Nonuniqueness Phenomena in Discontinuous Dynamical Systems and Their Regularizations

In a recent article by Guglielmi and Hairer [ SIAM J. Appl. Dyn. Syst. , 14 (2015), pp. 1454-1477], an analysis in the \varepsilon \rightarrow 0 limit was proposed of regularized discontinuous ODEs in co dimension -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 co dimension -2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.