Canards of Folded Saddle-Node Type I

Abstract

The canard phenomenon is robust in singular perturbation problems that have an at least 2-dimensional folded critical manifold. Canards are closely associated with folded singularities and, in the case of folded nodes, lead to a local twisting of invariant manifolds. Folded node canards, folded saddle canards, and their bifurcations have been studied extensively in $\mathbb{R}^3$, which is the minimal dimension for which the canard phenomenon is generic. The folded saddle-node of type I (FSN I) is the codimension-1 bifurcation that gives rise to folded nodes and folded saddles and has been observed in various applications, such as the forced Van der Pol oscillator and in models of neural excitability. Their dynamics, however, are not completely understood. In this work, we analyze the local dynamics near an FSN I by combining methods from geometric singular perturbation theory (blow-up) and the theory of dynamic bifurcations (analytic continuation into the plane of complex time). We prove the existence of canards, faux canards, and their concatenations near the FSN I. We also show that there is a delayed loss of stability and estimate the expected delay.