The Lyapunov exponent is a quantitative measure of how small differences in initial conditions grow over time in a dynamical system. It indicates whether a system is stable, predictable, or chaotic.
If the Lyapunov exponent is positive, it means that even tiny deviations between two starting points will exponentially diverge over time—this is a hallmark of chaotic behavior. A negative exponent implies convergence to stable behavior, and zero indicates neutral stability.
Key Features:
- Describes how fast nearby trajectories diverge or converge.
- Helps classify systems as chaotic, stable, or periodic.
- Often used in analyzing weather models, turbulent fluids, population dynamics, and planetary motion.
Examples:
- In a double pendulum, small changes in starting angle result in completely different motions—a system with a positive Lyapunov exponent.
- A simple harmonic oscillator has a zero exponent, showing predictable, repeating motion.
- In climate science, the Lyapunov exponent sets a theoretical limit on how far into the future weather can be accurately predicted.
The Lyapunov exponent is a powerful tool in understanding chaos theory, as it mathematically captures the unpredictability and complexity of many natural systems.