Qualitative Behavior Of Dynamical Vector Fields

Differential Equations come in two classes, deterministic and stochastic. The first part of this analysis establishes the stable properties of the set of all trajectories converging on a critical point in the real plane defined by two distinct negative eigenvalues. Also in the deterministic class I offer a method for finding closed-form primitives for a great variety of differential forms, through a reduction process facilitated by a Lyapunov-type Energy function. Many of these forms lie in classes which heretofore have not been shown to be solvable in closed form. The last part of this work outlines the appropriate procedures for calculating differentials and solutions for fields perturbed by random processes . In the final chapter a new theory of Laplace Transforms for stochastic calculations has been developed. The introduction of a Table of Transforms has been initiated, and shall eventually be enlarged. Applications are offered to demonstrate its utility.