Lyapunov exponents measure: rate of divergence of nearby trajectories in a dynamical system

The Lyapunov exponent (λ) measures the exponential rate at which trajectories diverge or converge in a dynamical system. This rate of separation is crucial for understanding the system's sensitivity to initial conditions and predicting long-term behavior.

In a dynamical system, two trajectories with an initial separation vector δ₀ diverge at a rate given by e^(λt)|δ₀| ≈ |δ(t)|. The exponent λ indicates how quickly trajectories move apart or come together over time t. Positive λ values suggest divergence, while negative λ values indicate convergence.

Understanding the Lyapunov exponent helps in predicting the stability of a system. Positive Lyapunov exponents are associated with chaotic systems where small differences in initial conditions can lead to vastly different outcomes. Negative exponents, on the other hand, indicate stable systems where trajectories converge to a fixed point or limit cycle.