Equilibria and stability

An equilibrium $f(x^\ast) = 0$ is stable if nearby states stay nearby (and asymptotically stable if they return). Linearize: if every eigenvalue of the Jacobian $J_f(x^\ast)$ has negative real part, it is asymptotically stable.

Real systems are always perturbed: a gust of wind, a rounding error, a noisy sensor. An equilibrium is only useful — an upright robot, a converged training run, a steady chemical reaction — if small perturbations die out.

A ball at the bottom of a bowl (stable), on top of a dome (unstable), on a saddle (stable in one direction, unstable in another). Near the equilibrium, the system behaves like its linearization $\dot e = Je$: each eigenvalue $\lambda$ is a mode that evolves like $e^{\lambda t}$, decaying if $\operatorname{Re}\lambda < 0$.

For $\dot x = f(x)$ with $f(x^\ast) = 0$ and $f \in C^1$: if $\operatorname{Re}\lambda < 0$ for every eigenvalue of $J_f(x^\ast)$, then $x^\ast$ is locally asymptotically stable; if some $\operatorname{Re}\lambda > 0$, it is unstable. For maps $x_{k+1} = g(x_k)$ the condition is $|\lambda| < 1$.

Control engineers design feedback to put the eigenvalues where they want (pole placement). Optimization theory reads minima of a loss as stable equilibria of the gradient flow. GAN training is notoriously hard partly because the game between generator and discriminator has rotational, not stable, dynamics.