| This dissertation is mainly a study of the center problem in the context of a family of three dimensional systems of ordinary differential equations of the form.;u˙ = - v + P( u,v,w), v˙ = u +Q( u,v,w), w˙ = - lambdaw + R(u,v,w),;for which the right-hand sides are polynomials and lambda is nonzero. Such systems are called polynomial systems. There is a two dimensional local center manifold through the origin. It is invariant under the flow. The problem is to decide whether there is a focus or a center at the origin for the flow restricted to the local center manifold. We first generalize ideas and methods used to study the center and cyclicity problems in the two-dimensional setting to the three-dimensional context. This will involve generalizing to this setting the concepts of the complexification of real systems, normal forms and the center variety, described for two-dimensional systems by Valery G. Romanovski and Douglas S. Shafer in the Center and Cyclicity Problems: A Computational Algebra Approach. We then apply our results to solve the center and cyclicity problems for the Moon-Rand family of systems that arise naturally in an engineering context. |
