Work

Nonsmooth folds as tipping points

Year: 2025

Type: article

Abstract: A nonsmooth fold occurs when 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 ordinary differential equations, 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 bifurcation structures of the truncated systems.

Source: Chaos An Interdisciplinary Journal of Nonlinear Science

DOI: 10.1063/5.0222291

Author: D. J. W. Simpson

Institution: Massey University

Language: English

Topic: Advanced Differential Equations and Dynamical Systems

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: Royal Society Te Apārangi, Marsden Fund

Award: G4233949480