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Lyapunov exponents quantify the average exponential rate at which nearby trajectories diverge or converge in each direction. An n-dimensional system has n Lyapunov exponents, ordered λ₁ ≥ λ₂ ≥ ... ≥ λₙ. A positive largest Lyapunov exponent (λ₁ > 0) is the definitive signature of chaos: it means nearby trajectories diverge exponentially on average. The full spectrum of exponents characterizes the attractor's geometry — stretching, neutral, and contracting directions — and determines its fractal dimension.
Sensitive dependence on initial conditions is the defining feature of chaos, but "nearby trajectories diverge" is a qualitative statement. Lyapunov exponents make it quantitative: they tell you exactly how fast divergence occurs, in which directions, and by how much. They are the numbers that separate chaos from everything else and that determine the practical prediction horizon of a chaotic system.
Consider two trajectories starting at x₀ and x₀ + δ₀, where δ₀ is tiny. After time t, the separation is approximately |δ(t)| ≈ |δ₀|eλ₁t, where λ₁ is the largest Lyapunov exponent. If λ₁ > 0, the separation grows exponentially — this is chaos. If λ₁ < 0, perturbations decay and the system is stable. If λ₁ = 0, perturbations neither grow nor decay — you're on the boundary, typically seeing periodic or quasiperiodic motion. The exponent λ₁ is computed as a time average: λ₁ = lim_{t→∞} (1/t) ln|δ(t)/δ₀|, where the perturbation is continuously renormalized to prevent it from growing so large that the linearization breaks down.
An n-dimensional system has n Lyapunov exponents, one for each independent direction in the tangent space. They measure the average exponential rates of stretching and compression along the principal axes of an infinitesimal ellipsoid of initial conditions as it evolves. The full Lyapunov spectrum {λ₁ ≥ λ₂ ≥ ... ≥ λₙ} characterizes the attractor completely. A fixed point: all λᵢ < 0. A stable limit cycle: λ₁ = 0 (the flow direction), all others negative. A quasiperiodic torus: λ₁ = λ₂ = 0, others negative. Chaos: at least one λᵢ > 0. For a continuous flow, one exponent is always exactly zero (perturbations along the trajectory direction neither grow nor shrink), so the minimum Lyapunov spectrum for a chaotic flow is (+, 0, -) in 3D.
The sum of all Lyapunov exponents equals the average rate of phase space volume contraction (or expansion). For dissipative systems, this sum is negative — volumes shrink. For Hamiltonian systems, it's zero — volumes are conserved (Liouville's theorem). The positive exponents create stretching, the negative ones create compression, and the net effect determines the attractor's dimension. The Kaplan-Yorke conjecture relates the Lyapunov spectrum to the fractal dimension: D_KY = j + (λ₁ + ... + λⱼ)/|λⱼ₊₁|, where j is the largest integer such that λ₁ + ... + λⱼ ≥ 0. For the Lorenz system, this gives D_KY ≈ 2 + 0.9/14.6 ≈ 2.06 — a fractal object slightly thicker than a surface.