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Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows: λ = lim t → ∞ lim | δ Z 0 | → 0 1 t ln | δ Z (t) | | δ Z 0 | {\displaystyle \lambda =\lim _{t\to \infty }\lim _{|\delta \mathbf {Z} _{0}|\to 0}{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}}

The limit | δ Z 0 | → 0 {\displaystyle |\delta \mathbf {Z} _{0}|\to 0} ensures the validity of the linear approximationat any time.

For discrete time system (maps or fixed point iterations) x n + 1 = f (x n) {\displaystyle x_{n+1}=f(x_{n})}, for an orbit starting with x 0 {\displaystyle x_{0}} this translates into: λ (x 0) = lim n → ∞ 1 n ∑ i = 0 n − 1 ln | f ′ (x i) | {\displaystyle \lambda (x_{0})=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\ln |f'(x_{i})|}