Fig. 5

Folded saddle geometry and associated trajectories. The geometry of a generic folded saddle. The folded saddle (purple) is denoted ${\mathbf{p}}_{\mathrm{fs}}$, while the fold curve (black dashed) is denoted F. a Folded saddle geometry according to the singular reduced problem. The folded singularity resembles an ordinary saddle equilibrium with stable and unstable invariant manifolds (red). The trajectories (blue) follow these invariant manifolds moving away from the stable manifold and toward the unstable. b Within the desingularized reduced problem the dynamics on the repelling surface, $S^r$, are reversed due to the rescaling of time (desingularization). Trajectories may pass through the folded saddle with non-zero velocity traveling either of the invariant manifolds. These trajectories correspond to singular canards; the stable invariant manifold to the true canard and the unstable invariant manifold to the faux canard. c Folded saddle geometry in 3D. The true canard acts as a separatrix on the attracting surface, $S^a$. If a trajectory lands within the region enclosed by the true and faux canards, then it is bound away from the fold curve. However, if the trajectory lands within the region enclosed by the fold curve and true canard it travels toward the fold curve. Here, the trajectory “jumps off” due to a blow up in finite time of the desingularized reduced problem, where subsequent dynamics are dictated by the layer problem. The region within which trajectories necessarily ‘jump off’ the critical manifold is indicated (purple shaded)