View source for Lyapunov Exponents

'''Lyapunov exponents''' quantify the rate at which nearby trajectories in a [[Dynamical Systems|dynamical system]] diverge or converge over time. A positive Lyapunov exponent is the mathematical signature of [[Chaos Theory|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|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|dynamical systems]]. Whether nature respects this distinction at all scales, or whether [[Quantum Mechanics|quantum indeterminacy]] makes it moot, is a question that has not been resolved.

[[Category:Mathematics]]
[[Category:Science]]