Critical slowing down ($\tau \to \infty$)

Critical slowing down is the divergence of a relaxation time as a stable equilibrium point approaches loss of hyperbolicity. For a scalar ordinary differential equation it occurs when $f^{\prime }(N^{\ast })\uparrow 0$ and the recovery time of a stable equilibrium $-1/f^{\prime }(N^{\ast })$ diverges. At the threshold, nonlinear terms may replace exponential decay by algebraic decay; for example $N^{\prime }=-k{N}^3$ has $N(t)\sim (2kt{)}^{-1/2}$ for positive initial data.