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 α and δ, α(z)∼(2z${)}^{\left(N\mathrm{-}1\right)/\left(z\mathrm{-}1\right)}$, δ/${\alpha }^z$=(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.

• Received 18 November 1985

DOI:https://doi.org/10.1103/PhysRevA.33.3292