Lyapunov Exponents

Lyapunov exponents quantify the rate at which nearby trajectories in a dynamical system diverge or converge over time. A positive Lyapunov exponent is the mathematical signature of chaos: it means that small differences in initial conditions grow exponentially, guaranteeing that finite measurement precision translates into a finite prediction horizon.

The largest Lyapunov exponent λ of a system measures how quickly two trajectories starting at nearby points separate: d(t) ≈ d(0)eˡᵗ. When λ > 0, the system is chaotic and long-run prediction is impossible for any observer with finite precision — including, as Laplace's Demon implies, any physical observer that is itself part of the universe.

The Lyapunov spectrum (all exponents together) describes the system's full geometry: positive exponents correspond to expanding directions in state space, negative exponents to contracting directions. The sum of all Lyapunov exponents equals the average rate at which the system's phase-space volume changes — in dissipative systems, this is negative, reflecting the collapse of trajectories onto attractors.

That a number — a single real value — can separate the predictable from the unpredictable is one of the stranger gifts of the mathematical theory of dynamical systems. Whether nature respects this distinction at all scales, or whether quantum indeterminacy makes it moot, is a question that has not been resolved.