of orbits beginning high up on $S^{a+}_{{ \varepsilon}}$. Fix a section $\Sigma = S^{a+}_{{ \varepsilon}}\cap \{x = 0.27\}$. Before the tangency occurs, trajectories lying on and sufficiently near $W^u$ must either escape to infinity or asymptotically approach $\Gamma$; these trajectories cannot jump to the attracting branches of the slow manifold, as they must first intersect $S^r_{{ \varepsilon}}$ before doing so. Trajectories beginning in $\Sigma$ first flow very close to $p_{eq}$. As shown in Figure 1, these trajectories then leave the region close to $W^u$. We observe that before the tangency, $W^u$ forms a boundary of the basin of attraction of $\Gamma$. Therefore, all trajectories sufficiently high up on $S^{a+}_{{ \varepsilon}}$ must lie inside the basin of attraction (Figure \[fig:tangbif\]a).

After the tangency has occurred, isolated trajectories lying in $W^u$ will also lie in $S^r_{{ \varepsilon}}$. These trajectories will bound sectors of trajectories which can now make large-amplitude passages. Trajectories within these sectors jump ‘to the left’ toward $S^{a-}_{{ \varepsilon}}$ or ‘to the right’ toward $S^{a+}_{{ \varepsilon}}$. Trajectories initialized in $\Sigma$ that leave neighborhoods of $p_{eq}$ near these sectors contain [*canard*]{} segments, which are solution segments lying along $S^r_{{ \varepsilon}}$. Examples of such trajectories are highlighted in green in Figure 1. 