Noise enables conditional recovery from collapse: Probabilistic persistence in threshold-activated systems

We report a paradoxical phenomenon where stochasticity reverses deterministic collapse in threshold-activated systems. By using a hybrid logistic-sigmoidal map, we show that weak noise alters phase-space topology, enabling probabilistic recovery from extinction. Lyapunov and quasipotential analyses reveal noise-induced metastability and stochastic robustness absent in deterministic frameworks. These results suggest that environmental variability can stabilize nonlinear systems, offering a counternarrative to classical extinction theory.