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In a recent paper by Guglielmi and Hairer (SIADS 2015), an analysis in the$\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs incodimension-2 switching domains; this was obtained by studying a certain2-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 ofwhich were labeled as ambiguous, and it was not clear whether or not theambiguity could be resolved. In this paper, we show that it cannot be resolvedin general. A first contribution of this paper is an illustration of thedependence of the limit solution on the form of the switching function.Considering the parameter dependence in the ambiguous class of discontinuoussystems, a second contribution is a bifurcation analysis, revealing a range ofpossible behaviors. Finally, we investigate the sensitivity of solutions in thetransition from codimension-2 domains to codimension-3 when there is a limitcycle in the hidden dynamics.

Nonuniqueness phenomena in discontinuous dynamical systems and their regularizations