Transition to Frozen Antiferromagnetic Pattern in Delayed Logistic Map

Abstract

We study nonlinearly coupled logistic map with feedback in the limit of large delay time ([Formula: see text]). In the [Formula: see text]’s, the mapping of delayed dynamical systems to the spatially extended system was pursued in the context of optical systems by Arecchi and coworkers. We map the delayed system to a coupled map lattice of size [Formula: see text] with a sequential update and strictly one-sided coupling. Apart from aiding visualization, this formal equivalence helps us identify “phases” in a delayed system. One of the easily identified phases is that of frozen antiferromagnetic phase which is abundant in phase space. As coupling parameter ([Formula: see text]) increases, the system goes from periodic cascades along the coarse-grain region to chaos. We use the number of domain walls as an order parameter for locating this transition. The domain walls undergo random walk with drift and annihilate each other. At critical parameter value, we observe the power-law decay of this parameter. The decay exponents fall within the directed Ising universality class in general. We also find the escape time distribution, i.e. time required for reaching frozen antiferromagnetic regime. It diverges as it approaches the critical point. The divergence of average laminar length shows features indicating crisis induced intermittency.