of $R$ ‘escape’. Obviously the trajectories with initial conditions inside $\cup_{i=0}^{\infty}R^{-i}(B_0)$ are contained within the basin of attraction of $\Gamma$, and furthermore the $j$-th iterate of the return map $R^j$ is defined only on the subset $\Sigma_0 - \cup_{i=0}^j R^{-j}(B_0)$. Finally, we abuse notation slightly and denote by $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) the intersections of the corresponding slow manifolds with $\Sigma_0$. We also refer to the intersection of $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) with $\Sigma_0$ as the [*attracting*]{} (resp. [*repelling*]{}) [*spiral*]{} due to its distinctive shape (see Figure \[fig:z0\]). The immediate basin of attraction $B_0$ is depicted by gray points in Figure \[fig:z0\](a). This result implies that the basin of attraction of the periodic orbit contains at least a thickened spiral which $S^{a+}_{{ \varepsilon}}$ intersects transversely in interval segments, accounting for the disconnected images of the one-dimensional return maps.

The slow manifolds also intersect transversely. Segments of the attracting spiral can straddle both $B_0$ and the repelling spiral. In Fig. \[fig:z0\](b), we color initial conditions based on the maximum $y$-coordinate achieved by the corresponding trajectory before its return to $\Sigma_0$. Due to the Exchange Lemma, only thin bands of trajectories are able to remain close enough to $S^r_{{ \varepsilon}}$ 