A phase portrait can be misleading near a sink. Several trajectories may initially move toward an equilibrium, yet a small perturbation can later escape to another region of the state space. Mere attraction is not enough: it says that some nearby initial conditions converge, but it does not protect the system from leaving a prescribed neighbourhood before convergence begins. Mere Lyapunov stability is not enough either: it keeps nearby solutions nearby, but it may allow them to circle forever without settling. Asymptotic stability is the point where these two requirements meet.

The basic question is this: when does a long-time prediction survive small errors in the initial condition? In an autonomous system, the object of interest is often an equilibrium $x^*$, but the same idea applies to invariant sets, semigroups, and PDE evolutions. We want nearby trajectories to remain controlled for all positive time and to approach the target as $t \to \infty$.

[example: Stability Without Attraction] Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be given by $f(x_1,x_2)=(-x_2,x_1)$. Since $f(0,0)=(0,0)$, the origin is an equilibrium. For initial value $x_0=(a,b)$, the proposed solution is \begin{align*} \varphi_t(x_0)=(a\cos t-b\sin t,\ a\sin t+b\cos t). \end{align*} At $t=0$ this gives \begin{align*} \varphi_0(x_0)=(a\cos 0-b\sin 0,\ a\sin 0+b\cos 0)=(a,b)=x_0. \end{align*} Differentiating componentwise, \begin{align*} \frac{d}{dt}\varphi_t(x_0)=(-a\sin t-b\cos t,\ a\cos t-b\sin t). \end{align*} On the other hand, \begin{align*} f(\varphi_t(x_0))=(-(a\sin t+b\cos t),\ a\cos t-b\sin t)=(-a\sin t-b\cos t,\ a\cos t-b\sin t), \end{align*} so $\varphi_t(x_0)$ solves $\dot{x}=f(x)$ with initial value $x_0$. Now compute its distance from the origin: \begin{align*} |\varphi_t(x_0)|^2=(a\cos t-b\sin t)^2+(a\sin t+b\cos t)^2. \end{align*} Expanding the two squares gives \begin{align*} (a\cos t-b\sin t)^2=a^2\cos^2 t-2ab\sin t\cos t+b^2\sin^2 t. \end{align*} Also, \begin{align*} (a\sin t+b\cos t)^2=a^2\sin^2 t+2ab\sin t\cos t+b^2\cos^2 t. \end{align*} Adding these identities cancels the mixed terms: \begin{align*} |\varphi_t(x_0)|^2=a^2(\cos^2 t+\sin^2 t)+b^2(\sin^2 t+\cos^2 t)=a^2+b^2=|x_0|^2. \end{align*} Hence $|\varphi_t(x_0)|=|x_0|$ for every $t\ge 0$. Therefore every initial condition close to $0$ remains equally close to $0$ for all positive time, but if $x_0\ne 0$ then $|\varphi_t(x_0)|=|x_0|>0$ for all $t$, so $\varphi_t(x_0)$ cannot converge to $0$. The system is stable in the no-escape sense, but it has no damping toward the equilibrium. [/example]

The rotating system has perfect control of errors but no loss of energy. The page builds the missing notion in stages: stability, attraction, their combination, the linear test, and then the Lyapunov-function method that lets us prove asymptotic stability without solving the equation.

Before naming any kind of stability, we also need to say exactly what the time-evolution maps are. The same symbol $\varphi_t$ is used throughout the page, and the definition below records both its domain of allowed initial data and its codomain of states reached at time $t$.

[definition: Forward Flow] Let $D \subset \mathbb{R}^n$ and let $f:D\to\mathbb{R}^n$ be a vector field. A forward flow generated by $f$ is a family of maps $(\varphi_t)_{t\ge 0}$ with \begin{align*} \varphi_t:\Omega_t &\to D, \end{align*} where $\Omega_t\subset D$ is the set of initial data whose unique solutions are defined at least up to time $t$, such that $\varphi_0=\operatorname{id}_D$ and, whenever $s,t\ge 0$ and both sides are defined, \begin{align*} \varphi_{t+s}(x_0)=\varphi_t(\varphi_s(x_0)). \end{align*} Moreover, whenever $x_0\in\Omega_t$, the curve $s\mapsto \varphi_s(x_0)$ is defined for $0\le s\le t$ and solves $\dot{x}=f(x)$ on the interval $[0,t]$ with initial value $x_0$. [/definition]

This convention keeps finite-time blow-up visible rather than hidden. If $x_0\notin\Omega_t$ for some $t$, then the trajectory has not survived long enough for a long-time stability conclusion.

## Controlled Targets and Invariance

To speak about stability, the target must be meaningful under the dynamics. If a solution starts in the target and instantly leaves it, then asking whether nearby solutions stay near that target confuses motion along the target with motion away from it. The first object is therefore invariance.

[definition: Positively Invariant Set] Let $D \subset \mathbb{R}^n$, let $f: D \to \mathbb{R}^n$ generate a forward flow $(\varphi_t:\Omega_t\to D)_{t\ge 0}$, and let $E \subset D$. The set $E$ is positively invariant if, for every $x_0 \in E$ and every $t\ge 0$, one has $x_0\in\Omega_t$ and $\varphi_t(x_0) \in E$. [/definition]

For an equilibrium, positive invariance is automatic once $f(x^*)=0$, since the constant solution stays at $x^*$. The next problem is error control: even when the target is invariant, a nearby initial condition might make a large excursion before returning. When the target is a set rather than a point, nearness is measured by the Euclidean distance to that set.

[definition: Distance to a Set] Let $D\subset\mathbb{R}^n$ and let $E\subset D$ be nonempty. The distance-to-set function associated to $E$ in the ambient Euclidean metric is the function $\operatorname{dist}(\cdot,E):D\to [0,\infty)$ defined by \begin{align*} \operatorname{dist}(x,E)=\inf_{y\in E}|x-y|. \end{align*} [/definition]

The distance function turns "near the target" into a numerical condition that can be checked at every future time. This is needed because finite-time continuity of the flow only controls a trajectory on a fixed time interval; stability asks for a single initial tolerance that prevents escape for all $t\ge 0$.

[definition: Lyapunov Stability of an Invariant Set] Let $D \subset \mathbb{R}^n$, let $f: D \to \mathbb{R}^n$ generate a forward flow $(\varphi_t:\Omega_t\to D)_{t\ge 0}$, and let $E \subset D$ be positively invariant. The set $E$ is Lyapunov stable if, for every $\varepsilon>0$, there exists $\delta>0$ such that every $x_0\in D$ with $\operatorname{dist}(x_0,E)<\delta$ satisfies $x_0\in\Omega_t$ and \begin{align*} \operatorname{dist}(\varphi_t(x_0),E)<\varepsilon \end{align*} for every $t\ge 0$. [/definition]

Lyapunov stability is a uniform-in-time statement, stronger than continuity of any single map $\varphi_t$. It still allows circular or oscillatory motion forever, so it does not answer the long-time question. The missing condition is that nearby trajectories should lose their displacement from the target as time increases.

[definition: Quasi-Asymptotic Stability] Let $D \subset \mathbb{R}^n$, let $f: D \to \mathbb{R}^n$ generate a forward flow $(\varphi_t:\Omega_t\to D)_{t\ge 0}$, and let $E \subset D$ be positively invariant. The set $E$ is quasi-asymptotically stable if there exists $r>0$ such that every $x_0\in D$ with $\operatorname{dist}(x_0,E)<r$ satisfies $x_0\in\Omega_t$ for every $t\ge 0$ and \begin{align*} \lim_{t\to\infty}\operatorname{dist}(\varphi_t(x_0),E)=0. \end{align*} [/definition]

## Definition

No-escape control and eventual convergence solve different problems. Lyapunov stability alone still permits persistent oscillation near the target, while attraction alone can allow large transient excursions before convergence. The useful combined notion requires both uniform control for all future time and convergence back to the invariant set.

[definition: Asymptotic Stability of an Invariant Set] Let $D \subset \mathbb{R}^n$, let $f: D \to \mathbb{R}^n$ generate a forward flow $(\varphi_t:\Omega_t\to D)_{t\ge 0}$, and let $E \subset D$ be positively invariant. The set $E$ is asymptotically stable if $E$ is Lyapunov stable and quasi-asymptotically stable with respect to this flow. [/definition]

This is the page's central definition. Everything that follows is a way of recognizing the two ingredients in concrete settings: first for equilibria and basins, then for linear systems, Lyapunov functions, and semigroup evolutions.