Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems

TL;DR

This paper reviews bifurcation phenomena unique to piecewise-smooth, continuous systems, focusing on border-collision bifurcations and codimension-two cases where nonlinearity plays a key role.

Contribution

It provides a comprehensive overview of bifurcations in non-smooth systems, emphasizing border-collision and codimension-two bifurcations with nonlinear effects.

Findings

Analysis of border-collision bifurcations in piecewise-smooth systems

Characterization of codimension-two bifurcations involving nonlinearity

Reduction to piecewise linearity near the border-collision surface

Abstract

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.