Collision of Feigenbaum cascades

The existence in dynamical systems of chaotic bands delimited on both sides by period-doubling cascades is a general two-parameter phenomenon. Here we show evidence that, whenever these chaotic regions disappear, the bifurcation convergence rate undergoes a slowing down and asymptotically approaches the square root of the universal number =4.6692. A simple renormalization-group analysis is performed to explain this critical behavior and its scaling properties. In particular a theoretical universal function describing the evolution of the convergence rate from 12, to is given and numerically verified.