Transition to Fully or Partially Arrested State in Coupled Logistic Maps on a Ladder

Abstract

Layered structures are an object of interest for theoretical and experimental reasons. In this work, we study coupled map lattice on a ladder. The ladder consists of two one-dimensional chains coupled at every point. We study linearly and nonlinearly coupled logistic maps in this system and study transition to nonzero persistence, in particular. We coarse-grain the variable value by assigning spin [Formula: see text] ([Formula: see text]) to sites that have value greater (less) than the fixed point and compute the number of sites that have not changed their spin values at all even times till the given time [Formula: see text]. The fraction of such sites at a given time [Formula: see text] is known as persistence. In our system, we observe a power-law of persistence at the critical value of coupling. This transition is also accompanied by long-range antiferromagnetic ordering for nonlinear coupling and long-range ferromagnetic ordering for linear coupling. The number of domain walls decay as [Formula: see text] at the critical point in both cases. The persistence exponent is 0.375 for a nonlinear case with two layers which is an exponent for the voter model on the ladder as well as for the Ising model at zero temperature or voter model in 1D. For linear coupling, we obtain a smaller persistence exponent.