Lyapunov Functions and Stability

Consider a nonlinear system with an equilibrium. Its linearization may have a purely imaginary eigenvalue. Or it may be too complicated to solve explicitly. Hartman-Grobman and Hyperbolicity tells us that hyperbolic equilibria behave like their linearizations. But it says nothing when hyperbolicity fails. Even when it holds, it gives no quantitative picture. It cannot tell us how far trajectories stray before returning. How, then, can we certify that an equilibrium is stable? Can we even show asymptotic stability without integrating the equations at all? Aleksandr Lyapunov's answer, developed in his 1892 doctoral thesis, is to mimic physical energy. Energy behaves in a specific way in a dissipative system. Suppose a scalar function decreases along every trajectory. Suppose it is otherwise positive away from equilibrium. Then the state is trapped in shrinking sublevel sets. It behaves like a ball rolling down a bowl. It must converge to the bottom. This idea converts a question about a system of differential equations. It becomes instead a question about the sign of a single derivative. It is an algebraic computation rather than an integration. The method's power is its universality. It applies to any dimension. It requires no closed-form solution. And it works precisely in the borderline cases. These include centers, saddle-node bifurcations, and systems with zero eigenvalues. In these cases, the Linearization and the Jacobian approach is silent. Lyapunov's method is the single most important tool for proving stability of nonlinear equilibria. This holds especially in the absence of an explicit flow. This article builds the formal definition of a Lyapunov function. It proves the stability, asymptotic stability, and instability theorems with full rigor. It shows how to construct candidate functions in practice. These include quadratic forms, energy functions, and variable-gradient methods. It closes by connecting Lyapunov theory to later topics. These are the conservative systems and limit-cycle analyses that follow in the study of nonlinear dynamics.

Key takeaways
  • A Lyapunov function is a scalar "energy" VV that is positive away from an equilibrium and non-increasing along trajectories.
  • If V˙≤0\dot V \le 0 near the equilibrium, the equilibrium is stable; if V˙<0\dot V < 0 (strictly, away from equilibrium), it is asymptotically stable.
  • No need to solve the ODE: only algebraic sign-checking of VV and its derivative along the vector field is required.
  • The method handles non-hyperbolic cases (e.g., centers) where linearization is inconclusive.
  • Construction is an art: quadratic forms, physical energy, and the variable-gradient method are standard techniques.
  • Key idea: trap the trajectory in shrinking sublevel sets of a scalar function whose value can only fall.

Where this lives today

Lyapunov functions are the backbone of modern control theory: feedback controllers for robots, aircraft, and chemical plants are routinely designed by first positing a candidate VV (often a quadratic cost or energy function) and choosing the control law to force V˙<0\dot V < 0, a technique called Lyapunov redesign or control-Lyapunov-function synthesis. In electrical engineering, stability of power grids and phase-locked loops is certified using energy-like Lyapunov functions built from the physical power and reactive-power structure of the network. In ecology and epidemiology, Lyapunov functions (often of the form ∑ci(xi−xi∗ln⁡xi)\sum c_i(x_i - x_i^* \ln x_i)) prove global stability of coexistence equilibria in Lotka–Volterra models and disease-free equilibria in SIR-type models. In numerical analysis and machine learning, Lyapunov-style arguments underlie convergence proofs for gradient descent and its variants, where a decreasing "potential function" (loss, or a regularized surrogate) plays exactly the role of VV. In cryptography and distributed computing, similar potential-function arguments certify convergence of consensus and gossip protocols. Within pure dynamics, the same idea reappears as the foundation for Conservative Systems and the Pendulum, where mechanical energy is itself a (non-strict) Lyapunov function explaining the neutral stability of a center, and for Limit Cycles and Closed Orbits, where Lyapunov-type arguments (Bendixson–Dulac and Poincaré–Bendixson reasoning) are used to rule out or trap periodic orbits.

Ask the tutor: Can you show me how to build a Lyapunov function step-by-step for a specific nonlinear system I'm working on?

DefinitionLyapunov function

Let x˙=f(x)\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) be an autonomous system with f:D→Rn\mathbf{f}: D \to \mathbb{R}^n continuously differentiable (or at least locally Lipschitz) on an open set D⊆RnD \subseteq \mathbb{R}^n containing the origin, and suppose f(0)=0\mathbf{f}(\mathbf{0}) = \mathbf{0}, so x=0\mathbf{x} = \mathbf{0} is an equilibrium. Let V:D→RV: D \to \mathbb{R} be continuously differentiable. Define the orbital derivative (derivative of VV along trajectories) by

V˙(x)  =  ∇V(x)⋅f(x)  =  ∑i=1n∂V∂xi(x) fi(x).\dot V(\mathbf{x}) \;=\; \nabla V(\mathbf{x}) \cdot \mathbf{f}(\mathbf{x}) \;=\; \sum_{i=1}^n \frac{\partial V}{\partial x_i}(\mathbf{x})\, f_i(\mathbf{x}).

This is exactly ddtV(x(t))\frac{d}{dt} V(\mathbf{x}(t)) evaluated along any solution x(t)\mathbf{x}(t) of the system, computed without solving the system, since by the chain rule ddtV(x(t))=∇V(x(t))⋅x˙(t)=∇V(x(t))⋅f(x(t))\frac{d}{dt}V(\mathbf{x}(t)) = \nabla V(\mathbf{x}(t))\cdot \dot{\mathbf{x}}(t) = \nabla V(\mathbf{x}(t)) \cdot \mathbf{f}(\mathbf{x}(t)). We say VV is a Lyapunov function for the equilibrium 0\mathbf{0} on a neighborhood U⊆DU \subseteq D of the origin if: 1. Positive definiteness: V(0)=0V(\mathbf{0}) = 0 and V(x)>0V(\mathbf{x}) > 0 for all x∈U∖{0}\mathbf{x} \in U \setminus \{\mathbf{0}\}; 2. Negative semidefiniteness of the orbital derivative: V˙(x)≤0\dot V(\mathbf{x}) \le 0 for all x∈U\mathbf{x} \in U. If in addition V˙(x)<0\dot V(\mathbf{x}) < 0 for all x∈U∖{0}\mathbf{x} \in U \setminus \{\mathbf{0}\}, VV is called a strict Lyapunov function, and V˙\dot V is called negative definite on UU.

Intuition

Picture VV as a bowl-shaped landscape sitting over the phase plane, with its unique minimum of value 00 located exactly at the equilibrium. The condition V˙≤0\dot V \le 0 says that as time moves forward, the state point can only slide to equal or lower "elevation" on this landscape — it can never climb. Since the level sets {V=c}\{V = c\} for small c>0c > 0 are closed curves shrinking down around the origin (this is a consequence of positive definiteness and continuity), a trajectory that starts inside such a level curve is trapped inside it forever: it cannot escape because escaping would require crossing to higher VV. This already gives stability: trajectories starting close to equilibrium stay close, forever, because they are confined to shrinking nested "energy shells." If V˙\dot V is strictly negative away from the origin, the value of V(x(t))V(\mathbf{x}(t)) must keep decreasing — it cannot merely level off — and this extra pressure forces the trajectory to be squeezed all the way down to the unique minimum, giving asymptotic stability. No formula for x(t)\mathbf{x}(t) was ever needed; only the sign of a dot product. The classical example is a damped mechanical oscillator, where VV is literal mechanical energy (kinetic plus potential): friction makes V˙≤0\dot V \le 0, and the energy bowl visibly funnels every nearby trajectory down to rest. This physical picture is why Lyapunov's method is often called the "energy method," even for systems, such as chemical reaction networks or population models, that carry no literal energy.

Figure
Level curves of a Lyapunov function trapping a spiraling trajectory
V(x,y) = x² + y², level sets and a trajectory V = 4 V = 2 V = 1 0 (equilibrium) start
TheoremLyapunov's Stability, Asymptotic Stability, and Instability Theorems

Let D⊆RnD \subseteq \mathbb{R}^n be open with 0∈D\mathbf{0} \in D, let f:D→Rn\mathbf{f}: D \to \mathbb{R}^n be locally Lipschitz with f(0)=0\mathbf{f}(\mathbf{0}) = \mathbf{0}, and consider x˙=f(x)\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}). Let U⊆DU \subseteq D be an open neighborhood of 0\mathbf{0} and V∈C1(U,R)V \in C^1(U, \mathbb{R}) with V(0)=0V(\mathbf{0}) = 0.

(a) Stability. If VV is positive definite on UU (i.e. V(x)>0V(\mathbf{x}) > 0 for x∈U∖{0}\mathbf{x} \in U\setminus\{0\}) and V˙(x)≤0\dot V(\mathbf{x}) \le 0 for all x∈U\mathbf{x} \in U, then x=0\mathbf{x} = \mathbf{0} is a stable equilibrium.

(b) Asymptotic stability. If in addition V˙(x)<0\dot V(\mathbf{x}) < 0 for all x∈U∖{0}\mathbf{x} \in U\setminus\{0\} (strict Lyapunov function), then x=0\mathbf{x} = \mathbf{0} is asymptotically stable.

(c) Instability (Chetaev-type). Suppose instead there is a sequence xk→0\mathbf{x}_k \to \mathbf{0} with V(xk)>0V(\mathbf{x}_k) > 0 for all kk, and V˙(x)>0\dot V(\mathbf{x}) > 0 for every x∈U\mathbf{x} \in U with V(x)>0V(\mathbf{x}) > 0. Then x=0\mathbf{x} = \mathbf{0} is unstable.

Proof

We use the technique of trapping trajectories in sublevel sets of VV, justified by continuity and by existence-uniqueness of solutions (Picard–Lindelöf, valid since f\mathbf{f} is locally Lipschitz).

Setup. Since UU is open and contains 0\mathbf{0}, choose r0>0r_0 > 0 small enough that the closed ball B‾(0,r0)⊂U\overline{B}(\mathbf{0}, r_0) \subset U. All balls below are understood to be intersected with DD and to lie inside UU.

(a) Stability. Fix ε∈(0,r0]\varepsilon \in (0, r_0]. We must produce δ>0\delta > 0 such that ∥x(0)∥<δ\|\mathbf{x}(0)\| < \delta implies ∥x(t)∥<ε\|\mathbf{x}(t)\| < \varepsilon for all t≥0t \ge 0 for which the solution exists (and we will see existence extends to all t≥0t\ge0). Let Sε={x:∥x∥=ε}S_\varepsilon = \{\mathbf{x} : \|\mathbf{x}\| = \varepsilon\} be the sphere of radius ε\varepsilon. This set is compact (closed and bounded in Rn\mathbb{R}^n) and does not contain 0\mathbf{0}, so by positive definiteness VV is strictly positive and continuous on SεS_\varepsilon; by the extreme value theorem,

m:=min⁡x∈SεV(x)>0.m := \min_{\mathbf{x} \in S_\varepsilon} V(\mathbf{x}) > 0.

By continuity of VV and V(0)=0V(\mathbf{0}) = 0, there exists δ∈(0,ε)\delta \in (0, \varepsilon) such that

∥x∥<δ  ⟹  V(x)<m.\|\mathbf{x}\| < \delta \implies V(\mathbf{x}) < m.

Now let x(t)\mathbf{x}(t) be the solution with ∥x(0)∥<δ\|\mathbf{x}(0)\| < \delta, defined on its maximal interval of existence [0,T)[0, T). Define

g(t)=V(x(t)).g(t) = V(\mathbf{x}(t)).

Then gg is differentiable wherever x(t)\mathbf{x}(t) is defined, with g′(t)=∇V(x(t))⋅f(x(t))=V˙(x(t))≤0g'(t) = \nabla V(\mathbf{x}(t)) \cdot \mathbf{f}(\mathbf{x}(t)) = \dot V(\mathbf{x}(t)) \le 0 by hypothesis, as long as x(t)\mathbf{x}(t) remains in UU. Hence gg is non-increasing while the trajectory stays in B‾(0,ε)⊂U\overline B(\mathbf 0,\varepsilon)\subset U, so

V(x(t))≤V(x(0))<mfor all such t.V(\mathbf{x}(t)) \le V(\mathbf{x}(0)) < m \quad \text{for all such } t.

We claim ∥x(t)∥<ε\|\mathbf{x}(t)\| < \varepsilon for all t∈[0,T)t \in [0,T). Suppose not: let t1t_1 be the first time (if any) with ∥x(t1)∥=ε\|\mathbf{x}(t_1)\| = \varepsilon, i.e., x(t1)∈Sε\mathbf{x}(t_1) \in S_\varepsilon. By continuity of x(⋅)\mathbf{x}(\cdot), the trajectory remains in B‾(0,ε)⊂U\overline{B}(\mathbf 0,\varepsilon) \subset U on [0,t1][0,t_1], so the inequality g(t)≤g(0)<mg(t) \le g(0) < m derived above applies at t=t1t = t_1. But x(t1)∈Sε\mathbf{x}(t_1) \in S_\varepsilon forces V(x(t1))≥mV(\mathbf{x}(t_1)) \ge m by definition of mm. This contradicts V(x(t1))<mV(\mathbf{x}(t_1)) < m. Hence no such t1t_1 exists, and ∥x(t)∥<ε\|\mathbf{x}(t)\| < \varepsilon for all t∈[0,T)t \in [0, T). Since the trajectory remains in the compact set B‾(0,ε)\overline{B}(\mathbf{0},\varepsilon) on its entire interval of existence, the standard extension theorem for ODEs (a solution can be continued as long as it stays in a compact subset of the domain of f\mathbf{f}) guarantees T=∞T = \infty. Thus ∥x(t)∥<ε\|\mathbf{x}(t)\| < \varepsilon for all t≥0t \ge 0, proving stability.

(b) Asymptotic stability. By part (a), 0\mathbf{0} is stable, so fix ε=r0\varepsilon = r_0 and let δ>0\delta > 0 be as constructed above; every trajectory starting in B(0,δ)B(\mathbf{0},\delta) remains in B(0,ε)⊂UB(\mathbf 0,\varepsilon)\subset U for all t≥0t \ge 0. We must show additionally that x(t)→0\mathbf{x}(t) \to \mathbf{0} as t→∞t \to \infty for such trajectories. Since V˙<0\dot V < 0 on U∖{0}U \setminus \{0\}, the function g(t)=V(x(t))g(t) = V(\mathbf{x}(t)) is strictly decreasing as long as x(t)≠0\mathbf{x}(t) \ne \mathbf{0}, and it is bounded below by 00 (positive definiteness). A monotone bounded function has a limit, so

g(t)→L≥0as t→∞,g(t) \to L \ge 0 \quad \text{as } t \to \infty,

for some constant LL. We claim L=0L = 0. Suppose for contradiction L>0L > 0. Since gg is decreasing to LL, we have V(x(t))≥L>0V(\mathbf{x}(t)) \ge L > 0 for all t≥0t \ge 0. Let K={x∈B‾(0,ε):V(x)≥L}K = \{\mathbf{x} \in \overline{B}(\mathbf{0},\varepsilon) : V(\mathbf{x}) \ge L\}. This set KK is closed (as the intersection of the closed ball and the closed sublevel-complement set {V≥L}\{V \ge L\}, using continuity of VV) and bounded, hence compact. Moreover 0∉K\mathbf{0} \notin K since V(0)=0<LV(\mathbf{0}) = 0 < L. Since V˙\dot V is continuous and strictly negative on the compact set K∖{0}=KK \setminus \{0\} = K (as 0∉K\mathbf 0\notin K), it attains a maximum on KK:

−η:=max⁡x∈KV˙(x)<0,η>0.-\eta := \max_{\mathbf{x} \in K} \dot V(\mathbf{x}) < 0, \qquad \eta > 0.

Since x(t)∈K\mathbf{x}(t) \in K for all t≥0t \ge 0 (because x(t)\mathbf{x}(t) stays in B‾(0,ε)\overline B(0,\varepsilon) and V(x(t))≥LV(\mathbf x(t))\ge L), we get g′(t)=V˙(x(t))≤−ηg'(t) = \dot V(\mathbf{x}(t)) \le -\eta for all t≥0t \ge 0. Integrating,

g(t)≤g(0)−ηt→−∞as t→∞,g(t) \le g(0) - \eta t \to -\infty \quad \text{as } t \to \infty,

contradicting g(t)≥L>0g(t) \ge L > 0 for all tt. This contradiction shows L=0L = 0, i.e., V(x(t))→0V(\mathbf{x}(t)) \to 0. Finally, we upgrade V(x(t))→0V(\mathbf x(t))\to 0 to x(t)→0\mathbf x(t) \to \mathbf 0. Suppose not: then there is ρ∈(0,ε)\rho \in (0,\varepsilon) and a sequence tk→∞t_k \to \infty with ∥x(tk)∥≥ρ\|\mathbf{x}(t_k)\| \ge \rho. Since also ∥x(tk)∥≤ε\|\mathbf x(t_k)\|\le \varepsilon, by compactness of the annulus ρ≤∥x∥≤ε\rho \le \|\mathbf x\| \le \varepsilon and positive definiteness of VV on it, min⁡ρ≤∥x∥≤εV(x)=m′>0\min_{\rho \le \|\mathbf x\| \le \varepsilon} V(\mathbf{x}) = m' > 0. Then V(x(tk))≥m′>0V(\mathbf{x}(t_k)) \ge m' > 0 for all kk, contradicting V(x(t))→0V(\mathbf{x}(t)) \to 0. Hence x(t)→0\mathbf{x}(t) \to \mathbf{0}, and asymptotic stability is proved.

(c) Instability. We argue by contradiction, following Chetaev's argument. Suppose 0\mathbf{0} were stable. Fix ε=r0\varepsilon = r_0; stability gives δ>0\delta > 0 such that ∥x(0)∥<δ  ⟹  ∥x(t)∥<ε\|\mathbf x(0)\|<\delta \implies \|\mathbf x(t)\| < \varepsilon for all t≥0t\ge 0. By hypothesis there is a point xk\mathbf{x}_k with ∥xk∥<δ\|\mathbf x_k\| < \delta (taking kk large enough, since xk→0\mathbf x_k \to 0) and V(xk)>0V(\mathbf{x}_k) > 0; call this initial condition x(0)=xk=:p\mathbf{x}(0) = \mathbf{x}_k =: \mathbf{p}, with a:=V(p)>0a := V(\mathbf p) > 0. By stability, x(t)∈B‾(0,ε)\mathbf{x}(t) \in \overline{B}(\mathbf{0},\varepsilon) for all t≥0t \ge 0. Let g(t)=V(x(t))g(t) = V(\mathbf{x}(t)). As long as x(t)\mathbf{x}(t) stays in the region {V>0}∩B‾(0,ε)\{V > 0\} \cap \overline B(0,\varepsilon), we have g′(t)=V˙(x(t))>0g'(t) = \dot V(\mathbf{x}(t)) > 0 by hypothesis, so gg is strictly increasing; in particular g(t)≥g(0)=a>0g(t) \ge g(0) = a > 0 for all such tt, so the trajectory never leaves {V>0}\{V > 0\} once it starts there (since gg never returns to 00), and hence gg is increasing for all t≥0t \ge 0. Consider the compact set K={x∈B‾(0,ε):V(x)≥a}K = \{\mathbf{x} \in \overline{B}(\mathbf{0},\varepsilon) : V(\mathbf{x}) \ge a\}. Since g(t)≥ag(t)\ge a for all t≥0t \ge 0 and x(t)∈B‾(0,ε)\mathbf x(t)\in \overline B(0,\varepsilon), we have x(t)∈K\mathbf{x}(t) \in K for all t≥0t \ge 0. Also 0∉K\mathbf 0 \notin K since V(0)=0<aV(\mathbf 0)=0<a, so V˙\dot V is continuous and strictly positive on the compact set KK, hence

η:=min⁡x∈KV˙(x)>0.\eta := \min_{\mathbf{x}\in K} \dot V(\mathbf{x}) > 0.

Then g′(t)≥ηg'(t) \ge \eta for all t≥0t \ge 0, so g(t)≥a+ηt→∞g(t) \ge a + \eta t \to \infty. But VV is continuous on the compact set B‾(0,ε)\overline{B}(\mathbf{0},\varepsilon), hence bounded there, say V(x)≤MV(\mathbf{x}) \le M for all x∈B‾(0,ε)\mathbf{x} \in \overline B(0,\varepsilon). This contradicts g(t)→∞g(t) \to \infty while g(t)=V(x(t))≤Mg(t) = V(\mathbf x(t)) \le M. This contradiction shows the assumption of stability is false: 0\mathbf{0} is unstable. ■\blacksquare

∎

Worked examples

Example 1 — an asymmetric planar system with no closed-form solution. Consider

x˙=−x+2y2,y˙=−y−xy.\dot x = -x + 2y^2, \qquad \dot y = -y - xy.

This is nonlinear (the y2y^2 and xyxy terms), and while it happens to be solvable in special cases, we treat it as if it were not, to illustrate the method's independence from explicit solutions. The origin is an equilibrium since both right-hand sides vanish at (0,0)(0,0). The linearization at the origin is x˙=−x\dot x = -x, y˙=−y\dot y = -y, with Jacobian (−100−1)\begin{pmatrix}-1 & 0\\0&-1\end{pmatrix}, eigenvalues −1,−1-1,-1 (a stable node — this equilibrium is actually already hyperbolic, so Hartman-Grobman and Hyperbolicity would suffice here; we use it deliberately as a controlled test case where we can check the Lyapunov approach against a known answer). Try the quadratic candidate V(x,y)=12x2+y2V(x,y) = \tfrac12 x^2 + y^2 (an asymmetric weighting, chosen to make the cross terms cancel — a symmetric x2+y2x^2+y^2 would leave a stray term, as we verify). Then VV is positive definite on all of R2\mathbb R^2 since it's a positive-definite quadratic form. Compute the orbital derivative:

V˙=xx˙+2yy˙=x(−x+2y2)+2y(−y−xy)=−x2+2xy2−2y2−2xy2.\dot V = x\dot x + 2y \dot y = x(-x + 2y^2) + 2y(-y - xy) = -x^2 + 2xy^2 - 2y^2 - 2xy^2.

The two 2xy22xy^2 terms cancel exactly, leaving

V˙(x,y)=−x2−2y2.\dot V(x,y) = -x^2 - 2y^2.

This is negative definite on all of R2\mathbb{R}^2: it vanishes only at (0,0)(0,0) and is strictly negative elsewhere, since it is −1-1 times a positive-definite quadratic form. By the theorem (part b) with U=R2U = \mathbb{R}^2, the origin is asymptotically stable — in fact, since UU can be taken arbitrarily large, this argument shows the basin of attraction is all of R2\mathbb{R}^2 (global asymptotic stability), a stronger conclusion than the local statement Hartman–Grobman alone would give. Notice the choice of coefficient 22 in front of y2y^2 was essential: with V=12(x2+y2)V = \tfrac12(x^2+y^2) we'd get V˙=−x2+xy2−y2−xy2=−x2−y2\dot V = -x^2+xy^2-y^2-xy^2 = -x^2-y^2 — actually that also happens to cancel in this particular example, but in general matching coefficients to the nonlinearity is what makes cross-terms vanish, and one should not expect the naive symmetric guess to always work.

Example 2 — a case where linearization is silent (a center-type nonlinear system). Consider the system

x˙=−y−x3,y˙=x−y3.\dot x = -y - x^3, \qquad \dot y = x - y^3.

The Jacobian at the origin is (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}, with eigenvalues ±i\pm i — purely imaginary. Hartman–Grobman does not apply (the equilibrium is non-hyperbolic), and the linearized system alone (a center) is inconclusive about the nonlinear system's actual behavior: nonlinear terms can turn a linear center into a stable or unstable focus. Try V(x,y)=12(x2+y2)V(x,y) = \tfrac12(x^2+y^2), positive definite on R2\mathbb R^2. Compute:

V˙=xx˙+yy˙=x(−y−x3)+y(x−y3)=−xy−x4+xy−y3⋅y.\dot V = x\dot x + y \dot y = x(-y - x^3) + y(x - y^3) = -xy - x^4 + xy - y^3\cdot y.

Carefully: yy˙=y(x−y3)=xy−y4y \dot y = y(x - y^3) = xy - y^4. So

V˙=(−xy−x4)+(xy−y4)=−x4−y4,\dot V = (-xy - x^4) + (xy - y^4) = -x^4 - y^4,

since the −xy-xy and +xy+xy cancel. This is negative definite: −x4−y4<0-x^4-y^4 < 0 whenever (x,y)≠(0,0)(x,y)\ne(0,0), and =0=0 only at the origin. By part (b) of the theorem, the origin is asymptotically stable, globally on R2\mathbb R^2 since V→∞V\to\infty as ∥(x,y)∥→∞\|(x,y)\|\to\infty and the sign conditions hold everywhere. This resolves, by direct algebra, a question that the linearization could not: the nonlinear cubic damping terms −x3,−y3-x^3, -y^3 tip the linear center into a genuine (nonlinear) stable focus, and we have proven this rigorously without ever solving the ODE.

Counterexampledropping strictness: semidefinite V˙\dot V need not give asymptotic stability

The asymptotic stability claim in part (b) of the theorem requires V˙\dot V to be strictly negative away from the origin, not merely ≤0\le 0. To see this hypothesis cannot be dropped, consider the planar linear system

x˙=0,y˙=−y.\dot x = 0, \qquad \dot y = -y.

The origin is an equilibrium. Take V(x,y)=12(x2+y2)V(x,y) = \tfrac12(x^2+y^2), positive definite everywhere. Compute

V˙=xx˙+yy˙=x⋅0+y(−y)=−y2.\dot V = x\dot x + y\dot y = x\cdot 0 + y(-y) = -y^2.

This is negative semidefinite on R2\mathbb{R}^2: it is ≤0\le 0 everywhere but vanishes whenever y=0y = 0, regardless of xx — not only at the origin. So VV satisfies the hypotheses of part (a) (stability) but not the strict hypothesis of part (b). Indeed, part (a) correctly certifies stability: solving directly, x(t)≡x(0)x(t) \equiv x(0) and y(t)=y(0)e−ty(t) = y(0)e^{-t}, so any trajectory starting near the origin stays near the origin for all t≥0t \ge 0 (in fact ∥(x(t),y(t))∥≤∥(x(0),y(0))∥\|(x(t),y(t))\| \le \|(x(0),y(0))\| for all tt). But the equilibrium is not asymptotically stable: the trajectory starting at (x0,0)(x_0, 0) with x0≠0x_0 \ne 0 satisfies x(t)≡x0↛0x(t) \equiv x_0 \not\to 0. The entire xx-axis consists of equilibria-adjacent orbits that never decay. This shows precisely why part (b) demands V˙<0\dot V < 0 off the origin: with only V˙≤0\dot V \le 0, the trajectory can get trapped forever on a level set (here, the line y=0y=0) without being squeezed further inward, so V(x(t))V(\mathbf x(t)) can plateau at a positive value instead of tending to 00. The proof of part (b) breaks exactly at the step invoking compactness of K={V≥L}∩B‾K = \{V \ge L\} \cap \overline B: on that set here, V˙\dot V is not bounded away from 00 (it equals 00 along the whole segment y=0y=0), so no η>0\eta > 0 can be extracted, and the contradiction argument collapses.

Scope, edge cases & common traps

SituationWhat to checkCommon pitfall
Only local positive definitenessV>0V>0 need only hold on a neighborhood UU, not globallyAssuming a locally valid VV automatically gives a global basin of attraction
V˙≤0\dot V \le 0 vs V˙<0\dot V < 0Semidefinite gives stability only; strict negativity needed for asymptotic stabilityCiting "LaSalle" conclusions without checking invariance conditions
Radial unboundednessNeeded for global asymptotic stability, not for local resultsClaiming global stability from a VV that is bounded (e.g. V=x2/(1+x2)V = x^2/(1+x^2))
Existence of VVThe theorem gives sufficient, not necessary conditions as statedConcluding instability just because you failed to find a Lyapunov function
Non-autonomous / time-varying systemsRequires a modified definition (V(t,x)V(t,\mathbf x), uniform bounds)Applying the autonomous theorem verbatim to time-dependent f\mathbf f
Multiple equilibriaUU must be a neighborhood on which VV is positive definite relative to that single equilibriumUsing one global VV naively when several equilibria compete for basins

The mathematician behind it

Origins · 1 minAleksandr Mikhailovich Lyapunov and how this was discovered
AM
Aleksandr Mikhailovich Lyapunov
1857–1918 · Imperial Russia; mechanics, stability theory, probability

Lyapunov trained under Chebyshev at St. Petersburg University and worked across mechanics, mathematical physics, and probability (he also proved a central limit theorem under general conditions). His 1892 doctoral dissertation, The General Problem of the Stability of Motion, introduced both the "first method" (stability from linearization, later formalized by Hartman and Grobman) and the "second" or "direct method" — the energy-function technique this article develops — allowing stability to be assessed without integrating the equations of motion. He held positions in Kharkov and later at the Russian Academy of Sciences in St. Petersburg. His work remained relatively unknown outside Russia until translations and the rise of control theory in the mid-20th century (notably through the work of engineers and mathematicians studying feedback stability) brought it to global prominence.

History
- 1857 — Lyapunov born in Yaroslavl, Russian Empire.
- 1892 — Doctoral thesis The General Problem of the Stability of Motion introduces the direct (second) method.
- 1918 — Lyapunov dies in Odesa, shortly after his wife's death, amid the upheavals following the Russian Revolution.
- 1949 — French translation of Lyapunov's thesis brings the method to wider Western attention.
- 1960s — Control theorists (including Kalman, LaSalle, and others) formalize Lyapunov methods for engineering feedback design, and prove converse theorems and invariance principles.
Do I need to find the unique Lyapunov function for a system?

No — Lyapunov functions are never unique, and the theorem gives sufficient, not necessary conditions in the form stated. Many different valid choices of VV can certify the same stability conclusion; conversely, failing to find one does not prove instability, it may just mean you haven't looked hard enough or in the right functional form (quadratic forms, weighted sums of squares, or physical energy are common starting guesses). Converse theorems guarantee existence of some strict Lyapunov function for any sufficiently smooth asymptotically stable equilibrium, but they do not tell you its formula; constructing one explicitly remains the practical challenge, often solved via the variable-gradient method or by exploiting known conserved/dissipated physical quantities.

How do I actually guess a Lyapunov function for a new system?

Three standard strategies: (1) try a quadratic form V=xTPxV = \mathbf x^T P \mathbf x with PP symmetric positive definite, tuning entries of PP to make cross-terms in V˙\dot V cancel against the nonlinearity, as in Example 1; (2) if the system arises from physics, use total mechanical or thermodynamic energy directly, since dissipative forces often make V˙≤0\dot V \le 0 automatically; (3) use the variable-gradient method, positing a general form for ∇V\nabla V, forcing V˙≤0\dot V \le 0 by design, then integrating to recover VV and checking it is a valid gradient (curl-free) and positive definite. Persistence and pattern-matching to the nonlinearity's structure (matching powers so terms cancel, as seen in both worked examples) is the main practical skill.

Can a Lyapunov function prove instability directly, without Chetaev's theorem?

Not usually with the same simple positive-definite VV: proving instability requires a different sign pattern, typically Chetaev's construction, where you find a region where V>0V>0 arbitrarily close to the equilibrium and V˙>0\dot V>0 throughout that region. This forces trajectories starting in that region to be pushed to ever-higher VV-values and hence away from the equilibrium, as the theorem's proof shows via a contradiction with boundedness of VV on a compact set. A single global positive-definite VV with V˙<0\dot V < 0 (a strict Lyapunov function) instead proves the opposite conclusion, asymptotic stability, so the two constructions are not interchangeable; they answer opposite questions and require different sign conditions.

Does the method work for discrete-time systems (maps) too?

Yes, with an analogous definition: for xk+1=F(xk)\mathbf x_{k+1} = \mathbf F(\mathbf x_k) with fixed point x∗\mathbf x^* , a Lyapunov function is positive definite near x∗\mathbf x^*, and stability/asymptotic stability follow from V(F(x))≤V(x)V(\mathbf F(\mathbf x)) \le V(\mathbf x) or V(F(x))<V(x)V(\mathbf F(\mathbf x)) < V(\mathbf x) respectively (a difference condition replacing the derivative condition). The proof structure is essentially identical, replacing the continuous monotonicity argument with a monotone decreasing sequence argument, and replacing "extension of solutions" with the trivial fact that iterating F\mathbf F is always defined. This discrete version underlies convergence proofs for many iterative numerical algorithms, including gradient descent, where VV is often the objective function itself or a distance-to-optimum function.

What is the relationship between Lyapunov functions and LaSalle's invariance principle?

LaSalle's invariance principle extends the method to handle semidefinite V˙\dot V (only V˙≤0\dot V \le 0), which is common in physical systems with partial damping. Instead of requiring V˙<0\dot V<0 everywhere off the origin, LaSalle shows trajectories converge to the largest invariant set contained in {x∈U:V˙(x)=0}\{\mathbf x \in U : \dot V(\mathbf x) = 0\}. If that invariant set is just {0}\{\mathbf 0\}, asymptotic stability follows even though V˙\dot V is only semidefinite — this resolves cases like a pendulum with friction only in one coordinate. It builds directly on the compactness and monotonicity arguments used in part (b) of the stability theorem above, and is typically covered alongside Limit Cycles and Closed Orbits since it is also the key tool for ruling out sustained oscillation.

Why does the theorem require f\mathbf f to be locally Lipschitz, and not just continuous?

Local Lipschitz continuity (or continuous differentiability, which implies it) guarantees existence and uniqueness of solutions to the initial value problem by the Picard–Lindelöf theorem, which the proof uses implicitly whenever it speaks of "the solution x(t)\mathbf x(t)" starting at a point. Without uniqueness, multiple trajectories could emanate from the same point, and the trapping argument (a single trajectory cannot cross a level curve) could fail because different branches of solution might behave inconsistently. Continuity alone (Peano's theorem) guarantees existence but not uniqueness, which is insufficient for the clean statement "the trajectory starting here stays here."

Is a Lyapunov function the same thing as an "energy function"?

Not always, though the terminology and intuition come from mechanical energy. In a genuinely conservative mechanical system, total energy is a valid Lyapunov function for stability (it's conserved, hence V˙≡0≤0\dot V \equiv 0 \le 0) but never a strict one, since V˙\dot V never becomes negative — this is exactly the situation explored further in Conservative Systems and the Pendulum, where centers are stable but not asymptotically stable. In dissipative or purely mathematical systems (chemical kinetics, population dynamics, algorithms), "Lyapunov function" is a purely formal scalar function satisfying the sign conditions, often with no physical energy interpretation at all — it's a mathematical device, not necessarily a physical quantity.

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