DUT Dynamics Seminar

We consider generalized standard maps, that is, maps $F_h:(x,y)\mapsto(x^{*},y^{*})$ of the form \[ \begin{cases} x^{*}=x+y^{*},\\ y^{*}=y+f(x;h), \end{cases} \] under quite general assumptions. These maps are area preserving. In particular, the Chirikov standard map, the area preserving Hénon map and the case $f(x;h)=\varepsilon(x-x^n)$, $n\geq 2$, where $\varepsilon=4\sinh^2(h/2)$ satisfy our hypotheses. For all these cases, the problem of the existence of transverse homoclinic points arises. The existence of these kind of orbits is one of the most celebrated methods to prove the existence of chaotic dynamics in a system. Indeed the Birkhoff-Smale theorem ensures that if there exist transversal intersections between the invariant manifolds of the same invariant object, the system is locally conjugate to a Smale horseshoe with infinite symbols, i.e., chaotic. We prove that, for a wide class of generalized standard maps, the primary homoclinic points are transverse, provided that some constant is different from 0. An algorithm, computer assisted, to check the non-vanishing character of this constant is obtained and applied to the above maps. This is joint work with Inmaculada Baldomà (UPC), Maciej Capinski (AGH), and Pau Martín (UPC).

We discuss a quasi-Töplitz approach to constructing full-dimensional invariant tori for higher-dimensional nonlinear Schrödinger equations with decaying convolution potentials. The focus is on small-divisor estimates and the use of affine lattice decompositions to control resonances involving two high frequencies.

Invariant circles organize the dynamics of twist maps and act as barriers to transport. This talk explains how geometric reasoning can turn questions about their nonexistence into finite, rigorously verifiable computations. The talk begins with an introduction to computer-assisted proofs, followed by the basic geometry of the standard map and its dissipative counterpart. Then I will briefly review Jungreis’s nonexistence result for the conservative standard map and its geometric ideas. The final part presents our nonexistence proof for the dissipative standard map using a different approach.

In this talk, we consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and an additional angular-momentum rotation term. This type of model arises in the Gross-Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and frequency around a classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action. This is joint work with Christof Sparber.

In this talk, we study the eigenvalues of planar 5-body central configurations. A self-similar periodic solution of the $N$-body problem is called a relative equilibrium, and the corresponding configuration is called a central configuration. We will describe central configurations from an algebraic perspective: their coordinate vectors are exactly eigenvectors of the Hessian of the Newtonian potential, briefly called the BWC matrix. We prove that in a relative equilibrium of 5 bodies the two nontrivial eigenvalues are strictly greater than the three trivial ones. This result improves published inequalities about the central configurations, which belong to two independent lines of research. One starts with Williams in 1938 and concerns constraints on the shape of the configuration. The other concerns the Hessian of the potential and its index, and applies to the linear stability of the relative equilibria. This talk is based on joint work with Alain Albouy.

This talk explains how the infinite-dimensional KAM theorem of Dolgopyat, Fayad, and Paradela applies to a concrete system of infinitely many weakly interacting particles. Starting from the physical Hamiltonian, we impose suitable conditions on the masses and potential, perform the heliocentric and scaling reductions, and pass to action–angle variables. The goal is to show how the concrete model is reduced step by step to the hypotheses of the abstract KAM theorem.

For the infinite Fermi–Pasta–Ulam (FPU) system, general solutions can be approximated by counter-propagating waves associated with solutions to the Korteweg–de Vries (KdV) equation as the lattice mesh size goes to zero. The Toda lattice is a special case of the FPU system. We show that, by exploiting the conservation of the FPU Hamiltonian, the continuum limit from the FPU system to the KdV equation with $L^2$-level initial data holds in appropriate norms on an arbitrary time interval, thereby answering an open question posed by Hong, Kwak, and Yang (2021). Moreover, for the local-in-time continuum limit of the FPU system to the KdV equation, we lower the required Sobolev regularity to $H^s$ with $s > -3/4$. To establish this low-regularity continuum limit, we prove key trilinear estimates that are sharp up to the endpoint by combining linear estimates, the transversality of characteristic curves, and multilinear dispersive smoothing properties of the linear FPU flow. This is joint work with Ruoyuan Liu.