local flow of full systems in neighborhoods of these folded singularities. Previous work by Guckenheimer[@guckenheimer2008chaos] analyzes the local flow maps and return maps of three-dimensional systems containing folded nodes and folded saddle-nodes. There, it is shown that the appearance of these folded singularities can give rise to complex and chaotic behavior. Characterizing the emergence of small-amplitude oscillations near a folded singularity has also been the subject of intense study. In the case of a folded node, Benoît[@benoit1990] and Wechselberger[@wechselberger2005] observed that the maximum number of small oscillations made by a trajectory passing through the folded node region is related to the ratio of eigenvalues of the folded node.

The present paper focuses on a dynamical system, defined in Sec. \[sec:shnf\], which contains folded singularities lying along a cubic critical manifold. The critical manifold serves as a global return mechanism. Parametric subfamilies of this dynamical system have served as important prototypical models of electrochemical oscillations, including the Koper model[@koper1992]. This system serves as a concrete, minimal example of a three-dimensional system having an $S$-shaped critical manifold as a global return mechanism. Trajectories leaving a neighborhood of the folded singularities do so by jumping between branches of the critical manifold, before ultimately 