Periodic Reporting for period 5 - HamInstab (Instabilities and homoclinic phenomena in Hamiltonian systems)
Reporting period: 2022-09-01 to 2023-12-31
SUMMARY OF THE CONTEXT AND OBJECTIVES OF THE PROJECT
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 Schrödinger, 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.
CONCLUSIONS OF THE ACTION
The results obtained during the six years of the ERC Starting grant have been outstanding in the different objectives of the project.
I have shown that instabilities appear widely in Celestial Mechanics models, in several different regions in the phase space and for a wide set in the parameter space. In particular, I have constructed oscillatory motions, chaotic motions and drifting orbits by means of an Arnold diffusion mechanism.
I have also analyzed transfer of energy phenomena in Hamiltonian Partial Differential Equation both from the quantitative and qualitative point of view and I have also shown transfer of energy for a model of weakly couple pendulums in an infinite lattice, which is the first example of an Arnold diffusion mechanism in infinite dimensions. Finally, I have shown the breakdown of small amplitude breathers in Klein-Gordon equations.
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 Schrödinger, 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.
CONCLUSIONS OF THE ACTION
The results obtained during the six years of the ERC Starting grant have been outstanding in the different objectives of the project.
I have shown that instabilities appear widely in Celestial Mechanics models, in several different regions in the phase space and for a wide set in the parameter space. In particular, I have constructed oscillatory motions, chaotic motions and drifting orbits by means of an Arnold diffusion mechanism.
I have also analyzed transfer of energy phenomena in Hamiltonian Partial Differential Equation both from the quantitative and qualitative point of view and I have also shown transfer of energy for a model of weakly couple pendulums in an infinite lattice, which is the first example of an Arnold diffusion mechanism in infinite dimensions. Finally, I have shown the breakdown of small amplitude breathers in Klein-Gordon equations.
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.
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.
All the progress achieved in the different problems addressed are well beyond the state of the art and concern fundamental problems in the study of unstable motions in Hamiltonian systems.
All the results concerning chaotic dynamics and oscillatory motions in Celestial Mechanics applied up to now to narrow ranges of parameters or restricted models. I have provided the first general result for the 3 body problem. I have also provided the first result on unstable motions in a Celestial mechanics model in a planetary regime, that is in a model of our Solar system. The constructed trajectories are such that the planets may have drastic changes in eccentricity, inclination and semimajor axis. Even if Newhouse domains were widely believed to exist in Celestial Mechanics, I have provided the first complete result of this fact.
I have constructed transfer of energy solutions in a pendulum lattice. This is first Arnold diffusion result in a Hamiltonian systems in infinite dimensions (also commonly called Hamiltonian systems with spatial structure). I have also shown for the first time chaotic-like behavior in Hamiltonian Partial Differential Equations. Finally, I have for the first time carried out the analysis of an exponentially small splitting of separatrices in infinite dimensions. This has allowed me to prove a conjecture on the breakdown of breathers for the Klein-Gordon equation.
All the results concerning chaotic dynamics and oscillatory motions in Celestial Mechanics applied up to now to narrow ranges of parameters or restricted models. I have provided the first general result for the 3 body problem. I have also provided the first result on unstable motions in a Celestial mechanics model in a planetary regime, that is in a model of our Solar system. The constructed trajectories are such that the planets may have drastic changes in eccentricity, inclination and semimajor axis. Even if Newhouse domains were widely believed to exist in Celestial Mechanics, I have provided the first complete result of this fact.
I have constructed transfer of energy solutions in a pendulum lattice. This is first Arnold diffusion result in a Hamiltonian systems in infinite dimensions (also commonly called Hamiltonian systems with spatial structure). I have also shown for the first time chaotic-like behavior in Hamiltonian Partial Differential Equations. Finally, I have for the first time carried out the analysis of an exponentially small splitting of separatrices in infinite dimensions. This has allowed me to prove a conjecture on the breakdown of breathers for the Klein-Gordon equation.