Lyapunov exponent

A Lyapunov exponent quantifies the average rate of separation of infinitesimally close trajectories in a dynamical system. It is the mathematical signature of predictability: a negative exponent means nearby trajectories converge, indicating stability; a positive exponent means they diverge exponentially, indicating chaos. The magnitude of the largest Lyapunov exponent determines the predictability horizon — the time beyond which forecast error exceeds a given threshold.

In a system with n degrees of freedom, there are n Lyapunov exponents forming the Lyapunov spectrum, which characterizes the geometry of the system's attractor. Strange attractors in chaotic systems are distinguished by having at least one positive Lyapunov exponent while remaining globally bounded. The Kaplan-Yorke conjecture relates the Lyapunov spectrum to the fractal dimension of the attractor, linking dynamical instability to geometric complexity.