Universality relations

By analyzing the functional equations governing N replication for real nonlinear maps on the (unit) interval (of order z), we have derived the asymptotic (N-->∞) behaviors of the scaling constants alpha and delta, alpha(z)~(2z)(N-1)/(z-1), delta/alphaz=(z-1)/(z-1/2). In addition, we have determined the limiting forms of the universal functions. The predictions are in excellent agreement with (separately compiled) numerical data over a wide range of z values and cycle structures.