Mireles-James, Jason D.

In this report we study the Aizawa field by first computing a Taylor series
expansion for the solution of an initial value problem. We then look for singularities
(equilibrium points) of the field and plot the set of solutions which lie in the linear
subspace spanned by the eigenvectors. Finally, we use the Parameterization Method
to compute one and two dimensional stable and unstable manifolds of equilibria for
the system.

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.