The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, the well-developed geometric theory does not apply. We present a method based on blow-up techniques, which leads to a rigorous geometric analysis of these problems. A detailed analysis of the extension of slow manifolds past fold points and canard points in planar systems is given. The efficient use of various charts is emphasized.

We give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields. The transition from small Hopf-type cycles to large relaxation cycles, which occurs in an exponentially thin parameter interval, is described as a perturbation of a family of singular cycles. The results are obtained by means of two blow-up transformations combined with standard tools of dynamical systems theory. The efficient use of various charts is emphasized. The results are applied to the van der Pol equation. Academic Press

We give a geometric analysis of canard solutions in three-dimensional singularly perturbed systems with a folded two-dimensional critical manifold. By analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold-curve. We classify these singularities, called canard points, as folded saddles, folded nodes, and folded saddle-nodes. We prove the existence of canard solutions in the case of the folded saddle. We show the existence of canards in the folded node case provided a generic non-resonance condition is satisfied and in a subcase of the folded saddle-node. The proof is based on the blow-up method.

The existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically. It is shown that near a singular periodic orbit a return map can be defined which has a one-dimensional slow manifold with a stable invariant foliation. Under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter. Additionally the existence of an invariant torus is proved for the periodically forced van der Pol oscillator. The analysis is based on methods from geometric singular perturbation theory. The blow-up method is used to analyze the dynamics near the fold-curves.

We consider the dynamics of singularly perturbed differential equations near points where the critical manifold has a transcritical or a pitchfork singularity. Our main tool is the recently developed blow-up method, which allows a detailed geometric analysis of such problems. A version of the Melnikov method to study transversality properties in this and related problems is developed.