Kalies, William D.

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.

We describe the lattice structure of attractors in a dynamical system and the lifting of sublattices of
attractors, which are computationally less accessible, to lattices of forward invariant sets and attracting
neighborhoods, which are computationally accessible. We also show how the use of these algebraic
structures of lattices to help us to capture the information about underlying dynamical system in a more
elegant way and with lesser computational cost. For example, they can be used to develop a much
efficient algorithm to compute a global lyapunov function
that describes the overall gradient dynamics.

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.

In this dissertation we present a computational approach to Conley's Decomposition
Theorem, which gives a global decomposition of dynamical systems, and
introduce an explicit numerical algorithm with computational complexity bounds
for computing global dynamical structures of a continous map including attractorrepeller
pairs and Conley's Lyapunov function. The approach is based on finite spatial
discretizations and combinatorial multivalued maps. The method is successful
in exhibiting approximations of attractor-repeller pairs, invariant sets, and Conley's
Lyapunov function. We used the C++ language to code the algorithm.

Ban and Kalies [3] proposed an algorithmic approach to compute attractor-
repeller pairs and weak Lyapunov functions based on a combinatorial multivalued
mapping derived from an underlying dynamical system generated by a continuous
map. We propose a more e cient way of computing a Lyapunov function for a Morse
decomposition. This combined work with other authors, including Shaun Harker,
Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes
the process of nding a global Lyapunov function for Morse decomposition very e -
cient. One of the them is to utilize highly memory-e cient data structures: succinct
grid data structure and pointer grid data structures. Another technique is to utilize
Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is
an essential step to compute a Lyapunov function. Finally, another major technique
in achieving a signi cant improvement in e ciency is the utilization of the lattice
structures of the attractors and attracting neighborhoods, as explained in [32]. The
lattice structures have made it possible to let us incorporate only the join-irreducible
attractor-repeller pairs in computing a Lyapunov function, rather than having to use
all possible attractor-repeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical
system allow for general algebraic treatment of global gradient-like dynamics. The
separation of these algebraic structures from underlying topological structure is the
basis for the development of algorithms to manipulate those structures, [32, 31].
There has been much recent work on developing and implementing general compu-
tational algorithms for global dynamics which are capable of computing attracting
neighborhoods e ciently. We describe the lifting of sublattices of attractors, which
are computationally less accessible, to lattices of forward invariant sets and attract-
ing neighborhoods, which are computationally accessible. We provide necessary and
su cient conditions for such a lift to exist, in a general setting. We also provide
the algorithms to check whether such conditions are met or not and to construct the
lift when they met. We illustrate the algorithms with some examples. For this, we
have checked and veri ed these algorithms by implementing on some non-invertible
dynamical systems including a nonlinear Leslie model.

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.

A second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for second-order Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.