cubic surface $C = \{y = x^2 + x^3\}$ having two fold lines $L_0 := S \cap \{x = 0\}$ and $L_{-2/3} := S \cap \{ x = -2/3\}$. When ${ \varepsilon}> 0 $ is sufficiently small, nonsingular portions of $C$ perturb to families of slow manifolds: near the branches $S\cap \{x > 0\}$ (resp. $S \cap \{x < -2/3\}$), we obtain the [*attracting slow manifolds*]{} $S^{a+}_{{ \varepsilon}}$ (resp. $S^{a-}_{{ \varepsilon}}$) and near the branch $S\cap\{-2/3<x<0\}$ we obtain the [*repelling slow manifolds*]{} $S^r_{{ \varepsilon}}$. Nearby trajectories are exponentially attracted toward $S^{a\pm}_{{ \varepsilon}}$ and exponentially repelled from $S^r_{{ \varepsilon}}$. One derivation of these estimates uses the Fenichel normal form.[@jones1994] Within each family, these sheets are $O(-\exp(c/ { \varepsilon}))$ close[@jones1994; @jones1995], so we refer to any member of a particular family as ‘the’ slow manifold. This convention should not cause confusion.

We focus on parameters where forward trajectories beginning on $S^{a+}_{{ \varepsilon}}$ interact with a ‘twist region’ near $L_0$, a saddle-focus equilibrium point $p_{eq}$, or both. A folded singularity $n =(0,0,0) \in L_0$ is the governing center of this twist region. The saddle-focus $p_{eq}$ has a two-dimensional unstable manifold $W^u$ and a one-dimensional stable manifold $W^s$. This notation disguises the dependence of 