This proposal seeks to develop several important new techniques and tools to further our understanding of stochastic dynamics and classical orbital dynamics. The specific problems are motivated by several key issues in astrophysics, in particular, the analysis of the dynamics of dark matter halos, galactic bulges, and extra-solar planetary systems. At the same time, this project will undertake new research directions concerning several classic problems of current interest in applied mathematics, including the analysis of the stochastically forced Hill's equation, analysis of the asymptotics of the eigenvalues of random matrices, and the dynamics of particles in non-Newtonian potentials. The proposal consists of three interrelated parts. The first pertains to the analysis of orbital instabilities that arise in the dynamics of test particles in extended mass distributions such as dark matter halos. Our analysis leads to a stochastically forced Hill's equation which can be studied by analyzing infinite products of random matrices. The second part discusses the role of turbulence in extra-solar planetary systems and leads to the study of stochastic pendulum problems. The long term dynamics of these systems can also be described by a discrete map with random parameters. The final part considers the phenomenon of tidal stripping in dark matter halos. This work involves the study of orbits of small halos as they fall into larger ones, including dynamical friction and its effects on orbital dynamics. Here, the overarching goal is to understand the nearly universal form found for the matter density profiles, and in particular how they are affected by smaller halos being assimilated into larger structures.
The field of astrophysics has experienced an unprecedented number of observational discoveries and theoretical breakthroughs in the past decade. These discoveries include super-massive black holes in galactic centers, the accelerating universe, measurement of the fluctuations in the cosmic microwave background, extra-solar planets, and brown dwarfs. The observational progress has been made possible through technological innovations, including detectors, telescopes, and spacecraft. Much of the theoretical progress has taken place through numerical simulations, which in turn have benefited from the ever-growing capabilities of computers. Unfortunately, however, the third pillar of this science --- the analytic understanding of these newly discovered astronomical objects and physical phenomena --- is lagging behind. One of the difficulties associated with analytic work in this area is the enormous complexity of the astrophysical systems under study. In particular, chaotic dynamics and sensitive dependence on initial conditions arise in many contexts and render it difficult to make progress through the traditional analytic methods used by astrophysicists. However, the development and application of new mathematical tools, as proposed herein, will facilitate progress on these astrophysical issues, and will be useful in many additional applications. This project will have educational impacts on several fronts, including the training of graduate students, the education of graduates and undergraduates, and reaching out to the general population through public lectures.
