Universality of period doubling in coupled maps.

We study the critical behavior of period doubling in two coupled onedimensional maps with a single maximum of order z. In particurlar, the effect of the maximum-order z on the critical behavior associated with coupling is investigated by a renormalization method. There exist three fixed maps of the period-doubling renormalization operator. For a fixed map associated with the critical behavior at the zero-coupling critical point, relevant eigenvalues associated with coupling perturbations vary depending on the order z, whereas they are independent of z for the other two fixed maps. The renormalization results for the zero-coupling case are also confirmed by a direct numerical method. PACS numbers: 05.45.+b, 03.20.+i, 05.70.Jk Typeset using REVTEX 1 Universal scaling behavior of period doubling has been found in one-dimensional (1D) maps with a single maximum of order z (z > 1), xi+1 = f(xi) = 1−A |xi| , z > 1. (1) For all z > 1, the 1D map (1) exhibits successive period-doubling bifurcations as the nonlinearity parameter A is increased. The period-doubling bifurcation points A = An(z) (n = 0, 1, 2, . . .), at which the nth period doubling bifurcation occurs, converge to the accumulation point A(z) on the A axis. The scaling behavior near the critical point A depends on the maximum-order z, i.e., the parameter and orbital scaling factors, δ and α, vary depending on z [1–4]. Therefore the order z determines universality classes. Here we study the critical behavior of period doubling in a map T consisting of two identical 1D maps coupled symmetrically: T : 