Abstract

When one studies a system of two coupled differential equations, complications may arise since the phase plane is unbounded. In some cases, one may fold the dynamics onto a compact, two-dimensional manifold (surface) in a meaningful way. The compactness of the new phase space allows for easier analysis. Flows on compact manifolds may also appear naturally, such as on an invariant torus occuring in linear undamped systems with two degrees of freedom.
Though the compactness of the manifold avoids some difficulties, classifying topologically different flows on two-manifolds is still challenging because of, among other things, the presence of periodic orbits. By introducing a representation of a flow (called the skeleton) which simplifies the periodic orbit structure, one finds there exist only a finite number of skeletons for a specified finite number of hyperbolic equilibria. Flows on the sphere form the basis of further work concerning flows on the Klein bottle, the projective plane, and the Moebius strip with a periodic boundary. An explicit classification for small numbers of equilibrium points on these manifolds is given.
This talk assumes only an introduction to systems of differential equations. Background material, such as the classification of compact two-manifolds, the Poincare-Bendixson Theorem and the Poincare Index Theorem, will be provided.