Lyapunov stability criterion

Let $𝑥 *$ be a Fixed point with neighbourhood $𝑈$ of a differential system $˙ 𝑥 = 𝑓 (𝑥)$ and let $𝑉 : 𝑈 \to ℝ 𝑛$ be differentiable with $𝑉 (𝑥 *) = 0$ as a minimum in $𝑈$ . Denote $˙ 𝑉 (𝑥) = 𝐷 𝑉 (𝑥) 𝑓 (𝑥)$ . Then¹² #m/thm/dynamics/flow

$𝑥 *$ is asymptotically stable if $˙ 𝑉 (𝑥) < 0$ for all $𝑥 \in 𝑈 ∖ {𝑥 *}$
$𝑥 *$ is Lyapunov stable if $˙ 𝑉 (𝑥) \leq 0$ for all $𝑥 \in 𝑈 ∖ {𝑥 *}$
$𝑥 *$ is Lyapunov unstable if $˙ 𝑉 (𝑥) \geq 0$ for all $𝑥 \in 𝑈 ∖ {𝑥 *}$
$𝑥 *$ is asymptotically unstable if $˙ 𝑉 (𝑥) > 0$ for all $𝑥 \in 𝑈 ∖ {𝑥 *}$

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §7.2, pp. 224–225 ↩
2021. MATH3021: Nonlinear dynamics & chaos, p. 49 ↩

Lyapunov stability criterion

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