Nonuniqueness phenomena in discontinuous dynamical systems and their regularizations

TL;DR

This paper explores the nonuniqueness and sensitivity of solutions in regularized discontinuous dynamical systems, demonstrating that ambiguities in limit solutions depend on switching functions and analyzing bifurcations and transitions in system codimensions.

Contribution

It shows that limit solutions in regularized discontinuous systems can be nonunique and depend on the switching function, providing bifurcation analysis and studying transitions between codimensions.

Findings

Limit solutions depend on the form of the switching function.

Bifurcation analysis reveals a range of behaviors in parameter dependence.

Solutions are sensitive when transitioning from codimension-2 to codimension-3 with limit cycles.

Abstract

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.