A Compendium of Hopf-Like Bifurcations in Piecewise-Smooth Dynamical Systems

TL;DR

This paper catalogs 20 geometric mechanisms for local limit cycle creation in 2D piecewise-smooth systems, comparing their scaling laws to classical Hopf bifurcations.

Contribution

It provides a comprehensive compendium of Hopf-like bifurcations specific to piecewise-smooth dynamical systems, including new mechanisms and their scaling behaviors.

Findings

Identifies 20 geometric bifurcation mechanisms for limit cycles.

Compares scaling laws of these bifurcations to classical Hopf bifurcations.

Provides a unified framework for understanding bifurcations in piecewise-smooth systems.

Abstract

This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous systems, and limit cycles created from folds and by the addition of hysteresis or time-delay. Scaling laws for the amplitude and period of the limit cycles are compared to (classical) Hopf bifurcations.