NSF Award
2307987
Hamiltonian systems play an essential role in mathematical physics, where they are used to model the behavior of a wide variety of physical systems ranging from electrons in semiconductors to the motions of stars and planets in galaxies. One major difficulty with these systems is that, since there is no friction, there is nothing in the model to smooth out and isolate fine structure. Instead, the evolution of a Hamiltonian system is organized by geometric objects like equilibrium solutions, periodic orbits, and invariant tori. This research project provides new insights into the fine structure of Hamiltonian systems using a blend of computational and analytical tools. Indeed, the project combines computational and analytical techniques in novel ways, providing both new techniques for proving mathematical theorems about the properties of the models, and new computational techniques for understanding their practical behavior. These ideas are used to answer outstanding questions in Celestial Mechanics, and to explore the behavior of gravitating systems of particles at a finer resolution than ever before. Explicit applications of this research include the design/discovery of new low energy transfer orbits for space flights in the solar system.<br/><br/>This project introduces new analytical and computational methods for studying phase space structure in nonlinear systems. Applications to Hamiltonian systems are stressed, as in this case there are no attractors to organize the dynamics. The project has two main parts. The first part is local and expands existing high order techniques for computing hyperbolic invariant objects to encompass the parabolic case. The second part is more global and develops computational methods for growing invariant manifold atlases comprised of high order chart maps for stable/unstable manifolds. An important application is the computation of intersections between such objects, and efficient search strategies are discussed. These intersections (or webs) play a critical role in organizing Hamiltonian systems as they determine transport between different regions of phase space and generate complicated recurrent dynamics. The project involves graduate research activities and facilitates collaborations between researchers in the areas of resurgence theory and computational dynamics.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
