Feigenbaum Function

Consider an arbitrary 1-D Map

$\begin{displaymath} x_{n+1}=F(x_n) \end{displaymath}$

(1)

at the onset of Chaos. After a suitable rescaling, the Feigenbaum function

$\begin{displaymath} g(x)=\lim_{n\to\infty} {1\over F^{(2^n)}(0)} F^{(2^n)}(xF^{(2^n)}(0)) \end{displaymath}$

(2)

is obtained. This function satisfies

$\begin{displaymath} g(g(x))=-{1\over\alpha} g(\alpha x), \end{displaymath}$

(3)

with $\alpha=2.50290\ldots$ , a quantity related to the Feigenbaum Constant.

References

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.