Nonsmooth folds as tipping points

TL;DR

This paper investigates nonsmooth folds in dynamical systems, showing that after bifurcation, the system typically lacks bounded invariant sets, leading to a tipping point where attractors change state.

Contribution

It provides a rigorous analysis of nonsmooth fold bifurcations, demonstrating the absence of local invariant sets post-bifurcation and explaining the global bifurcation structure.

Findings

Post-bifurcation systems lack bounded invariant sets.

Nonsmooth fold bifurcations cause attractors to tip to new states.

Results apply to Filippov, hybrid, and piecewise-smooth systems.

Abstract

A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation the leading-order truncation to the system in general has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ODEs, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus if the equilibrium or limit cycle is attracting the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of the bifurcation structures of the truncated systems.