Noise-controlled critical slowing down in forced neural and climate models

Slowly forced nonlinear systems are known to exhibit delayed transitions beyond static bifurcation thresholds, a phenomenon that can be substantially modified by additive noise. In this work, we investigate how stochastic fluctuations modulate bifurcation delay and tipping dynamics in two representative models: the FitzHugh-Nagumo (FHN) neuronal excitable system and the Stommel model of Atlantic Meridional Overturning Circulation collapse. Both systems are analyzed within a unified first-passage framework. For the FHN model, we combine deterministic delay scaling with channel-resolved first-passage statistics and eigenvector-based noise projection. Our analysis reveals that noise acting on the fast and slow variables reduces the bifurcation delay through qualitatively distinct mechanistic pathways. Fast-variable noise induces a sharp onset window for pre-Hopf activation, whereas slow-variable noise acts via projection onto the slow manifold over the relaxation timescale, producing a broader onset region in the phase map. These differences give rise to distinct rate-noise crossover regimes that reflect the underlying geometry of the attractor. For the stochastic Stommel model, the same first-passage framework identifies a sharp crossover from delayed tipping to noise-induced early collapse. Near the crossover noise amplitude, the mean delay (-2.9 yr), median delay (3.4 yr), and interquartile range (2.05 kyr) provide markedly different characterizations of the same first-passage distribution. This striking discrepancy exposes the limitations of single-statistic summaries in saddle-node geometries and underscores the need for multistatistical descriptions of tipping times. Across both systems, we construct rate-noise phase maps and first-passage observables that establish a common basis for comparing delayed Hopf bifurcations (oscillation onset) with delayed saddle-node bifurcations (circulation collapse). The analytical framework developed here, including delay scaling laws, activation windows, and crossover estimates, is directly testable against numerical simulations and is readily extendable to other slowly forced bifurcations, including pitchfork and transcritical cases.