Math — Chaos & the Logistic Map
Period doubling and the Lyapunov exponent.
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x_{n+1} = r\,x_n(1 - x_n)
\text{Onset of chaos at } r \approx 3.5699
\lambda = \lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1} \ln\left| f'(x_i) \right|
\text{Chaotic when } \lambda > 0