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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - HamInstab (Instabilities and homoclinic phenomena in Hamiltonian systems)

Teaser

Summary

A fundamental problem in the study of dynamical systems is to ascertain whether the effect of a perturbation on an integrable Hamiltonian system accumulates over time and leads to a large effect (instability) or it averages out (stability). Instabilities in nearly integrable systems, usually called Arnold diffusion, take place along resonances and by means of a framework of partially hyperbolic invariant objects and their homoclinic and heteroclinic connections. The goal of this project is to develop new techniques, relying on the role of invariant manifolds in the global dynamics, to prove the existence of physically relevant instabilities and homoclinic phenomena in several problems in celestial mechanics and Hamiltonian Partial Differential Equations.

The N -body problem models the interaction of N puntual masses under Newton gravitational force. Astronomers have deeply analyzed the role of resonances in this model. Nevertheless, mathematical results showing instabilities along them are rather scarce. I plan to develop a new theory to analyze the transversal intersection between invariant manifolds along mean motion and secular resonances to prove the existence of Arnold diffusion. I will also apply this theory to construct oscillatory motions.

Several Partial Differential Equations such as the nonlinear SchrÃ¶dinger, the Klein-Gordon and the wave equations can be seen as infinite dimensional Hamiltonian systems. Using dynamical systems techniques and understanding the role of invariant manifolds in these Hamiltonian PDEs, I will study two type of solutions: transfer of energy solutions, namely solutions that push energy to arbitrarily high modes as time evolves by drifting along resonances; and breathers, spatially localized and periodic in time solutions, which in a proper setting can be seen as homoclinic orbits to a stationary solution.

Work performed

Concerning the N body problem, the members of the team have been working 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 (andgoes back to Siegel). IWe 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. In parallel, we are working on the existence of 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 simplification of the motion of comets. Finally, we have also started working on the homoclinic and chaotic phenomena around the Lagrange point L3, which corresponds to the mean motion resonances 1:1.

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 SchrÃ¶dinger equation on the 2 dimensional torus. As a consequence, we prove 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 are currently working on the existence of local instabilities (Smale horseshoe type behavior) in the Hartree and beam equation on the 2 dimensional torus. In parallel, we have started working on the transfer of energy phenomenon (Arnold diffusion) in a pendulum lattice and 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.

Final results

All the progress achieved in the different problems adressed are well beyond the state of the art and concern fundamental problems in the study of unstable motions in Hamiltonian systems.