Transition to almost periodic patterns in circle map with delay: persistence as order parameter.

We study delayed circle map. A previously proposed analogy between delayed map and spatiotemporal system [F. T. Arecchi et al., Phys. Rev. A 45, R4225 (1992)] is employed to study this system. In the phase diagram, we observe laminar phase, travelling defect phase, and standing defect phase. We push this analogy further; and in this pseudo-spatiotemporal system, we investigate "phases" and define "order parameter" to describe transition between phases. We find that persistence (defined as the probability that a given site has not deviated even once from its coarse grained initial state upto time t) works as an "order parameter" for the transition from a travelling wave phase to standing wave phase. We observe an interesting finite "size" scaling and off-critical scaling above the critical point.