Quantifying Resilience in Non-Autonomous and Stochastic Earth System Dynamics with Application to Glacial-Interglacial Cycles

Abstract. Understanding Earth resilience—the capacity of the Earth system to absorb and regenerate from perturbations—is key to assessing risks from anthropogenic pressures and sustaining a safe operating space for humanity within planetary boundaries. Classical resilience indicators are designed for autonomous systems with fixed attractors, but the Earth system is fundamentally non-autonomous and out of equilibrium, calling for new ways of defining and quantifying resilience.

Here, we introduce a path-resilience approach that assesses how perturbations deviate from and return to a reference trajectory of a conceptual climate model replicating the glacial-interglacial cycles of the Late Pleistocene. We generate two types of perturbation ensembles: a stochastic ensemble and a single-event ensemble, and compute two complementary metrics: the Reference Adherence Ratio (RAR)—defined as the fraction of stochastic trajectories that remain within a narrow band around the unperturbed trajectory—and return time—defined as the time a single perturbed trajectory takes to return to the reference path. Together, these metrics reveal strong temporal variation in resilience across the glacial-interglacial cycles. We find that RAR increases markedly during deglaciations and peaks in interglacial periods, while return times generally shorten as the system approaches interglacial conditions—indicating that certain phases of the cycles act as convergence zones and potential anchors of Earth system stability. As the Earth system departs from such stable interglacial regimes under ongoing anthropogenic forcing, understanding the resilience of these trajectories—and what it may take to return to them—becomes increasingly important.

These results highlight that resilience in non-autonomous systems is inherently path-dependent and illustrate a promising first step toward its quantification. Further research is needed to develop more general resilience indicators suitable for complex, forced dynamical systems.