across branches of $C$. Trajectory segments lying near the slow manifolds converge to solutions of the slow flow as ${ \varepsilon}$ tends to 0.

\(a) ![image](1apng.png){width="45.00000%"} (b) ![image](1bpng.png){width="45.00000%"}

We now focus on the case of two slow variables and one fast variable ($m=1$, $n=2$). The critical manifold $C$ is two-dimensional and folds of $C$ form curves. Points on fold curves are called [*folded singularities*]{}. when the slow flow is two-dimensional we use the terms “folded node”, “folded focus”, and “folded saddle” to denote folded singularities of node-, focus-, and saddle-type, respectively. In analogy to classical bifurcation theory, folded saddle-nodes are folded singularities having a zero eigenvalue. When they exist, folded saddle-nodes are differentiated by whether they persist as equilibria in the full system of equations. We are interested here in folded-saddle nodes of type II (FSNII), which are true equilibria of the full system. It can be shown that [*singular Hopf bifurcations*]{} occur generically at distances $O({ \varepsilon})$ from the FSNII bifurcation in parameter space.[@guckenheimer2008siam] At this bifurcation, a pair of eigenvalues of the linearization of the flow crosses the imaginary axis, and a small-amplitude periodic orbit is born at the bifurcation point.

Normal forms are used to study the 