Duck Traps: Two-dimensional Critical Manifolds in Planar Systems

In this work we consider two-dimensional critical manifolds in planar fast-slow systems near fold and so-called canard (=`duck') points. These higher-dimension, and lower-codimension, situation is directly motivated by the case of hysteresis operators limiting onto fast-slow systems as well as by systems with constraints. We use geometric desingularization via blow-up to investigate two situations for the slow flow: generic fold (or jump) points, and canards in one-parameter families. We directly prove that the fold case is analogous to the classical fold involving a one-dimensional critical manifold. However, for the canard case, considerable differences and difficulties appear. Orbits can get trapped in the two-dimensional manifold after a canard-like passage thereby preventing small-amplitude oscillations generated by the singular Hopf bifurcation occurring in the classical canard case, as well as certain jump escapes.