Tipping Points & Resilience

Fold bifurcation: dx/dt = rx−x³+μ has two stable equilibria for μ < 0 and one for μ > 0. As the stress parameter μ approaches the bifurcation point (μ=0), the basin of attraction shrinks — the system is less resilient to perturbations. Critical slowing down occurs: the recovery rate λ = −∂f/∂x → 0, causing the system to respond sluggishly. This manifests as rising variance (σ²∝1/λ) and rising lag-1 autocorrelation AR(1) → 1. These are generic early warning signals (EWS) for tipping points in ecosystems, climate, and finance. The system can tip catastrophically to the alternative state when the stable branch disappears at the fold.