Lyapunov stability

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the flow of $x' = f(x)$, and let $x^* \in U$ be an equilibrium point. The point $x^*$ is Lyapunov stable if for each $\varepsilon > 0$ there is a $\delta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \delta$, the solution $\varphi_t(x_0)$ exists for all $t \ge 0$, and $\|\varphi_t(x_0) - x^*\| < \varepsilon$ for all $t \ge 0$. The point $x^*$ is unstable if it is not Lyapunov stable.

Remarks

For $x' = -x$ and for $x' = 0$ on $\mathbb{R}$, the equilibrium $0$ is stable. For $x' = x$, it is unstable. The centre of the harmonic oscillator is stable but not asymptotically stable: nearby solutions stay near, but they do not approach it. A Lyapunov function gives a way to prove stability without solving the equation.

Explanations

L1 human

An equilibrium is stable if solutions that start close to it stay close to it forever. To stay within distance $\varepsilon$, it is enough to start within a smaller distance $\delta$.