On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

TL;DR

This paper investigates how small perturbations like hysteresis, delay, and noise can resolve ambiguities in the evolution of two-fold singularities in piecewise-smooth vector fields, leading to a probabilistic description of dynamics.

Contribution

It introduces a framework for perturbing two-fold singularities to achieve well-defined forward evolution and characterizes the limiting probabilistic dynamics for different perturbation types.

Findings

Perturbations resolve ambiguity in two-fold evolution.

Limit dynamics are described probabilistically.

Different perturbations lead to distinct probabilistic behaviors.

Abstract

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. The purpose of this paper is to explore the concept of perturbing the vector field so that forward evolution is well-defined, and characterising the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. Three forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. In each case, the limit leads to a novel probabilistic notion of forward evolution from the two-fold.