- #1
mliuzzolino
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Homework Statement
Calculate the Lyapunov exponent for the linear map xn+1= rxn.
Homework Equations
λ = Lyapunov Exponent
λ = [itex]\lim_{n \rightarrow \infty} \begin{bmatrix}\dfrac{1}{n} \sum_{i = 0}^{n - 1} ln|f'(x_i)| \end{bmatrix} [/itex]
The Attempt at a Solution
f'(x) = r.
λ = [itex]\lim_{n \rightarrow \infty} \begin{bmatrix}\dfrac{1}{n} \sum_{i = 0}^{n - 1} ln|r| \end{bmatrix} [/itex]
= [itex]\lim_{n \rightarrow \infty} \begin{bmatrix}\dfrac{ln(r)}{n} \end{bmatrix} [/itex]
This is where I'm a bit lost. Is λ = ∞, or is λ = ln(r)?
In another example, with the tent function [itex] f(x) =\begin{cases}rx, \hspace{4mm} 0 \leq x \leq \dfrac{1}{2} \\ r - rx, \hspace{4mm} \dfrac{1}{2} \leq x \leq 1 \end{cases} [/itex]
λ = ln(r).
Is the Lyapunov exponent for both of these systems the same?