Stochastic Theory of Synchronization Transitions in Extended Systems

Abstract

We propose a general Langevin equation describing the universal properties of synchronization transitions in extended systems. By means of theoretical arguments and numerical simulations we show that the proposed equation exhibits, depending on parameter values, either: i) a continuous transition in the bounded Kardar-Parisi-Zhang universality class, with a zero largest Lyapunov exponent at the critical point; ii) a continuous transition in the directed percolation class, with a negative Lyapunov exponent, or iii) a discontinuous transition (that is argued to be possibly just a transient effect). Cases ii) and iii) exhibit coexistence of synchronized and unsynchronized phases in a broad (fuzzy) region. This phenomenology reproduces almost all the reported features of synchronization transitions of coupled map lattices and other models, providing a unified theoretical framework for the analysis of synchronization transitions in extended systems. [7]. In particular, coupled map lattices (CMLs) [8], initially introduced as simple models of spatio-temporal chaos [11], have received a great deal of attention, as models of synchronization in spatially extended systems (another possibility is to analyze discrete cellular automata [9,10]). In this context, it was observed that distant patches of a given CML can oscillate in phase [11], a phenomenon related to the Kardar-Parisi-Zhang (KPZ) equation [12] describing the roughening of growing interfaces [13]. Afterwards, it was also realized that when two different replicas of a same CML are locally coupled, they become synchronized if the coupling strength is large enough [6,9,14]. Also globally coupled CML can achieve mutual synchronization [15] with interesting implications in neuro-science [16]. Last but not least, different replicas of a CML can be synchronized if they are coupled to a sufficiently large common external random noise, even if they are not directly coupled to each other [17,18,19].In all these examples, there is a transition from a chaotic or unsynchronized phase in which perturbations grow, and two replicas evolve independently, to a synchronized phase in which memory of the initial difference is asymptotically lost and replicas synchronize. This synchronization transition (ST) resembles very much other non-equilibrium critical phenomena, particularly transitions into absorbing states [20], and the determination of its universal properties has become the subject of many recent studies [9,19,22,23] as well as the main motivation of this Letter. Within this context, a major contribution was made by Pikovsky and Kurths (PK) who proposed a stochastic model for the dynamics of perturbations in locally coupled CML, in which, under very general conditions, the difference-between-two-replicas field can be described by the so-called multiplicative noise (MN) Langevin equation [22]:where φ(x, t) is the difference field (or "synchronization error"), a, b, σ 2 , and D > 0 are parameters, and η a delta correlated Gaussian white noise with zero average. It is worth ...

Critical behavior of a bounded Kardar-Parisi-Zhang equation

“…single-site) like behavior. However, a similar problem recently studied in the context of synchronization has revealed inconsistencies probably due to insufficient statistics [2,13]. In this subsection improved simulation results are provided upon revisiting the analysis reported in [8] for larger system sizes and longer sampling times.…”

“…This is particularly interesting in the context of synchronization [2,13]. This possibility will also be discussed in the paper.…”

“…While the MN1 class has been extensively studied, especially after its connections with nonequilibrium wetting [10][11][12] and with the problem of synchronization in extended systems were established [2,13], the MN2 class remains poorly studied. Furthermore, recent numerical analyses have revealed that the preliminary critical exponent values reported in [14] might be far from their true asymptotic values.…”

Some recent developments in models with absorbing states

“…However, for a certain class of dynamical systems ST has a different critical behaviour, namely it belongs to the DP universality class [25]. Despite some attempts, theoretical understanding of the mechanism that changes the critical behaviour is still missing [26][27][28]. An interesting related problem is a nature of a multicritical point that possibly joins the critical lines of DP and BKPZ type.…”

“…But there are some other arguments suggesting that the critical behaviour at ST typically is different and belongs to the bounded Kardar-Parisi-Zhang universality class [24]. A possible crossover between these two universality classes has recently been intensively studied [26][27][28]. …”