Fig. 6

The desingularised reduced problem. The desingularized reduced problem, (15), near the lower fold is projected onto $(v,s)$-space. Here, ${\tau }_s=15$, although the basic structure shown here is common to all values of ${\tau }_s$ analyzed. The lower fold curve, $F^{-}$ (gray dashed), is indicated. The folded saddle, ${\mathbf{p}}_{\mathrm{fs}}$ (purple), lies on the fold curve and gives rise to two invariant manifolds: a stable invariant manifold, $W^s({\mathbf{p}}_{\mathrm{fs}})$ (red), and an unstable invariant manifold, $W^u({\mathbf{p}}_{\mathrm{fs}})$ (blue). The dynamics in $(v,s)$-space above $F^{-}$ are reversed, and so the stable and unstable manifolds have reversed stability properties above $F^{-}$. Arrows indicate motion along the invariant manifolds. Each manifold terminates at a stable node equilibrium, within the reduced flow. The stable manifold terminates at ${\mathbf{eq}}_2$ on $S_a^{-}$ (orange) and the unstable manifold terminates at ${\mathbf{eq}}_3$ on $S_r^{-}$ (green). Note, if a singular trajectory lands onto the shaded region of $S_0$, it eventually undergoes a rebound spike. Inset: A magnification of the desingularized reduced problem near the folded saddle