A topological perspective on singular canards for critical sets with transverse intersections

Abstract

This paper gives a new perspective on singular canards, which is topological in flavour. One key feature is that our construction does not rely on coordinates; consequently, the conditions for the existence of singular canards that we provide are purely geometric. The singularities we study originate at the self-intersection of curves of equilibria of the unperturbed system. Our contribution even allows us to consider degenerate cases of multiple pairwise transverse intersecting branches of the critical set. We employ stratification theory and algebraic geometric properties to provide sufficient conditions leading to the presence of singular canards. By means of two examples, we corroborate our findings using the well-known blow-up technique.