Early warning signals: variance and AR(1) increase before tipping points
r(t): –
Variance: –
AR(1): –
Tipped: No
Critical slowing down: near a fold bifurcation, the dominant eigenvalue λ → 0, so perturbations
decay slower (recovery time τ = -1/λ → ∞). This manifests as increasing variance and
lag-1 autocorrelation AR(1) → 1. Both are generic early warning signals (EWS) of tipping.
System: dx = (rx - x³)dt + σ dW. As r decreases from r₀=1 toward r_c=0, the stable equilibrium
x* = √r loses stability. EWS can signal tipping months/years before the transition.