Enhanced delay to bifurcation

Abstract

We present an example of slow-fast system which displays a full open set of initial data so that the corresponding orbit has the property that given any $\epsilon$ and $T$, it remains to a distance less than $\epsilon$ from a repulsive part of the fast dynamics and for a time larger than $T$. This example shows that the common representation of generic fast-slow systems where general orbits are pieces of slow motions near the attractive parts of the critical manifold intertwined by fast motions is false. Such a description is indeed based on the condition that the singularities of the critical set are folds. In our example, these singularities are transcritical.