  In a recent paper by Guglielmi and Hairer (SIADS 2015), an analysis in the
$\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in
codimension-2 switching domains; this was obtained by studying a certain
2-dimensional system describing the so-called hidden dynamics. In particular,
the existence of a unique limit solution was not proved in all cases, a few of
which were labeled as ambiguous, and it was not clear whether or not the
ambiguity could be resolved. In this paper, we show that it cannot be resolved
in general. A first contribution of this paper is an illustration of the
dependence of the limit solution on the form of the switching function.
Considering the parameter dependence in the ambiguous class of discontinuous
systems, a second contribution is a bifurcation analysis, revealing a range of
possible behaviors. Finally, we investigate the sensitivity of solutions in the
transition from codimension-2 domains to codimension-3 when there is a limit
cycle in the hidden dynamics.
