the last case, one-dimensional approximations of return maps were used to analyze a Shilnikov bifurcation in a system which exhibits singular Hopf bifurcation.

In this paper, we use a similar technique to analyze a tangency of invariant manifolds. Our starting point is a study by Guckenheimer and Meerkamp[@guckenheimer2012siam], which comprehensively classifies local and global unfoldings of singular Hopf bifurcation. We describe the changes in the phase space as the unstable manifold of the saddle-focus equilibrium point crosses the repelling slow manifold of the system. Our approach takes for granted the complicated crossings of these two-dimensional manifolds, instead focusing directly on the influence of these crossings on the global dynamics. The main tool in our analysis is the approximation of the two-dimensional return map by a map on an interval, which parametrizes trajectories beginning on the attracting slow manifold. We show that in the presence of a small-amplitude stable periodic orbit, the one-dimensional return map has a rich topology. The domain of the map is disconnected, with components separated by finite-length gaps. Intervals where the return map is undefined correspond to bands of initial conditions in the full system whose forward trajectories asymptotically approach the small-amplitude stable periodic orbit without making 