Mathematical physics.

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.

A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the
standard horseshoe, but where the set of points that escape are now reinjected and contribute to
richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing
an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from
classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic
topology to show that for suitable parameters the flow contains a strange attractor.