A pendulum released near its lowest point may remain near that point forever; a ball balanced at the top of a hill may move away after an arbitrarily small perturbation. Both positions are equilibria: if the system starts exactly there, it stays there. The difference is not the existence of an equilibrium, but how nearby initial conditions behave under the flow for all future time.

Lyapunov stability is the language for this difference. It does not ask whether a trajectory converges to the equilibrium. It asks a more basic question: if the initial error is small enough, can the future error be kept below any prescribed tolerance? This is a uniform-in-time question, and that is why it is stronger than continuity of solutions on a finite time interval.

The subject begins with a warning. Local existence theorems say that nearby initial data produce nearby trajectories for short time. They do not prevent small errors from accumulating. Stability theory measures whether the flow itself contains a mechanism that keeps perturbations controlled.

[example: Unstable Growth in a Linear Equation] Consider the scalar equation \begin{align*} \frac{dx}{dt}=x \end{align*} with initial value \begin{align*} x(0)=x_0. \end{align*} The function \begin{align*} x(t)=e^t x_0 \end{align*} satisfies \begin{align*} \frac{d}{dt}(e^t x_0)=e^t x_0=x(t) \end{align*} and \begin{align*} x(0)=e^0x_0=x_0. \end{align*} In particular, when $x_0=0$, the solution is $x(t)=e^t0=0$ for every $t\ge 0$, so $0$ is an equilibrium. We show that this equilibrium is not Lyapunov stable. Fix $\varepsilon>0$, and let $\delta>0$ be arbitrary. Choose $x_0$ with \begin{align*} 0<|x_0|<\min(\delta,\varepsilon). \end{align*} Along the solution, \begin{align*} |x(t)|=|e^t x_0|=e^t|x_0|. \end{align*} Since $0<|x_0|<\varepsilon$, \begin{align*} \frac{\varepsilon}{|x_0|}>1 \end{align*} and therefore \begin{align*} \log\left(\frac{\varepsilon}{|x_0|}\right)>0. \end{align*} If \begin{align*} t>\log(\varepsilon/|x_0|), \end{align*} then exponentiating both sides gives \begin{align*} e^t>\varepsilon/|x_0|. \end{align*} Multiplying by the positive number $|x_0|$ gives \begin{align*} e^t|x_0|>\varepsilon. \end{align*} Thus this initial condition satisfies $|x_0|<\delta$ but eventually has $|x(t)|>\varepsilon$. No matter how small the initial neighbourhood is, some nonzero initial error inside it is later amplified beyond the fixed tolerance $\varepsilon$. [/example]

The previous example is the simplest form of the phenomenon: the vector field vanishes at the equilibrium, yet the flow expands nearby errors. Lyapunov stability isolates the opposite behaviour, where every tolerance determines a smaller initial neighbourhood whose trajectories never leave the tolerance neighbourhood.

## Definition

The central definition should say exactly what it means for small errors to remain small. We assume uniqueness of solutions so that the phrase "the solution starting at $x_0$" is unambiguous; without uniqueness, stability would have to specify whether all possible futures or some possible future stay near the equilibrium.

[definition: Lyapunov Stable Equilibrium] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $x^*\in U$ satisfy $f(x^*)=0$. Consider the differential equation \begin{align*} \frac{dx}{dt}=f(x). \end{align*} The equilibrium $x^*$ is Lyapunov stable if for every $\varepsilon>0$ there exists $\delta>0$ such that whenever $x_0\in U$ satisfies $|x_0-x^*|<\delta$, the solution $x(t;x_0)$ exists for all $t\ge 0$ and satisfies \begin{align*} |x(t;x_0)-x^*| < \varepsilon \end{align*} for all $t\ge 0$. [/definition]

The quantifiers matter. The number $\delta$ is chosen before the trajectory begins, and the same $\delta$ must work for every future time. This is why the equation

\begin{align*} \frac{dx}{dt}=x \end{align*}

fails: every initial error, however small, is eventually magnified past a fixed tolerance.

## Equations and Equilibria

The definition above is compact, but it rests on the standard language of autonomous equations. We isolate that language because stability is a property of a flow generated by a time-independent vector field, not merely of a formula written at one instant.

### Autonomous Dynamics

To discuss stability, we need the future motion to be determined only by the current state, not by an external clock. This lets nearby initial conditions be compared under the same fixed rule $f$, so that the question "do small errors stay small?" has a precise dynamical meaning.

[definition: Autonomous Differential Equation] Let $U \subset \mathbb R^n$ be open and let $f:U\to \mathbb R^n$ be continuous. An autonomous differential equation on $U$ is an equation of the form \begin{align*} \frac{dx}{dt} = f(x), \end{align*} where a solution with initial condition $x_0 \in U$ is a differentiable map $x:I\to U$, defined on an interval $I\subset \mathbb R$ with $0\in I$, such that $x(0)=x_0$ and $x'(t)=f(x(t))$ for all $t\in I$. [/definition]

The autonomous equation gives a rule, but stability needs a distinguished state that the unperturbed system preserves. For an equilibrium, the system can sit still indefinitely, so the only question is what happens after a small displacement.

[definition: Equilibrium] Let $U\subset \mathbb R^n$ be open and let $f:U\to \mathbb R^n$ be continuous. For the autonomous differential equation \begin{align*} \frac{dx}{dt}=f(x), \end{align*} a point $x^*\in U$ is an equilibrium if \begin{align*} f(x^*) = 0. \end{align*} [/definition]

### Stable Sets and Flow Tubes

If $f$ is locally Lipschitz, the solution starting from $x_0$ is unique on its maximal interval of existence, and this justifies the notation $x(t;x_0)$. For many systems the point equilibrium is too narrow. Periodic orbits, invariant tori, and conserved-energy surfaces are not fixed points, but they can still be stable as sets: nearby trajectories may remain close to the whole set without tracking any particular point on it. To speak about such objects, we need the flow map with its domain made explicit, since solutions may not exist forever from every initial condition.