Scaling pattern of period doubling in four dimensions

Bicritical behavior of period doublings in unidirectionally coupled maps

We study the scaling behavior of period doublings in two unidirectionally coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization-group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization-group analysis agree well with those obtained by a direct numerical method.