Stability

3 min read#foundations

Whether small disturbances to an equilibrium grow or decay — the difference between a valley and a hilltop.

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Stability

Stability asks a simple question: if you nudge a system away from an equilibrium, does it return or run away? A marble in a bowl rolls back — the bottom is stable. A marble balanced on a dome rolls off — the top is unstable. The same distinction governs every Fixed Point of a Dynamical System, and it decides which equilibria you ever actually observe, since unstable ones are destroyed by the slightest perturbation.

The Lyapunov picture

Aleksandr Lyapunov gave the idea its precise form. A fixed point x^* is stable if trajectories that start nearby stay nearby, and asymptotically stable if they additionally return to it as t \to \infty. The intuition is energy: if you can find a quantity that only ever decreases along trajectories and bottoms out at x^* — a Lyapunov function, the mathematical bowl — then the state is forced to slide downhill into equilibrium.

Linearization and eigenvalues

For most purposes you can read stability straight off the linearization. Near x^*, a displacement u obeys \dot u \approx A u with A = Df(x^*), and perturbations evolve like e^{\lambda t} where \lambda are the eigenvalues of A:

\operatorname{Re}(\lambda) < 0 \ \text{for all } \lambda \;\Longrightarrow\; \text{stable}, \qquad \operatorname{Re}(\lambda) > 0 \ \text{for some } \lambda \;\Longrightarrow\; \text{unstable}.
(1)

The sign of the real part is everything: negative means the perturbation decays, positive means it explodes. The imaginary part only sets whether the approach is a smooth slide (real eigenvalues, a node) or a ringing spiral (complex eigenvalues).

Decay versus growth

The plot contrasts two perturbations: one to a stable equilibrium, e^{-0.7x}, which shrinks away, and one to an unstable equilibrium, e^{0.7x}, which blows up. Same starting size, opposite fates — set entirely by the sign of the exponent.

Stable (decaying) vs. unstable (growing) perturbation
Lower curve: Re(lambda) < 0. Upper curve: Re(lambda) > 0.

Beyond equilibria

Stability is not only about fixed points. A whole trajectory can be stable or unstable, and the rate at which neighbors converge or diverge along it is measured by the Lyapunov Exponent. When that rate is positive on an Attractor, you have the stretching that defines Chaos.

See also