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We study one-dimensional coupled logistic maps with delayed linear ornonlinear nearest-neighbor coupling. Taking the nonzero fixed point of the mapx* as reference, we coarse-grain the system by identifying values above x* withthe spin-up state and values below x* with the spin-down state. We definepersistent sites at time T as the sites which did not change their spin stateeven once for all even times till time T. A clear transition from asymptoticzero persistence to non-zero persistence is seen in the parameter space. Thetransition is accompanied by the emergence of antiferromagnetic, orferromagnetic order in space. We observe antiferromagnetic order for nonlinearcoupling and even delay, or linear coupling and odd delay. We observeferromagnetic order for linear coupling and even delay, or nonlinear couplingand odd delay. For symmetric coupling, we observe a power-law decay ofpersistence. The persistence exponent is close to 0.375 for the transition toantiferromagnetic order and close to 0.285 for ferromagnetic order. The numberof 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 thecritical point, in the presence of asymmetry.

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