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 δ=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 δ¹², to δ is given and numerically verified.