Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map. Consider the autonomous first-order system

$z^{\mathrm{\prime }}(t)=F(z(t)),$

read as the first-order system of First-order systems, initial value problems, and solutions on intervals with $D=\mathbb{R}\times U$ and right-hand side $(t,z)\mapsto F(z)$. A solution below means a solution of this system on an interval containing $0$; a solution whose interval is $[0,\mathrm{\infty })$ or contains it is said to be defined for all $t\ge 0$.

A solution $\tilde{z}:J\to U$ extends $z:I\to U$ when $I\subseteq J$ and $\tilde{z}{\mathrm{\mid }}_I=z$; the extension is proper if $I⊊J$. A solution is maximal when it has no proper extension (Extensions, maximal solutions, maximal intervals, and global solutions). All stability predicates below quantify over maximal solutions whose intervals contain $0$; their local restrictions remain solutions. This convention asserts neither existence nor uniqueness of maximal solutions for an arbitrary map $F$.

A point $z_{\ast }\in U$ is an equilibrium (also fixed point or zero) of $F$ when $F(z_{\ast })=0$; then the constant function $z(t)=z_{\ast }$ is a solution on every interval containing $0$.

Let $z_{\ast }\in U$ be an equilibrium.

1. $z_{\ast }$ is Lyapunov stable (or simply stable) when for every Euclidean ball $B_{\epsilon }(z_{\ast })$ with $\overline{B_{\epsilon }(z_{\ast })}\subseteq U$ (Open ball, closed ball and sphere in a metric space) there is $\delta \in (0,\epsilon ]$ such that every maximal solution $z$ with $\mathrm{\parallel }z(0)-z_{\ast }{\mathrm{\parallel }}_2<\delta$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$) is defined for all $t\ge 0$ and satisfies

$\mathrm{\parallel }z(t)-z_{\ast }{\mathrm{\parallel }}_2<\epsilon (t\ge 0)\mathrm{.}$

2. $z_{\ast }$ is asymptotically stable when it is Lyapunov stable and in addition there is ${\delta }_0>0$ such that every maximal solution $z$ with $\mathrm{\parallel }z(0)-z_{\ast }{\mathrm{\parallel }}_2<{\delta }_0$ is defined for all $t\ge 0$ and satisfies $\mathrm{\parallel }z(t)-z_{\ast }{\mathrm{\parallel }}_2\to 0$ as $t\to \mathrm{\infty }$.

An equilibrium which is not Lyapunov stable is unstable. An equilibrium is globally asymptotically stable when it is asymptotically stable and every maximal solution of the system is defined for all $t\ge 0$ and converges to it as $t\to \mathrm{\infty }$.

• The two clauses are not equivalent. The equation $z^{\mathrm{\prime }}=0$ on $\mathbb{R}$ has every point Lyapunov stable and no point asymptotically stable. Conversely, convergence of nearby solutions to $z_{\ast }$ does not by itself imply stability, which is why clause 2 requires stability explicitly; a standard planar witness is recorded in Problem 6.16 of the cited source. No claim about any physical system is made by either clause.

• The quantification in clause 1 is invariant under replacing Euclidean balls by arbitrary neighbourhoods of $z_{\ast }$: if the ball form holds then the neighbourhood form holds by choosing a ball inside the given neighbourhood, and conversely. Requiring $\overline{B_{\epsilon }(z_{\ast })}\subseteq U$ costs no generality, because $\epsilon$ is only a test radius and $U$ is open.

• Whether the definition is non-vacuous for a given $F$ depends on an existence theory for solutions. Under the continuity and local-Lipschitz hypotheses of Picard-Lindelof, every initial value problem has a unique maximal solution (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval), and clause 1 can be read as forbidding finite-time breakdown before any boundary of the ball is reached. No such hypothesis is built into this definition.

• Linear stability and orbital stability are defined separately in Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium; hyperbolicity, limit cycles and homoclinic trajectories in Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories. The stability of a limit cycle is orbital stability of its image, not Lyapunov stability of any of its points.