Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map; retain the convention of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system that a solution means a solution of $z^{\mathrm{\prime }}(t)=F(z(t))$, read as a system of First-order systems, initial value problems, and solutions on intervals, and that it is defined for all $t\ge 0$ when its interval contains $[0,\mathrm{\infty })$. A maximal solution means one with no proper extension, as defined there; the invariance and stability predicates below quantify over maximal solutions through time $0$, not their arbitrary local restrictions. No existence or uniqueness is asserted for an arbitrary $F$.

Invariant sets. A set $S\subseteq U$ is forward invariant (for this system) when every maximal solution $z$ with $z(0)\in S$ is defined for all $t\ge 0$ and satisfies $z(t)\in S$ for all $t\ge 0$.

Distance to a set. For a nonempty $S\subseteq U$ define

$\mathrm{dist}(z,S):=\inf \{\mathrm{\parallel }z-s{\mathrm{\parallel }}_2:s\in S\}(z\in {\mathbb{R}}^n),$

the infimum of a nonempty set of nonnegative reals that is bounded below, computed for the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Define the Euclidean closure of $S$ by

$\bar{S}:=\{z\in {\mathbb{R}}^n:\text{for every }\epsilon >0\text{ there is }s\in S\text{ with }\mathrm{\parallel }z-s{\mathrm{\parallel }}_2<\epsilon \}\mathrm{.}$

The map $z\mapsto \mathrm{dist}(z,S)$ is nonnegative and vanishes exactly on $\bar{S}$: by the defining property of the infimum, $\mathrm{dist}(z,S)=0$ exactly when every positive-radius Euclidean ball about $z$ meets $S$. When $S=\{z_{\ast }\}$ is a singleton, $\mathrm{dist}(z,S)=\mathrm{\parallel }z-z_{\ast }{\mathrm{\parallel }}_2$.

Orbital stability. Let $S\subseteq U$ be nonempty and forward invariant. Then $S$ is orbitally stable when for every $\epsilon >0$ there is $\delta >0$ such that every maximal solution $z$ with $\mathrm{dist}(z(0),S)<\delta$ is defined for all $t\ge 0$ and satisfies

$\mathrm{dist}(z(t),S)<\epsilon (t\ge 0)\mathrm{.}$

It is asymptotically orbitally stable when it is orbitally stable and there is ${\delta }_0>0$ such that every maximal solution with $\mathrm{dist}(z(0),S)<{\delta }_0$ is defined for all $t\ge 0$ and satisfies $\mathrm{dist}(z(t),S)\to 0$ as $t\to \mathrm{\infty }$.

Linear stability. Let $z_{\ast }\in U$ be an equilibrium of $F$ and suppose $F$ is totally differentiable at $z_{\ast }$ in the sense of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, with total derivative $DF(z_{\ast }):{\mathbb{R}}^n\to {\mathbb{R}}^n$; write $L:=DF(z_{\ast })$. The linearized system at $z_{\ast }$ is the autonomous linear system

$y^{\mathrm{\prime }}(t)=Ly(t)\mathrm{.}$

The equilibrium $z_{\ast }$ is linearly stable when every maximal solution $y$ of the linearized system (First-order systems, initial value problems, and solutions on intervals) is defined for all $t\ge 0$ and is bounded on $[0,\mathrm{\infty })$, that is, there is a real constant $C\ge 0$ with $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\le C$ for every $t\ge 0$. It is linearly asymptotically stable when every maximal solution of the linearized system is defined for all $t\ge 0$ and satisfies $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\to 0$ as $t\to \mathrm{\infty }$.

• Orbital stability of $S$ is a statement about the set $S$, not about any chosen point of it: it controls the distance of a whole solution to $S$ and does not assert that a solution starting near one point of $S$ stays near that point. Lyapunov stability of an equilibrium is the special case $S=\{z_{\ast }\}$; orbital stability of a limit cycle concerns its periodic image $S$ (Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories) when that image is forward invariant in the sense of the Definition. Under the arbitrary-$F$ hypotheses here, a periodic solution alone does not ensure that every maximal solution through its image remains there; forward invariance must be established separately.

• Linear stability is a property of the linear system alone. It is not equivalent to Lyapunov stability of $z_{\ast }$ for the nonlinear system; linearization transfers stability only under additional spectral hypotheses, which this definition does not supply.

• The constant $C$ in the definition of linear stability is not required to be uniform over a family of systems, and no rate of decay is required; linear asymptotic stability is a strictly stronger property than linear stability. Both are defined here for the concrete Euclidean finite-dimensional case only.