Simpson DJW2025-02-122025-02-122025-02-05Simpson DJW. (2025). Nonsmooth folds as tipping points.. Chaos. 35. 2. (pp. 023125-).1054-1500https://mro.massey.ac.nz/handle/10179/72488A 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.(c) The author/shttps://creativecommons.org/licenses/by-nc-nd/4.0/Journal article10.1063/5.02222911089-7682CC BY-NC-NDjournal-article023125-https://www.ncbi.nlm.nih.gov/pubmed/399085590231253333879