimecres, 19 de maig de 2021
Lloc: ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Jean-Pierre Marco (Université Pierre et Marie Curie)
Títol: Attracted by an elliptic fixed point
Resum: This talk will be dedicated to the construction of an example of close-to-identity C∞ symplectic diffeomorphism on ℝ2n, with an elliptic fixed point at the origin, which admits an orbit which converges to the origin. We will also mention some more recent results in this direction.
This is a joint work with Bassam Fayad and David Sauzin.
Last updated: Monday, 28-Jun-2021 08:42:09 CEST