Mathematical models that describe the physical reality often exhibit erratic behavior, and include small noise accounting for external perturbations. A fundamental problem is to understand the long term behavior of such systems. The origins of this problem go back to one of the oldest questions in dynamics- "Is the Solar System stable?"-which remains largely unsolved to this day. Instability turns out to be a rather typical regime. A conjecture formulated by V.I. Arnold in 1964 asserts that even simple mechanical systems, modeled as integrable Hamiltonian systems, start to exhibit chaotic and unstable behaviors when they become subjected to small, external perturbations. Despite recent progress in the understanding of this problem, a comprehensive chart of the routes yielding instability and chaos is far from completion. Most current approaches apply only to perturbations of special types, and of very small sizes. This limitation makes it difficult to analyze realistic models. The objective of this project is to develop new methods to analyze the effects of realistic perturbations on integrable Hamiltonian systems. The proposed research considers general types of perturbations, which are not necessarily of an extremely small size, and aims to formulate rigorous results that can be applied to concrete models. Particularly, the conditions considered on these systems are explicit and verifiable, rather than of generic type. The approach intends to expand and innovate techniques from differential, algebraic, low dimensional, and symplectic topology. This machinery will be used in tandem with methods based on normal hyperbolicity, perturbation theory, KAM and Aubry-Mather theories, pseudo-holomorphic curves, and random dynamics.
A significant strength of the proposed approach is its applicability to practical situations. This research yields recipes to increase the energy of physical systems with small forcing, with potential applications to particle accelerators, plasma confinement devices, chemical reactions, astrodynamics, and dynamical astronomy. The results from the proposed investigation on instability in the three-body problem can be applied to design fuel efficient trajectories for spacecrafts to explore the Solar System, or to change the orbits of satellites in specific fashions. As an example, one of the methods discussed in this research has been used to design the trajectory of the current NASA's GRAIL mission to the Moon, which begun on January 1st, 2012. The study of random perturbations of deterministic systems proposed in this project plays a key role in modeling of climate change. In addition, this project will pro-actively engage students, including members of underrepresented groups, in educational and research activities, and will contribute to the professional growth of K-12 educators.
