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. [definition: Forward Flow] Let $U\subset \mathbb R^n$ be open and let $f:U\to \mathbb R^n$ be locally Lipschitz. For each $t\ge 0$, let \begin{align*} D_t = \{x_0\in U : \text{the solution } x(s;x_0) \text{ exists for every } s\in[0,t]\}. \end{align*} The forward flow at time $t$ is the map $\varphi_t:D_t\to U$ defined by \begin{align*} \varphi_t(x_0)=x(t;x_0). \end{align*} [/definition] A set should count as dynamically preserved only when every trajectory that starts in it remains in it for all future times. This is the set-valued analogue of an equilibrium staying fixed, and it is the minimal hypothesis needed before asking whether nearby trajectories stay near the set. [definition: Invariant Set] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $\varphi_t:D_t\to U$ denote the forward flow. A set $E\subset U$ is positively invariant if for every $x_0\in E$ and every $t\ge 0$, one has $x_0\in D_t$ and $\varphi_t(x_0)\in E$. [/definition] Once a set is invariant, stability should not require a trajectory to remain near its starting point. It should remain near the set. The right neighbourhoods are metric neighbourhoods, because the distance from a point to a set measures the smallest perturbation needed to land in the set. [definition: Neighbourhood of a Set] Let $E\subset \mathbb R^n$ be nonempty. The distance-to-$E$ function is the map $\operatorname{dist}(\cdot,E):\mathbb R^n\to [0,\infty)$ defined by \begin{align*} \operatorname{dist}(x,E)=\inf_{y\in E}|x-y|. \end{align*} For $r>0$, the open $r$-neighbourhood of $E$ is \begin{align*} N_r(E) = \{x\in \mathbb R^n : \operatorname{dist}(x,E)<r\}. \end{align*} [/definition] Distance neighbourhoods give a clean way to measure closeness to a set, but their geometry depends on the set. For a closed invariant set, the condition $\operatorname{dist}(x,E)<r$ really means that $x$ lies in a metric tube around the object itself. For nonclosed sets, the same formula cannot distinguish $E$ from its closure, so the stability notion may silently describe closeness to limit points not lying in $E$. The standard set version is therefore best stated for closed positively invariant sets, with compact invariant sets forming the most common phase-portrait examples. [definition: Lyapunov Stable Invariant Set] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, let $\varphi_t:D_t\to U$ denote the forward flow, and let $E\subset U$ be a nonempty closed positively invariant set. The set $E$ is Lyapunov stable if for every $\varepsilon>0$ there exists $\delta>0$ such that whenever $x_0\in U\cap N_\delta(E)$, one has $x_0\in D_t$ for every $t\ge 0$ and \begin{align*} \varphi_t(x_0)\in U\cap N_\varepsilon(E) \end{align*} for all $t\ge 0$. [/definition] The point definition is recovered by taking $E=\{x^*\}$. For closed invariant sets, Lyapunov stability means that every small metric neighbourhood of the invariant object contains a smaller metric neighbourhood whose trajectories remain in the larger one. The same $\delta$ must work for every initial condition in the whole tube $N_\delta(E)$, so the quantifier is uniform along all of $E$. When $E$ is compact this matches the geometric picture of nested tubes; without compactness, the condition is still measured by distance to $E$, but it may not provide the same visual thickness in every geometric direction. ## Linear Systems and Spectral Tests Linear equations are the testing ground for stability theory. They expose the difference between bounded oscillation, exponential decay, and polynomial growth caused by non-diagonal Jordan blocks. They also provide the linear approximation used near nonlinear equilibria. ### Uniform Boundedness A linear system has a global flow written using the matrix exponential. The stability question becomes a question about whether the family of matrices $e^{tA}$ is uniformly bounded for $t\ge 0$. [definition: Linear Autonomous System] Let $A\in \mathbb R^{n\times n}$. The linear autonomous system generated by $A$ is \begin{align*} \frac{dx}{dt} = Ax, \end{align*} where the state is $x(t)\in\mathbb R^n$. [/definition] For initial condition $x_0\in \mathbb R^n$, the solution is \begin{align*} x(t) = e^{tA}x_0 \end{align*} For a linear system, stability of the origin can be phrased without restricting to a small neighbourhood, because scaling an initial condition scales the whole trajectory. The missing test is not whether a single solution stays bounded, but whether the entire family of solution operators has one bound that works for every time. The next criterion turns the epsilon-delta definition into exactly that uniform estimate, replacing infinitely many neighbourhood tests by one inequality valid for every initial vector. [quotetheorem:7899] This criterion is useful because it separates the dynamical issue from the choice of initial condition. If the operators $e^{tA}$ have a common bound, every sufficiently small initial vector remains small for all future time; if no such bound exists, the linear flow contains directions whose amplification eventually defeats any fixed Lyapunov neighbourhood. The remaining question is how to detect this operator bound from the matrix $A$ itself. ### Eigenvalues and Jordan Blocks The boundedness criterion reduces a dynamical question to an operator bound. The spectral form of that bound explains why eigenvalues with negative real part are stabilising, why eigenvalues with positive real part are destabilising, and why purely imaginary eigenvalues require attention to Jordan blocks. [quotetheorem:7900] The condition on Jordan blocks is not cosmetic. A zero real part eigenvalue produces no exponential growth, but a nontrivial Jordan block produces polynomial growth. That polynomial growth is enough to violate the uniform-in-time requirement. [example: Jordan Block with Zero Eigenvalue] Let $e_1,e_2$ be the standard basis of $\mathbb R^2$, and define $A\in \mathbb R^{2\times 2}$ by \begin{align*} Ae_1=0, \qquad Ae_2=e_1. \end{align*} For a vector $v=\alpha e_1+\beta e_2$, linearity gives \begin{align*} Av=\alpha Ae_1+\beta Ae_2=\beta e_1. \end{align*} Applying $A$ once more gives \begin{align*} A^2v=A(\beta e_1)=\beta Ae_1=0. \end{align*} Thus $A^2=0$, and the matrix exponential series stops after the linear term: \begin{align*} e^{tA}v=\sum_{k=0}^{\infty}\frac{t^kA^k v}{k!}=v+tAv. \end{align*} In particular, \begin{align*} e^{tA}e_1=e_1+tAe_1=e_1 \end{align*} and \begin{align*} e^{tA}e_2=e_2+tAe_2=e_2+te_1. \end{align*} Now take an initial condition in the second coordinate, $x_0=ae_2$ with $a\ne 0$. The solution of the linear system is \begin{align*} x(t)=e^{tA}x_0=a(e_2+te_1)=tae_1+ae_2. \end{align*} Therefore, in coordinates, \begin{align*} x(t)=(ta,a). \end{align*} Its Euclidean norm satisfies \begin{align*} |x(t)|=\sqrt{t^2a^2+a^2}\ge |ta|=t|a|. \end{align*} To see failure of Lyapunov stability, fix the tolerance $\varepsilon=1$. Given any $\delta>0$, choose \begin{align*} a=\min(\delta/2,1/2). \end{align*} Then $0<a<\delta$, so $|x_0|=a<\delta$. If \begin{align*} t>\frac1a, \end{align*} then \begin{align*} |x(t)|\ge ta>1=\varepsilon. \end{align*} Thus every neighbourhood of the origin contains an initial condition whose trajectory eventually leaves the unit ball. Finally, if $Av=\lambda v$ with $v\ne 0$, then $0=A^2v=\lambda^2v$, so $\lambda=0$; the only eigenvalue is $0$, but the origin is not Lyapunov stable. [/example] The stable alternatives are also visible in dimension two. Centers rotate without shrinking and are Lyapunov stable, while sinks shrink and are asymptotically stable. Stability does not require decay. [example: Center versus Sink] For the system $\dot x_1=-x_2$ and $\dot x_2=x_1$, let $x_0=(a,b)$. Define \begin{align*} x(t)=(a\cos t-b\sin t,\ a\sin t+b\cos t). \end{align*} Then \begin{align*} x_1'(t)=-a\sin t-b\cos t=-x_2(t) \end{align*} and \begin{align*} x_2'(t)=a\cos t-b\sin t=x_1(t). \end{align*} Also $x(0)=(a,b)=x_0$, so this is the solution with initial condition $x_0$. Its norm is constant because \begin{align*} |x(t)|^2=(a\cos t-b\sin t)^2+(a\sin t+b\cos t)^2 \end{align*} and expanding gives \begin{align*} |x(t)|^2=a^2\cos^2t-2ab\sin t\cos t+b^2\sin^2t+a^2\sin^2t+2ab\sin t\cos t+b^2\cos^2t. \end{align*} The mixed terms cancel, and $\sin^2t+\cos^2t=1$, so \begin{align*} |x(t)|^2=a^2+b^2=|x_0|^2. \end{align*} Given $\varepsilon>0$, choose $\delta=\varepsilon$. If $|x_0|<\delta$, then $|x(t)|=|x_0|<\varepsilon$ for every $t\ge 0$. Thus the origin is Lyapunov stable, but unless $x_0=0$ the norm never tends to $0$, so nearby nonzero solutions do not converge to the origin. For the system $\dot x_1=-x_1$ and $\dot x_2=-2x_2$, let $x_0=(a,b)$ and set \begin{align*} x(t)=(e^{-t}a,\ e^{-2t}b). \end{align*} Then \begin{align*} x_1'(t)=-e^{-t}a=-x_1(t) \end{align*} and \begin{align*} x_2'(t)=-2e^{-2t}b=-2x_2(t), \end{align*} with $x(0)=(a,b)=x_0$. Since $0<e^{-2t}\le e^{-t}\le 1$ for $t\ge 0$, \begin{align*} |x(t)|^2=e^{-2t}a^2+e^{-4t}b^2\le a^2+b^2=|x_0|^2. \end{align*} Hence $|x(t)|\le |x_0|$ for all $t\ge 0$, and the same choice $\delta=\varepsilon$ proves Lyapunov stability. Moreover, \begin{align*} \lim_{t\to\infty}e^{-t}a=0 \end{align*} and \begin{align*} \lim_{t\to\infty}e^{-2t}b=0, \end{align*} so $x(t)\to 0$. The sink is therefore stable and attracting, while the center is stable without attraction. [/example] This distinction leads to a hierarchy of stability notions. Lyapunov stability controls escape; [asymptotic stability](/page/Asymptotic%20Stability) adds convergence; exponential stability gives a quantitative decay rate. ## Attraction and Stronger Stability A stable equilibrium may fail to attract. A center in the plane keeps nearby orbits nearby, but a nearby orbit does not converge to the center. For applications, especially dissipative systems, one often needs both persistence near the equilibrium and eventual convergence to it. Convergence by itself measures the final destination of a trajectory, not the size of its intermediate excursion. It is still useful to isolate attraction before combining it with stability. [definition: Attracting Equilibrium] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $x^*\in U$ be an equilibrium. The equilibrium $x^*$ is attracting if there exists $r>0$ such that for every $x_0\in U$ with $|x_0-x^*|<r$, the solution $x(t;x_0)$ exists for all $t\ge 0$ and satisfies \begin{align*} \lim_{t\to\infty}|x(t;x_0)-x^*| = 0. \end{align*} [/definition] Attraction alone is not enough for the usual notion of stable convergence. A trajectory might converge after first making a large excursion, so attraction does not protect the system during the transient part of the motion. The next definition combines the two requirements needed in applications: nearby trajectories must stay nearby for all future time, and they must also converge to the equilibrium as time tends to infinity. [definition: Asymptotically Stable Equilibrium] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $x^*\in U$ be an equilibrium. The equilibrium $x^*$ is asymptotically stable if it is Lyapunov stable and attracting. [/definition] Asymptotic stability still gives no rate, so it may be too weak for estimates under forcing, discretisation, or modelling error. A quantitative version asks for uniform exponential decay, with constants that work throughout a whole neighbourhood of the equilibrium. [definition: Exponentially Stable Equilibrium] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $x^*\in U$ be an equilibrium. The equilibrium $x^*$ is exponentially stable if there exist constants $r>0$, $C>0$, and $\alpha>0$ such that whenever $|x_0-x^*|<r$, the solution exists for all $t\ge 0$ and satisfies \begin{align*} |x(t;x_0)-x^*| \le C e^{-\alpha t}|x_0-x^*| \end{align*} for all $t\ge 0$. [/definition] The hierarchy should be checked rather than merely asserted, because the definitions combine different types of control: uniform boundedness, convergence, and a decay rate. Exponential stability gives a numerical estimate, but asymptotic stability is an epsilon-delta plus limit statement. The next result verifies that the numerical estimate supplies both the no-escape condition and convergence to the equilibrium. [quotetheorem:7901] The distinction is not a technicality. The scalar equation with a cubic restoring force converges, but its decay is slower than exponential near the equilibrium. [example: Asymptotic Stability without Exponential Decay] Consider the scalar equation \begin{align*} \frac{dx}{dt}=-x^3 \end{align*} on $\mathbb R$. The origin is an equilibrium because $f(0)=-0^3=0$. If $x_0=0$, then $x(t)=0$ for all $t\ge 0$. Now suppose $x_0\ne 0$. On any interval where $x(t)\ne 0$, the chain rule gives \begin{align*} \frac{d}{dt}\bigl(x(t)^{-2}\bigr)=-2x(t)^{-3}x'(t)=-2x(t)^{-3}(-x(t)^3)=2. \end{align*} Integrating from $0$ to $t$ gives \begin{align*} x(t)^{-2}-x_0^{-2}=2t. \end{align*} Hence \begin{align*} x(t)^2=\frac{1}{x_0^{-2}+2t}=\frac{x_0^2}{1+2tx_0^2}. \end{align*} The denominator $1+2tx_0^2$ is positive for every $t\ge 0$, so the solution exists for all future time and satisfies \begin{align*} |x(t)|=\frac{|x_0|}{\sqrt{1+2tx_0^2}}. \end{align*} Since \begin{align*} \sqrt{1+2tx_0^2}\ge 1, \end{align*} we have \begin{align*} |x(t)|\le |x_0| \end{align*} for every $t\ge 0$. Therefore, given $\varepsilon>0$, choosing $\delta=\varepsilon$ ensures that $|x_0|<\delta$ implies $|x(t)|<\varepsilon$ for all $t\ge 0$, so the origin is Lyapunov stable. Also, for $x_0\ne 0$, the denominator in the displayed formula for $|x(t)|$ tends to infinity as $t\to\infty$, and therefore \begin{align*} \lim_{t\to\infty}|x(t)|=\lim_{t\to\infty}\frac{|x_0|}{\sqrt{1+2tx_0^2}}=0. \end{align*} Thus the origin is attracting as well as Lyapunov stable, so it is asymptotically stable. It is not exponentially stable. Suppose, for contradiction, that there were constants $r>0$, $C>0$, and $\alpha>0$ such that \begin{align*} |x(t)|\le Ce^{-\alpha t}|x_0| \end{align*} for all $t\ge 0$ whenever $|x_0|<r$. Choose a nonzero $x_0$ with $|x_0|<r$. Substituting the explicit solution and dividing by $|x_0|>0$ would give \begin{align*} \frac{1}{\sqrt{1+2tx_0^2}}\le Ce^{-\alpha t}. \end{align*} Equivalently, \begin{align*} \frac{e^{\alpha t}}{\sqrt{1+2tx_0^2}}\le C. \end{align*} But the left-hand side is unbounded as $t\to\infty$, because its square is \begin{align*} \frac{e^{2\alpha t}}{1+2tx_0^2}, \end{align*} and the exponential numerator grows faster than the linear denominator. This contradicts the fixed bound $C$. The example shows that asymptotic stability can occur with only algebraic decay, here with inverse-square-root decay along each nonzero trajectory. [/example] The next failure goes in a different direction. Convergence can coexist with a loss of uniform control if trajectories first move far away. This is why the definition of asymptotic stability includes Lyapunov stability instead of merely attraction. [remark: Attraction Does Not Replace Stability] There are planar systems with an attracting equilibrium such that arbitrarily nearby initial conditions make large excursions before returning; standard constructions appear in texts on qualitative stability, where thin sectors near the equilibrium guide selected trajectories outward before the global vector field bends them back. In such systems the limit $x(t;x_0)\to x^*$ holds for nearby $x_0$, but the equilibrium is not Lyapunov stable. The Lyapunov condition is the no-overshoot part of stable convergence. [/remark] ## Lyapunov Functions The epsilon-delta definition is exact, but it can be hard to verify from the solution formula because nonlinear systems rarely have explicit solutions. Lyapunov's method replaces explicit integration by a scalar quantity that behaves like an energy. If the energy cannot increase along trajectories, then trajectories cannot cross to higher energy levels. To state this method, we first need a function that measures displacement from the equilibrium without assigning negative values. It should vanish at the equilibrium and be positive nearby. [definition: Positive Definite Function] Let $U\subset \mathbb R^n$ be an open neighbourhood of $x^*\in \mathbb R^n$. A [continuous function](/page/Continuous%20Function) $V:U\to \mathbb R$ is positive definite at $x^*$ if \begin{align*} V(x^*) = 0 \end{align*} and \begin{align*} V(x)>0 \end{align*} for every $x\in U\setminus\{x^*\}$. [/definition] A positive definite function is useful for stability when its sublevel sets form small cages around the equilibrium. To test whether those cages are respected by the dynamics, we need the rate of change of the function along the vector field, not its derivative in an arbitrary direction. [definition: Derivative of a Lyapunov Function Along a Vector Field] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be continuous, and let $V\in C^1(U;\mathbb R)$. The derivative of $V$ along $f$ is the function $\dot V_f:U\to \mathbb R$ defined by \begin{align*} \dot V_f(x)=\nabla V(x)\cdot f(x). \end{align*} [/definition] The notation $\dot V_f$ records the chain rule along solutions. If $x(t)$ solves \begin{align*} \frac{dx}{dt}=f(x), \end{align*} then the derivative of $V(x(t))$ with respect to time is $\dot V_f(x(t))$. The derivative along the vector field tells us whether an energy candidate rises or falls on actual trajectories, but by itself it does not say whether the candidate measures displacement from the equilibrium. The next definition packages both features: the function must be positive away from the equilibrium, and its value must not increase along the dynamics. This is exactly the structure needed to trap trajectories inside nested sublevel sets without solving the differential equation explicitly. [definition: Lyapunov Function] Let $U\subset \mathbb R^n$ be an open neighbourhood of an equilibrium $x^*$ for the differential equation \begin{align*} \frac{dx}{dt}=f(x), \end{align*} where $f:U\to \mathbb R^n$ is continuous. A function $V\in C^1(U;\mathbb R)$ is a Lyapunov function for $x^*$ on $U$ if $V$ is positive definite at $x^*$ and \begin{align*} \dot V_f(x) \le 0 \end{align*} for every $x\in U$. [/definition] The Lyapunov function theorem is the basic engine of the subject. It turns the geometric idea of nested invariant sublevel sets into an epsilon-delta conclusion. [quotetheorem:7603] The theorem does not require $\dot V_f$ to be negative away from the equilibrium. Nonincrease already prevents escape from sufficiently small sublevel sets. Strict decrease gives more, but nonincrease is enough for stability. When $\dot V_f$ vanishes on a larger set, the direct method by itself usually proves trapping rather than convergence; the standard refinement is LaSalle's invariance principle, which studies the invariant part of the zero-derivative set. [example: Energy for the Harmonic Oscillator] Consider the harmonic oscillator \begin{align*} \dot x_1 = x_2, \end{align*} \begin{align*} \dot x_2 = -x_1. \end{align*} For \begin{align*} V(x_1,x_2)=x_1^2+x_2^2, \end{align*} we have $V(0,0)=0$, and if $(x_1,x_2)\ne (0,0)$, then at least one of $x_1^2$ and $x_2^2$ is positive, so $V(x_1,x_2)>0$. Thus $V$ is positive definite at $(0,0)$. Let $f(x_1,x_2)=(x_2,-x_1)$. Since \begin{align*} \nabla V(x_1,x_2)=(2x_1,2x_2), \end{align*} the derivative of $V$ along the vector field is \begin{align*} \dot V_f(x_1,x_2)=\nabla V(x_1,x_2)\cdot f(x_1,x_2)=(2x_1,2x_2)\cdot(x_2,-x_1). \end{align*} Expanding the dot product gives \begin{align*} \dot V_f(x_1,x_2)=2x_1x_2-2x_1x_2=0. \end{align*} Therefore, along any solution $x(t)=(x_1(t),x_2(t))$, \begin{align*} \frac{d}{dt}V(x(t))=\dot V_f(x(t))=0, \end{align*} so $V(x(t))=V(x(0))$ for all $t\ge 0$. Equivalently, \begin{align*} x_1(t)^2+x_2(t)^2=x_1(0)^2+x_2(0)^2. \end{align*} Given $\varepsilon>0$, choose $\delta=\varepsilon$. If $|x(0)|<\delta$, then \begin{align*} |x(t)|^2=V(x(t))=V(x(0))=|x(0)|^2<\varepsilon^2 \end{align*} for every $t\ge 0$, hence $|x(t)|<\varepsilon$. Thus the origin is Lyapunov stable. The equality $x_1(t)^2+x_2(t)^2=x_1(0)^2+x_2(0)^2$ also shows that every nonzero trajectory stays on a circle of positive radius, so its distance from the origin is constant and cannot tend to $0$. [/example] To prove convergence rather than boundedness, the energy must lose value along every nonzero trajectory near the equilibrium. This leads to a stronger condition. [definition: Negative Definite Derivative Along a Vector Field] Let $U\subset \mathbb R^n$ be an open neighbourhood of $x^*\in \mathbb R^n$, let $f:U\to \mathbb R^n$ be continuous, and let $V\in C^1(U;\mathbb R)$. The derivative $\dot V_f$ is negative definite at $x^*$ if \begin{align*} \dot V_f(x^*) = 0 \end{align*} and \begin{align*} \dot V_f(x)<0 \end{align*} for every $x\in U\setminus\{x^*\}$. [/definition] Strict decrease of a positive definite Lyapunov function should rule out persistent motion on a nonzero energy level. The next theorem is the local convergence statement behind that intuition: if every non-equilibrium point near $x^*$ loses Lyapunov energy, then trajectories starting sufficiently close cannot merely remain trapped; they must approach the equilibrium. [quotetheorem:7603] The theorem is local. It says that trajectories starting sufficiently close converge. It does not assert that all initial conditions in a large domain converge unless the Lyapunov function has sublevel sets that control the entire domain. ## Linearisation and Nonlinear Equilibria Near a differentiable equilibrium, the first-order part of the vector field often decides stability. The nonlinear system behaves like its linearisation for small perturbations, except when the linearisation sits on the boundary between decay and growth. The relevant matrix is the Jacobian matrix at the equilibrium, not merely a symbolic derivative. It records the best linear approximation of the vector field in standard coordinates. [definition: Linearisation at an Equilibrium] Let $U\subset \mathbb R^n$ be open, let $f\in C^1(U;\mathbb R^n)$, and suppose $x^*\in U$ is an equilibrium for the differential equation \begin{align*} \frac{dx}{dt}=f(x). \end{align*} The linearisation of the system at $x^*$ is the linear system \begin{align*} \frac{dy}{dt} = Jf_{x^*}y, \end{align*} where $Jf_{x^*}\in \mathbb R^{n\times n}$ is the Jacobian matrix with entries \begin{align*} (Jf_{x^*})_{ij} = \frac{\partial f_i}{\partial x_j}(x^*). \end{align*} [/definition] If the linearisation has all eigenvalues strictly in the left half-plane, the nonlinear terms are smaller than the linear restoring part near the equilibrium. The question is whether this linear decay survives after the higher-order error is restored to the equation. The standard local test answers yes, and it is stronger than mere convergence: it gives exponential decay for all sufficiently small perturbations, making the nonlinear equilibrium behave like its stable linear model at leading order. [quotetheorem:7902] The opposite sign gives a complementary test. A positive real part creates an expanding linear direction, and near the equilibrium the higher-order terms are too small to remove that expansion. This gives a practical way to prove failure of Lyapunov stability without explicitly finding escaping nonlinear trajectories. [quotetheorem:7903] The tests leave open the critical case where eigenvalues lie on the imaginary axis. Then nonlinear terms decide the answer, and examples with the same linearisation can behave differently. [example: Same Linearisation, Different Stability] Consider the scalar equations \begin{align*} \dot x=-x^3 \end{align*} and \begin{align*} \dot y=y^3. \end{align*} For $f(x)=-x^3$ and $g(y)=y^3$, we have $f(0)=0$ and $g(0)=0$, so the origin is an equilibrium for both equations. Their linearisations at the origin are both zero, because \begin{align*} f'(0)=-3\cdot 0^2=0 \end{align*} and \begin{align*} g'(0)=3\cdot 0^2=0. \end{align*} For the equation $\dot x=-x^3$, the solution with $x_0=0$ is $x(t)=0$. If $x_0\ne 0$, then on any interval where $x(t)\ne 0$, \begin{align*} \frac{d}{dt}\bigl(x(t)^{-2}\bigr)=-2x(t)^{-3}x'(t)=-2x(t)^{-3}(-x(t)^3)=2. \end{align*} Integrating from $0$ to $t$ gives \begin{align*} x(t)^{-2}-x_0^{-2}=2t. \end{align*} Hence \begin{align*} x(t)^2=\frac{1}{x_0^{-2}+2t}. \end{align*} Multiplying numerator and denominator by $x_0^2$ gives \begin{align*} x(t)^2=\frac{x_0^2}{1+2tx_0^2}. \end{align*} Since $1+2tx_0^2\ge 1$ for $t\ge 0$, we get $|x(t)|\le |x_0|$ for all $t\ge 0$. Given $\varepsilon>0$, choosing $\delta=\varepsilon$ gives $|x_0|<\delta$ implies $|x(t)|<\varepsilon$ for every $t\ge 0$, so the origin is Lyapunov stable. Also, \begin{align*} \lim_{t\to\infty}\frac{|x_0|}{\sqrt{1+2tx_0^2}}=0, \end{align*} so every sufficiently close solution tends to $0$. Thus the origin is asymptotically stable for $\dot x=-x^3$. For the equation $\dot y=y^3$, take $y_0>0$. While $y(t)>0$, \begin{align*} \frac{d}{dt}\bigl(y(t)^{-2}\bigr)=-2y(t)^{-3}y'(t)=-2y(t)^{-3}y(t)^3=-2. \end{align*} Integrating from $0$ to $t$ gives \begin{align*} y(t)^{-2}-y_0^{-2}=-2t. \end{align*} Thus \begin{align*} y(t)^2=\frac{1}{y_0^{-2}-2t}. \end{align*} Multiplying numerator and denominator by $y_0^2$ gives \begin{align*} y(t)^2=\frac{y_0^2}{1-2ty_0^2}. \end{align*} This formula is valid as long as \begin{align*} 1-2ty_0^2>0, \end{align*} that is, for \begin{align*} 0\le t<\frac{1}{2y_0^2}. \end{align*} Now fix any $\varepsilon>0$. Given any $\delta>0$, choose $y_0$ with $0<y_0<\min(\delta,\varepsilon)$. The time \begin{align*} t_*=\frac{1}{2y_0^2}-\frac{1}{2\varepsilon^2} \end{align*} is positive because $y_0<\varepsilon$, and it is less than the upper endpoint of the interval above. At this time, \begin{align*} 1-2t_*y_0^2=1-2y_0^2\left(\frac{1}{2y_0^2}-\frac{1}{2\varepsilon^2}\right)=\frac{y_0^2}{\varepsilon^2}. \end{align*} Therefore \begin{align*} y(t_*)^2=\frac{y_0^2}{y_0^2/\varepsilon^2}=\varepsilon^2. \end{align*} For any $t$ with \begin{align*} t_*<t<\frac{1}{2y_0^2}, \end{align*} the denominator satisfies \begin{align*} 0<1-2ty_0^2<\frac{y_0^2}{\varepsilon^2}. \end{align*} Hence $y(t)^2>\varepsilon^2$, so $|y(t)|>\varepsilon$. Thus arbitrarily small positive initial conditions leave the $\varepsilon$-neighbourhood of the origin before their maximal existence time ends. The two systems have the same zero linearisation at the origin, but one origin is asymptotically stable and the other is not Lyapunov stable; in this critical case, the nonlinear terms decide the stability. [/example] This is the conceptual role of Lyapunov functions in nonlinear theory. They can resolve stability when the first-order approximation is inconclusive, and they can express conserved or dissipated quantities that are invisible from eigenvalues alone. ## Invariant Sets and Phase Portraits Equilibria are only the first invariant objects encountered in dynamics. Periodic orbits, closed annuli of conserved energy, and attractors are also sets whose stability matters. The set formulation of Lyapunov stability is therefore not an optional generalisation; it is the language needed for phase portraits. For a periodic solution, the right question is not whether a nearby solution remains near the same point at the same time. A small phase shift may move a trajectory along the orbit. Stability of the orbit asks only that the trajectory remain close to the closed curve. [definition: Periodic Orbit] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $\varphi_t:D_t\to U$ denote the forward flow. A periodic orbit is a set \begin{align*} \Gamma = \{\varphi_t(x_0):0\le t<T\} \end{align*} for some $x_0\in U$ and $T>0$ such that $\varphi_T(x_0)=x_0$ and $\varphi_t(x_0)\ne x_0$ for $0<t<T$. [/definition] The set $\Gamma$ is positively invariant. Its Lyapunov stability says that a sufficiently small perturbation remains in a tube around the orbit for all future time, even if its phase along the orbit changes. [example: Stable Invariant Circles in the Harmonic Oscillator] For the harmonic oscillator \begin{align*} \dot x_1=x_2 \end{align*} and \begin{align*} \dot x_2=-x_1, \end{align*} fix $r>0$ and let \begin{align*} E_r=\{(x_1,x_2)\in\mathbb R^2:x_1^2+x_2^2=r^2\}. \end{align*} The solution starting from $(r,0)$ is \begin{align*} x(t)=(r\cos t,-r\sin t), \end{align*} because \begin{align*} x_1'(t)=-r\sin t=x_2(t) \end{align*} and \begin{align*} x_2'(t)=-r\cos t=-x_1(t). \end{align*} Also $x(0)=(r,0)$ and $x(2\pi)=(r,0)$, while $x(t)\ne(r,0)$ for $0<t<2\pi$. As $t$ runs from $0$ to $2\pi$, the identity \begin{align*} x_1(t)^2+x_2(t)^2=r^2\cos^2t+r^2\sin^2t=r^2 \end{align*} shows that this periodic orbit is exactly the circle $E_r$. Now let $x(t)=(x_1(t),x_2(t))$ be any solution. For the energy \begin{align*} V(x_1,x_2)=x_1^2+x_2^2, \end{align*} the chain rule gives \begin{align*} \frac{d}{dt}V(x(t))=2x_1(t)x_1'(t)+2x_2(t)x_2'(t). \end{align*} Substituting $x_1'(t)=x_2(t)$ and $x_2'(t)=-x_1(t)$ gives \begin{align*} \frac{d}{dt}V(x(t))=2x_1(t)x_2(t)-2x_2(t)x_1(t)=0. \end{align*} Hence $V(x(t))=V(x(0))$ for every $t\ge 0$, so every trajectory remains on the circle determined by its initial radius. It remains to translate conservation of radius into stability of the set $E_r$. For any point $z\in\mathbb R^2$, write $\rho=|z|$. If $\rho>0$, then \begin{align*} y=\frac r\rho z \end{align*} lies in $E_r$, and \begin{align*} |z-y|=\left|z-\frac r\rho z\right|=|\rho-r|. \end{align*} For every $y\in E_r$, the [reverse triangle inequality](/theorems/2300) gives \begin{align*} |\rho-r|=\bigl||z|-|y|\bigr|\le |z-y|. \end{align*} Taking the infimum over $y\in E_r$ gives $\operatorname{dist}(z,E_r)=|\rho-r|$; the same formula also holds at $z=0$, since then $\operatorname{dist}(0,E_r)=r=|0-r|$. Given $\varepsilon>0$, choose $\delta=\varepsilon$. If $\operatorname{dist}(x(0),E_r)<\delta$, then \begin{align*} \bigl||x(0)|-r\bigr|<\delta. \end{align*} Since $|x(t)|^2=V(x(t))=V(x(0))=|x(0)|^2$, we have $|x(t)|=|x(0)|$ for all $t\ge 0$. Therefore \begin{align*} \operatorname{dist}(x(t),E_r)=\bigl||x(t)|-r\bigr|=\bigl||x(0)|-r\bigr|<\varepsilon \end{align*} for every $t\ge 0$. Thus each circle $E_r$ is Lyapunov stable as an invariant set: nearby initial points may rotate with a different phase and radius, but their distance from the original circle stays uniformly small for all future time. [/example] Set stability also clarifies why conserved quantities are powerful. A conserved function partitions phase space into invariant level sets; if nearby level sets stay geometrically close, Lyapunov stability follows for the level set. [remark: Conserved Levels and Stable Tubes] Let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, let $H\in C^1(U;\mathbb R)$ satisfy $\nabla H(x)\cdot f(x)=0$ for every $x\in U$, and let $E\subset U$ be contained in the level set $\{x\in U:H(x)=c\}$ for some $c\in \mathbb R$. A useful stability template is the following: for every $\varepsilon>0$, find $\eta>0$ and $\delta>0$ such that $x\in U\cap N_\delta(E)$ implies $|H(x)-c|<\eta$, the tube $T_\eta=\{x\in U:|H(x)-c|<\eta\}$ is contained in $N_\varepsilon(E)$, and the initial data under consideration have forward-complete trajectories that remain in $U$ while they stay in $T_\eta$. Since $H$ is conserved along those forward-complete trajectories, this pair of inclusions traps nearby initial data inside the prescribed tube around $E$ for all future time. [/remark] This observation is a template rather than a universal theorem. It explains why geometric information about level sets is needed in addition to the conservation law. A conserved quantity controls motion only through the geometry of its level sets. ## Constructing and Reading Lyapunov Functions Finding a Lyapunov function is part analysis and part modelling. In mechanical systems it may be energy. In gradient-like systems it may be the objective function. In linear systems it can be constructed from a positive definite quadratic form. Quadratic Lyapunov functions are the bridge between matrix stability and nonlinear estimates. They encode ellipsoids rather than Euclidean balls, but all norms on $\mathbb R^n$ are locally equivalent, so their sublevel sets still define neighbourhoods of the equilibrium. [definition: Positive Definite Matrix] A symmetric matrix $P\in \mathbb R^{n\times n}$ is positive definite if \begin{align*} x^\top P x >0 \end{align*} for every $x\in \mathbb R^n\setminus\{0\}$. [/definition] Positive definite matrices give the standard quadratic candidates $V(x)=x^\top Px$. Along the linear system \begin{align*} \frac{dx}{dt}=Ax, \end{align*} the derivative of this candidate is governed by $A^\top P+PA$. To construct a certificate rather than guess one, we want to prescribe a negative quadratic derivative and solve for the matrix $P$ that produces it. The Lyapunov equation is the matrix equation that encodes this construction. [definition: Continuous Lyapunov Equation] Let $A\in \mathbb R^{n\times n}$ and let $Q\in \mathbb R^{n\times n}$ be symmetric. The continuous Lyapunov equation for $A$ and $Q$ is \begin{align*} A^\top P+PA = -Q, \end{align*} where the unknown is a symmetric matrix $P\in \mathbb R^{n\times n}$. [/definition] The Lyapunov equation is more than a computational device, but it is useful only if the matrix it produces is itself positive definite. Otherwise the quadratic form would not measure distance from the origin. The next theorem gives the missing guarantee: for a Hurwitz matrix, every positive definite choice of $Q$ produces a unique positive definite $P$, so spectral decay becomes an explicit energy whose derivative is negative definite. [quotetheorem:6369] In practice, the candidate function must also be read correctly. If $\dot V_f\le 0$, stability may follow. If $\dot V_f<0$ away from the equilibrium, convergence is expected under local hypotheses. If $\dot V_f$ changes sign, the candidate fails as a Lyapunov certificate even if the system might still be stable. [example: A Quadratic Certificate] Consider the linear system $\dot x_1=-2x_1+x_2$ and $\dot x_2=-x_2$. For $x=(x_1,x_2)$, the vector field is \begin{align*} Ax=(-2x_1+x_2,-x_2). \end{align*} Take $P=I$ and define \begin{align*} V(x)=x^\top Ix=x_1^2+x_2^2. \end{align*} Then $V(0,0)=0$, and if $x\ne 0$, at least one of $x_1^2$ and $x_2^2$ is positive, so $V(x)>0$. Since \begin{align*} \nabla V(x_1,x_2)=(2x_1,2x_2), \end{align*} the derivative of $V$ along the vector field is \begin{align*} \dot V_A(x)=\nabla V(x)\cdot Ax=(2x_1,2x_2)\cdot(-2x_1+x_2,-x_2). \end{align*} Expanding the dot product gives \begin{align*} \dot V_A(x)=2x_1(-2x_1+x_2)+2x_2(-x_2). \end{align*} Distributing each term gives \begin{align*} \dot V_A(x)=-4x_1^2+2x_1x_2-2x_2^2. \end{align*} The inequality $2x_1x_2\le 2|x_1x_2|$ and the elementary estimate $2|x_1x_2|\le x_1^2+x_2^2$ imply \begin{align*} 2x_1x_2\le x_1^2+x_2^2. \end{align*} Substituting this upper bound into the expression for $\dot V_A$ gives \begin{align*} \dot V_A(x)\le -4x_1^2+x_1^2+x_2^2-2x_2^2. \end{align*} Combining like terms yields \begin{align*} \dot V_A(x)\le -3x_1^2-x_2^2. \end{align*} Thus $\dot V_A(0)=0$, while for every $x\ne 0$ the quantity $-3x_1^2-x_2^2$ is strictly negative, so $\dot V_A(x)<0$. Therefore $V$ is positive definite and has negative definite derivative along the vector field. By the *Lyapunov Asymptotic Stability Theorem*, the origin is asymptotically stable. [/example] The method is flexible because the function need not be quadratic. For nonlinear systems, the geometry of $V$ should match the nonlinear mechanism that prevents escape. ## Beyond and Connected Topics Lyapunov stability is a local language for robustness of motion, but it sits inside a larger theory of qualitative dynamics. The next step in [Cambridge II Dynamical Systems](/page/Cambridge%20II%20Dynamical%20Systems) is to combine stability with phase portraits, invariant manifolds, and bifurcations, where equilibria change type as a parameter varies. The analytic foundations come from [Cambridge IA Analysis Notes](/page/Cambridge%20IA%20Analysis%20Notes) and [Cambridge IB Analysis and Topology](/page/Cambridge%20IB%20Analysis%20and%20Topology): continuity, compactness, norms, and metric neighbourhoods are the language behind the epsilon-delta definition. Stability is a dynamical version of uniform control over time. Linear stability also connects to complex eigenvalues and exponentials of matrices. The appearance of $\operatorname{Re}(\lambda)$ is parallel to growth and decay in complex exponentials, a theme that also appears in [Cambridge IB Complex Analysis](/page/Cambridge%20IB%20Complex%20Analysis) through analytic functions and contour methods. For more advanced work, Lyapunov functions lead toward LaSalle's invariance principle, input-to-state stability, control Lyapunov functions, and stability of invariant manifolds. These topics keep the same central idea but replace pointwise energy decay by invariance, forcing, feedback, or infinite-dimensional phase spaces. ## References Androma, [Cambridge II Dynamical Systems](/page/Cambridge%20II%20Dynamical%20Systems). Androma, [Cambridge IB Analysis and Topology](/page/Cambridge%20IB%20Analysis%20and%20Topology). Androma, [Cambridge IB Complex Analysis](/page/Cambridge%20IB%20Complex%20Analysis). Androma, [Cambridge IA Analysis Notes](/page/Cambridge%20IA%20Analysis%20Notes). Hirsch, Smale, and Devaney, *Differential Equations, Dynamical Systems, and an Introduction to Chaos* (2013). Khalil, *Nonlinear Systems* (2002). LaSalle and Lefschetz, *Stability by Liapunov's Direct Method with Applications* (1961). Perko, *Differential Equations and Dynamical Systems* (2001). Bhatia and Szegő, *Stability Theory of Dynamical Systems* (1970).