Kepley, Shane

Gravitational N-body problems are central in classical mathematical physics.
Studying their long time behavior raises subtle questions about the interplay between
regular and irregular motions and the boundary between integrable and chaotic dynamics.
Over the last hundred years, concepts from the qualitative theory of dynamical
systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles,
KAM theory, and whiskered invariant tori, have come to play an increasingly important
role in the discussion. In the last fty years the study of numerical methods for
computing invariant objects has matured into a thriving sub-discipline. This growth
is driven at least in part by the needs of the world's space programs.
Recent work on validated numerical methods has begun to unify the computational
and analytical perspectives, enriching both aspects of the subject. Many
of these results use computer assisted proofs, a tool which has become increasingly
popular in recent years. This thesis presents a proof that the circular restricted four
body problem is non-integrable. The proof of this result is obtained as an application
of more general rigorous numerical methods in nonlinear analysis.

The Cauchy Green strain tensor provides an effective tool for understanding unsteady flows. In
particular, the dominant eigenvalue of this tensor has been seen to be a reliable estimator of the finite
time Lyapunov exponent. We propose a new method for computing the CG strain tensor using a local
quadratic regression LOESS technique. We compare this LOESS method with several classical
methods using closed form flows, noisy flows, and simulated time series. In each case, the CG strain
tensor produced by the LOESS method is remarkably
accurate and robust compared to classical methods.