Lyapunov Exponent

Quantifying the divergence or convergence of trajectories in chaotic systems

Results

Lyapunov Exponent λ: --
System State: --
λ = lim→∞ (1/t) · ln(|δx(t)/δx(0)|)

What is the Lyapunov Exponent?

The Lyapunov Exponent is a key metric for quantifying the sensitivity of trajectories to initial conditions in dynamical systems. It describes the average exponential rate at which nearby trajectories separate in phase space over time.

Mathematical Formula

λ = limt→∞ (1/t) · ln(|δx(t)/δx(0)|)
  • λ: Lyapunov exponent, representing the average rate of trajectory separation
  • δx(t): Separation distance between two trajectories at time t
  • δx(0): Initial separation distance between two trajectories

Interpretation

λ > 0: Chaotic System

Trajectories diverge exponentially, showing extreme sensitivity to initial conditions. Even tiny differences in initial conditions lead to rapidly separating trajectories, exhibiting the 'butterfly effect'.

Typical systems: Logistic map (r > 3.57), Lorenz system, Rossler system

λ ≤ 0: Stable System

Trajectories converge or exhibit periodic motion. Nearby trajectories do not diverge, and the system has predictability.

Typical systems: Damped harmonic oscillator, maps converging to fixed points, periodic orbits

Applications

Calculation Method

For discrete maps x(n+1) = f(x(n)), the Lyapunov exponent can be approximated by:

λ ≈ (1/N) · Σ ln(|f'(xi)|)

Where N is the number of iterations and f'(x) is the derivative of the map function.