Universal Bicritical Behavior of Period Doublings in Unidirectionally Coupled Oscillators

We study the bicritical behavior of period doublings in unidirectionally coupled oscillators to confirm the universality of the bicriticality in an abstract system of two unidirectionally coupled one-dimensional (1D) maps. A transition to hyperchaos occurs (i.e., a hyperchaotic attractor with two positive Lyapunov exponents appears) when crossing a bicritical point where two Feigenbaum critical lines of a period-doubling transition to chaos in the two subsystems meet. Using both a “residue-matching” renormalization group method and a direct numerical method, we make an analysis of the scaling behavior near the bicritical point. It is thus found that the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the usual Feigenbaum critical state. Note that the bicritical scaling behavior is the same as that in the unidirectionally coupled 1D maps. We thus suppose that bicriticality may be observed generally in real systems, consisting of period-doubling subsystems with a unidirectional coupling.