RUI: Hamiltonian Instability

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This research is devoted to dynamical systems, with emphasis on Arnold instability and chaotic phenomena in Hamiltonian systems. A quintessential example of instability in Hamiltonian systems was described by Arnold in 1964: it represents a simple mechanical system, consisting in a pendulum and two rotators with a weak coupling, whose trajectories wander `wildly' and `arbitrarily far' from their points of departure in the phase space. This example gave rise to a conjecture, referred as Arnold instability, stating that such a behavior occurs in rather general systems. The first objective of this project is to investigate this conjecture on some general models and identify various geometric and topological mechanisms of instability. Perturbations of integrable Hamiltonian systems will be considered. Under certain non-degeneracy assumptions, there exist families of KAM tori that posses invariant manifolds; the invariant manifolds give rise to connecting orbits between nearby tori, but separating gaps also appear. The main goal is to find diffusing trajectories that move along the connections that link these tori and also move across the gaps, traveling a uniform distance as the size of the perturbation tends to zero. Some of these trajectories should be able to make chaotic excursions. The methodology of this research will be based on a topological technique of correctly aligned windows, reinforced with geometric approximation theory and variational methods. The main goals of this research are to overcome the large gap problem, obtain optimal estimates on the diffusion time, relax the transversality and non-degeneracy assumptions on the system, and study the existence of diffusion for large perturbations. The second objective of this project is to apply the knowledge gained from the study of Arnold instability to the three-body problem. The goal is to understand the geometric mechanisms that produce chaotic motions and find optimal trajectories with prescribed itineraries. The objectives of this project will be addressed both theoretically and numerically.The general context of this research is the study of the cumulative effect of small perturbations applied periodically to a stable system. This study originated from a question that goes back to Netwon, Lagrange and Laplace, whether the Solar System in the distant future will keep the same form as it is now, or will undergo a catastrophic change. If in some perturbed systems, like the Solar System, stability is predominant, in some other systems instability is typical. The Arnold instability can be viewed as a recipe on how to increase the energy of physical systems with small periodic forcing. Arnold instability ideas applied to the three-body problem have potential applications to spacecraft dynamics, in designing fuel efficient space missions exploring various regions of the Solar System. The techniques outlined in this project can also be used in studying the dynamics of planets orbiting systems of binary stars, and of mass transfers in tight binary star systems (e.g. Algol in Betta Persei). Other possible applications include heart pacemaking, plasma confinement, and accelerator physics. Additionally, this project will enhance the knowledge and professional development of students and K-12 educators. It will directly engage students, including members of underrepresented groups, in research activities and mathematics education.