The Effect of Geometry on the Dynamics of a Three Timescale System

Abstract

Ordinary differential equations with multiple timescales are frequently used to model systems with processes that take place on different timescales. Geometric singular perturbation theory (GSPT) provides methods for the analysis of such systems, including determining their bifurcation structures and the types of oscillations that can occur. This thesis uses ideas from GSPT to analyse the dynamics of a system of three ordinary differential equations in which there are three distinct timescales. We are particularly interested in the effect on the dynamics of the relative positions of two geometric objects: the critical manifold and the superslow manifold. The system of interest has a cubic-shaped critical manifold and also a cubic-shaped superslow manifold contained within the critical manifold. When both folds of the superslow manifold lie on one attracting sheet of the critical manifold, we find there are four Hopf bifurcations for the system. This configuration gives spike adding transitions in the orbit arising from one of the Hopf bifurcation branches. When we adjust the left fold of the critical manifold so that it is closer to the right fold of the critical manifold the overall bifurcation structure is largely unchanged, but the nature of solutions on one of the branches of periodic solutions changes: small amplitude oscillations appear in the large amplitude periodic orbit. When the left fold of the superslow manifold is shifted to the middle repelling branch of the critical manifold only two Hopf bifurcations occur in the system, although spike adding still occurs along one branch of periodic orbits and the mechanism for this seems similar to that seen in the earlier case.When the left and right folds of the superslow manifold are located on different attracting branches of the critical manifold, two Hopf bifurcations are observed, but the periodic orbits occurring in the system are quite different from before. In particular, no spike adding is observed and the oscillations mostly look like relaxation oscillations. From these case studies, we conclude that the relative positions of the critical and superslow manifolds can have a strong influence on the dynamics of a three timescale system.