Dynamical properties of the synchronization transition

TL;DR

This paper investigates the dynamical properties of the synchronization transition in one-dimensional coupled map lattices, revealing different universality classes and critical behaviors depending on the map type and parameters.

Contribution

It identifies the universality classes of the synchronization transition for various maps and configurations, including DP and BKPZ classes, and explores the effects of asymmetry.

Findings

Synchronization transition belongs to DP class for Bernoulli map configurations.

Spreading exponents differ from DP but their sum matches DP exponents in certain cases.

For the tent map, the transition belongs to the BKPZ universality class.

Abstract

We study the dynamics of the synchronization transition (ST) of one-dimensional coupled map lattices. For the Bernoulli map it was recently found by Ahlers and Pikovsky (Phys. Rev. Lett. {\bf 88}, 254101 (2002)) that the ST belongs to the directed percolation (DP) universality class. Spreading dynamics confirms such an identification, only for a certain class of synchronized configurations. For homogeneous configurations spreading exponents $\eta$ and $\delta$ are different than DP exponents but their sum equals to the corresponding sum of DP exponents. Such a relation is typical to some models with infinitely many absorbing states. Moreover, we calculate the spreading exponents for the tent map for which the ST belongs to the bounded Kardar-Parisi-Zheng (BKPZ) universality class. Our estimation of spreading exponents are consistent with the hyperscaling relation. Finally, we examine the asymmetric tent map. For small asymmetry the ST remains of the BKPZ type. However, for large asymmetry a different critical behaviour appears with exponents being relatively close to the ones of DP.