Learning objectives
By the end of this lesson, you should be able to:
- Interpret the largest Lyapunov exponent.
- Relate its sign to contraction or divergence.
- Connect exponential separation to a predictability horizon.
Prerequisites: Sensitivity to Initial Conditions
EXPERIMENT Interactive dynamics laboratory
Perturb → Evolve → CompareChange one parameter at a time. Predict the qualitative behavior before running or moving the control.
Quantifying divergence
The Lyapunov exponent measures the average exponential growth or decay of an infinitesimal perturbation. Informally, \(\delta(t)\approx\delta_0e^{\lambda t}\).
Interpreting the sign
A negative exponent indicates average contraction, an exponent near zero indicates neutral behavior, and a positive largest exponent is a standard signature of sensitive dependence in many systems.
A finite-time estimate
Numerical estimates fluctuate with trajectory length and state-space sampling, so convergence and uncertainty should be examined rather than treating a single number as absolute proof.
Key takeaways
What should stay with you
- λ<0 usually indicates contraction toward stable behavior.
- λ≈0 is associated with neutral directions or marginal behavior.
- λ>0 indicates average exponential growth of infinitesimal perturbations.
Knowledge check
What does a positive largest Lyapunov exponent indicate?
A positive largest Lyapunov exponent is a standard indicator of sensitive dependence and chaos in many systems.