Lyapunov Exponent
This applet calculates the Lyapunov exponent h(x) for the logistic map g(x) = ax(1-x).
g (x) = ax(1-x) x1 = g(x0)
g'(x) = a - 2ax x2 = g(x1)
|g'(x)| = absolute-value(g'(x)) x3 = g(x2), etc.
h (x) = [ |g'(x1)| * ... * |g'(xn)| ] ^ (1/n)
= [ln(|g'(x1)|) + ... + ln(|g'(xn)|)] / n
g'(x) = a - 2ax x2 = g(x1)
|g'(x)| = absolute-value(g'(x)) x3 = g(x2), etc.
h (x) = [ |g'(x1)| * ... * |g'(xn)| ] ^ (1/n)
= [ln(|g'(x1)|) + ... + ln(|g'(xn)|)] / n
Note that if any g'(x) = 0, then the Lyapunov exponent is 0, and the result is meaningless.
Lyapunov Exponent Graph
This applet calculates the Lyapunov-exponent graph h(x) for the logistic map g(x) = ax(1-x).
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