A discrete-time benchmark for assessing critical slowing down indicators

Abstract

We introduce a low-dimensional, discrete-time benchmark for evaluating critical slowing down indicators and early-warning signals in the presence of complex dynamics. Starting from a map-based attention-deficit-disorder model, we add a bias offset to obtain a modified system with a controllable, hysteresis-like coexistence band. Within this band, forward and backward parameter sweeps follow distinct branches, and abrupt switching can occur alongside periodic windows and chaotic regimes. We characterize the dynamics using state-space portraits, bifurcation diagrams, and Lyapunov exponents. We then evaluate four metric-based indicators-lag-1 autocorrelation, variance, skewness, and kurtosis-using a period-aware computation designed for regimes beyond period-one. We find that variance exhibits the most consistent warning trend near the coexistence boundaries, whereas autocorrelation is more susceptible to spurious spikes. Higher-order moments are generally less reliable, particularly in intermittently chaotic regions. Overall, the benchmark is computationally efficient and provides a practical testbed for stress-testing early-warning methods and for quantifying sensitivity to analysis choices.