Dynamic tipping and cyclic folds, in a one-dimensional non-smooth dynamical system linked to climate models

We study the behaviour at tipping points close to (smoothed) non-smooth fold bifurcations in one-dimensional oscillatory forced systems with slow parameter drift. The focus is the non-smooth fold in the Stommel 2-box, and related climate models, which are piecewise-smooth continuous dynamical systems, modelling thermohaline circulation. These exhibit non-smooth fold bifurcations which arise when a saddle-point and a focus meet at a border collision bifurcation. By using techniques from the theory of non-smooth dynamical systems we are able to provide precise estimates for the general tipping behaviour close to the non-smooth fold as parameters vary. These are different from the usual tipping point estimates at a saddle-node bifurcation, with advanced tipping apparent in the non-smooth case when compared to the behaviour near smoothed approximations to this fold. We also see rapid, and non-monotone, transitions in the tipping points for oscillatory forced systems close to both non-smooth folds and saddle-node bifurcations due to the effects of both phase changes and non-smoothness. These variations can have implications for the prediction of tipping in climate systems, particularly in close proximity to a non-smooth fold.