saddle-nodes[@guckenheimer2008chaos; @krupa2010]. Where the return map is defined, trajectories beginning in different components of the domain of $R$ make different numbers of small turns before escaping the local region. These subsets are somewhat analogous to the rotation sectors arising from twists due to a folded node.[@wechselberger2005] However, in the present case there is a folded singularity as well as a saddle-focus as well as a small-amplitude periodic orbit. Each of these local objects plays a role in the twisting of trajectories that enter neighborhoods of the fold curve $L_0$.

When the small-amplitude stable periodic orbit exists, the domain of the return map is now disconnected, with components separated by finite-length gaps (Figure \[fig:retmap\]). The gaps where $R$ is undefined correspond to those trajectories beginning on $S^{a+}_{{ \varepsilon}}$ that asymptotically approach $\Gamma$ without making a large-amplitude oscillation. The second difference concerns the extreme nonlinearity near the boundaries of the disconnected intervals where $R$ is defined (Figure \[fig:retmap2\]a). Portions of the image lie below the local minima in these local concave segments, resulting in tiny regions near the boundaries where the derivative changes rapidly. These points arise from canard segments of trajectories resulting in a jump from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$ 