\varepsilon}}$,

-   a segment (green) that flows from $S^{a+}_{{ \varepsilon}}$ to $S^{r}_{{ \varepsilon}}$, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,

-   a segment (blue) that flows back down into the region near $p_{eq}$, making small oscillations around $W^s$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,

-   a segment (magenta) with similar dynamics to the green segment, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$, and

-   a segment (black) making a large-amplitude excursion by jumping left to $S^{a-}_{{ \varepsilon}}$, flowing to the fold $L_{-2/3}$, and then jumping to $S^{a+}_{{ \varepsilon}}$.

A linearized flow map can be constructed[@glendinning1984; @silnikov1965] in small neighborhoods of the saddle-focus $p_{eq}$, which can be used to count the number of small-amplitude oscillations contributed by orbit segments approaching the equilibrium point. However, the small-amplitude periodic orbit and the twist region produce additional twists, as observed in the green and magenta segments of the example above.

(a)![\[fig:z0\] (a) Geometry in the section $\Sigma_0 = \{(x,y,z): x \in \left[-0.07,0.11\right], y \in \left[-0.005,0.01\right], z=0\}$. Gray points sample the subset of $\Sigma_0$ whose corresponding 