Lyapunov exponents

Lyapunov exponents measure how sensitive a system is to initial conditions, indicating whether small differences grow or shrink over time. If the exponent is positive, tiny variations quickly amplify, making the system unpredictable and chaotic. If it's negative, differences fade, and the system stabilizes. Zero indicates neutral behavior. Think of a weather pattern: small changes can lead to vastly different outcomes if the Lyapunov exponent is positive, reflecting chaos, whereas a stable system resists change. These exponents quantify the rate of divergence or convergence of nearby trajectories in a system's state space, helping us understand predictability and stability.