Ulam Map

$\begin{figure}\begin{center}\BoxedEPSF{UlamMap.epsf}\end{center}\end{figure}$

$\begin{displaymath} f(x)=1-2x^2 \end{displaymath}$

for $x\in[-1,1]$ . Fixed points occur at

, 1/2, and order 2 fixed points at $x=(1\pm\sqrt{5}\,)/4$ . The Invariant Density of the map is

$\begin{displaymath} \rho(y)={1\over\pi\sqrt{1-y^2}}. \end{displaymath}$

References

Beck, C. and Schlögl, F. Thermodynamics of Chaotic Systems: An Introduction. Cambridge, England: Cambridge University Press, p. 194, 1995.