Geometry and Gevrey asymptotics of two-dimensional turning points

Abstract

This thesis deals with singular perturbation problems in the plane, or more generally on 2-manifolds. A differential system (or vector field) with a small parameter ε is called singularly perturbed if the order of the differential system is lower for ε = 0 than for ε ≠ 0. Roughly, the reduced set of differential equations (the equations for ε = 0) are easier to study, while the addition of small perturbations causes a significant increase in complexity. The perturbation can|no matter how small|have a drastic effect on the solutions of the differential equations. Small perturbations can accumulate and have an effect on longer time scales. One says that the differential system exhibits different time scales: a fast time scale, on which small perturbations have little or no effect, and a slow time scale. This kind of "slow{fast" systems have a large number of applications in both sciences and industry. Applications of singular perturbation theory such as in the study of chemical reactions, specific electric circuits, biological eco-models, . . . are well-documented in the literature. In this thesis, we are interested in the study of so-called \turning points". On the fast time scale the dynamics is determined by a fast attraction towards some stable equilibrum state. The stability of this equilibrum can loose its strength on a slower time scale, and can eventually become unstable at some (turning) point. One expects that this instability will ensure that the solutions that were close to the equilibrum state before the turning point will immediately be repelled from the equilibrum state after passing near the turning point. However, it appears that sometimes the loss of stability is delayed: solutions stay near the equilibrum state for some time before the loss of stability becomes dominant. ...