Analytic Solutions of the Cvitanović–Feigenbaum and Feigenbaum–Kadanoff–Shenker Equations

The Cvitanovic–Feigenbaum (CF) equation arising in the universal scaling theory of iterated maps of the real line has strong links with the classical Schroder and Abel functional equations. This link is exploited to obtain information about the analytic solutions, and specifically the singular solution, of the CF equation, providing an alternative description of the latter to that of Eckmann and Wittwer. We obtain an accurate numerical approximation to this singular solution, using special techniques to handle the divergent series. This accuracy is a substantial improvement on previous estimates of the solution, and of the associated asymptotic feigenvalues α and δ. The solutions of the Feigenbaum–Kadanoff–Shenker equation for universal scaling in circle maps are shown to yield to the same analysis, producing accurate numerical values for the associated α and δ.