 Recovery time of a stable equilibrium

For a scalar ordinary differential equation $N^{\prime }=f(N)$ and an equilibrium point $N^{\ast }$ with $f^{\prime }(N^{\ast })<0$, the linearization gives $n^{\prime }=f^{\prime }(N^{\ast })n$. Thus small perturbations decay like $e^{-t/\tau }$ with $\tau =-1/f^{\prime }(N^{\ast })$. This is a local exponential recovery time. In several dimensions, the slowest decaying stable eigenvalue typically controls asymptotic recovery, subject to additional polynomial factors for nontrivial Jordan blocks.