Transition to Coarse-Grained Order in Coupled Logistic Maps: Effect of Delay and Asymmetry

TL;DR

This paper investigates how delay and asymmetry influence the transition to coarse-grained order in coupled logistic maps, revealing distinct magnetic-like ordering and persistence behaviors at critical points.

Contribution

It introduces a coarse-graining approach to coupled logistic maps and characterizes the transition to ordered states influenced by delay and asymmetry.

Findings

Transition from zero to non-zero persistence observed in parameter space.

Antiferromagnetic order for nonlinear coupling with even delay or linear coupling with odd delay.

Persistence decay follows a stretched exponential at criticality.

Abstract

We study one-dimensional coupled logistic maps with delayed linear or nonlinear nearest-neighbor coupling. Taking the nonzero fixed point of the map x* as reference, we coarse-grain the system by identifying values above x* with the spin-up state and values below x* with the spin-down state. We define persistent sites at time T as the sites which did not change their spin state even once for all even times till time T. A clear transition from asymptotic zero persistence to non-zero persistence is seen in the parameter space. The transition is accompanied by the emergence of antiferromagnetic, or ferromagnetic order in space. We observe antiferromagnetic order for nonlinear coupling and even delay, or linear coupling and odd delay. We observe ferromagnetic order for linear coupling and even delay, or nonlinear coupling and odd delay. For symmetric coupling, we observe a power-law decay of persistence. The persistence exponent is close to 0.375 for the transition to antiferromagnetic order and close to 0.285 for ferromagnetic order. The number of domain walls decays with an exponent close to 0.5 in all cases as expected. The persistence decays as a stretched exponential and not a power-law at the critical point, in the presence of asymmetry.