r = -1.00
τ_theory = 1.00
AC(1) = -
Mode: Saddle-node
About: Near a bifurcation (tipping point), dynamical systems slow down — perturbations decay ever more slowly as the stable fixed point approaches the bifurcation. For the saddle-node ẋ = r + x², the relaxation time near x* = √(-r) is τ = 1/(2√(-r)) ∝ |r|^(-1/2). This "critical slowing down" causes rising autocorrelation (AC1) and variance — observable early warning signals before ecosystem collapse, climate tipping points, and disease outbreaks.