Concerning the N body problem, the members of the team have worked in different problems. We have analyzed the asymptotic density of collisions in the Restricted Planar Circular 3 Body Problem. The question of density of collisions was raised by Alekseev (and goes back to Siegel). We have also worked in the stochastic behavior along mean motion resonances in the 3 body problem. Those resonances are physically relevant, since they correspond to the Kirkwood gaps present in the Asteriod belt in the Solar system. We have also constructed hyperbolic invariant sets with symbolic dynamics for the 3 body problem which contain, in particular, oscillatory motions for the 3 body problem with any choice of masses. Those are orbits such that the bodies ''oscillate'' between infinity and a compact region of phase space, and can be seen as a simplification of the motion of comets. The existence of such hyperbolic sets also imply positive topological entropy for the 3 body problem. Close to these orbits we have also constructed trajectories which undergo a large drift in angular momentum. We have also proven the breakdown of homoclinic connections to the Lagrange point L3, which corresponds to the mean motion resonances 1:1. Associated to this breakdown, we have constructed coorbital chaotic dynamics (Smale horseshoes) and Newhouse domains. Finally, we have constructed Arnold diffusion solutions for the 4 body problem. More precisely, we construct solutions of the 4 body problem such that one of the bodies can change freely its osculating eccentricity, semimajor axis and inclination. The results apply to the planetary regime, which models the Solar system. This is the analytic result of unstable motions in a Solar system model.
On the second area of research, we have also worked in several problems. We have analyzed the problem of transfer of energy (growth of Sobolev norms) close to certain invariant tori of the cubic defocusing nonlinear Schrödinger equation on the 2 dimensional torus. As a consequence, we have proven that those tori are Lyapunov unstable in certain Sobolev spaces in a very strong sense: these norms can grow by an arbitrarily large factor. We have constructed transfer of energy phenomenon (Smale horseshoe type behavior) in the nonlinear Wave, Hartree, Beam and Kirchoff equations. This gives the existence of chaotic-like beating solutions for these PDEs. We have also proven the existence of growth of Sobolev norms solutions for the cubic defocusing nonlinear Schrödinger equation on irrational tori. We have also constructed transfer of energy solutions (Arnold diffusion) in a model of weakly coupled pendulums in an infinite lattice. We have proven the existence of quasiperiodic invariant tori for the Degasperis-Procesi equation. Finally, we have studied the interaction between kinks and deffects in finite dimensional reductions of the perturbed sine-Gordon equation, and we have proven the breakdown of small amplitude breathers in the odd Klein-Gordon equation.
These results have been published in several journals and expained in conferences and workshops. Quanta magazine has written an outreach article about the results on unstable motions for the 4 body problem in the planetary regime.
