A criterion for safe overshoot in coupled tipping systems

Abrupt transitions are a central concern in climate and ecological research and may arise when critical thresholds known as tipping points are crossed. However, previous work has shown that finite-time overshoots of tipping points can be safe, and that such behavior is captured by an inverse-square-law criterion when overshoots are sufficiently small and slow. So far studied in isolated systems with external drivers, (un)safe overshoots may also emerge from interactions between subsystems. Here, we investigate safe-overshoot phenomena in unidirectionally coupled slow-fast systems featuring both nonlinear interactions and coupling through time derivatives. Specifically, we derive a criterion for the occurrence of safe overshoots analogous to the inverse-square law for isolated systems, but adapted to interactive settings, and expressed explicitly in terms of the timescale separation and coupling strength between subsystems. We illustrate these results using two conceptual models in which the Atlantic Meridional Overturning Circulation interacts with either the Amazon rainforest or the Greenland Ice Sheet.