Figure 8

Bistable slow–fast phase portraits as reduction of a larger dimensional model. Left: critical manifolds obtained as the intersection of a higher-dimensional one (green) with a surface (gray). Right: corresponding phase plane with the critical manifold obtained (green) and a possible nullcline for the slow variable (dashed) that completes the dynamics. Top: Hindmarsh–Rose model can be obtained constraining the dynamics to a plane, the critical manifold in the phase plane is the classical N-shaped one, but presents nontrivial dynamics leading to rest–spike bistability. Bottom: the transcritical model obtained constraining the dynamics to a surface. The transcritical bifurcation is obtained when this surface is tangent to a line of folds at a point. This bifurcation is responsible for a singular homoclinic trajectory in the planar reduction