Tipping mechanisms in a carbon cycle model.

This article investigates rate-induced tipping (R-tipping) and noise-induced tipping (N-tipping) in a marine carbonate cycle model originally proposed by Rothman, focusing on transitions away from a stable fixed point within a bistable regime. The study demonstrates that rapid increases in the external carbon dioxide (CO₂) injection rate can cause R-tipping, pushing the system from the basin of attraction of a stable fixed point to that of a stable periodic orbit before any bifurcation occurs. For N-tipping, the authors analyze stochastic perturbations with additive white noise and identify that escape trajectories concentrate on a specific region of an unstable periodic orbit forming the basin boundary, rather than cycling indefinitely as classical theory predicts. Using geometric dynamical systems methods, including computations of invariant manifolds, heteroclinic orbits, and the Maslov index, combined with the Onsager–Machlup functional, the work selects a most probable escape path that aligns with Monte Carlo simulation results. The findings highlight the susceptibility of the marine carbonate system to critical transitions driven by both rapid anthropogenic forcing and stochastic variability, emphasizing the importance of advanced mathematical tools for assessing tipping risks in climate-relevant models.