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 $\delta =4.6692\mathrm{}\dots$. 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 ${\delta }^{\frac{1}{2}}$, to $\delta$ is given and numerically verified.