Lyapunov Exponents and Chaos

Key idea

Sensitivity to initial conditions.

In depth

The Lyapunov exponent λ = lim (1/n) Σ ln|f′(xᵢ)| measures how fast nearby orbits separate. Positive λ means sensitive dependence on initial conditions: chaos. Long-term prediction becomes impossible.

Worked example

The logistic map xₙ₊₁ = 4xₙ(1 − xₙ) has λ = ln 2 > 0, so is chaotic.

Real-world uses

• Measuring unpredictability in weather models and the stability of orbits

Common mistake

Confusing chaos with randomness; it is deterministic.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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