Understanding synchronization induced by “common noise”

Noise-induced synchronization refers to the phenomenon where two uncoupled, independent nonlinear oscillators can achieve synchronization through a “common” noisy forcing. Here, “common” means identical. However, “common noise” is a construct which does not exist in practice. Noise by nature is unique and two noise signals cannot be exactly the same. How to justify and understand this central concept in noiseinduced synchronization? What is the relation between noise-induced synchronization and the usual chaotic synchronization? Here we argue and demonstrate that noise-induced synchronization is closely related to generalized synchronization as characterized by the emergence of a functional relation between distinct dynamical systems through mutual interaction. We show that the same mechanism applies to the phenomenon of noise-induced (or chaos-induced) phase synchronization.  2005 Published by Elsevier B.V. PACS: 05.45.Xt; 05.45.-a In the past twenty years, chaotic synchronization, including complete synchronization, generalized synchronization, phase synchronization and lag synchronization, has been intensively investigated. Among them, generalized synchronization discovered by Rulkov et al. [1–4] is an interesting phenomenon. It refers to the existence of some functional relation between coupled but nonidentical chaotic oscillators. To detect generalized synchronization, Abarbanel et al. proposed the interesting idea of considering an auxiliary response system and examining the conditional stability of typical trajectories in the driven system [2]. In particular, suppose one wishes to determine whether there is a generalized synchronization between two uni-directionally coupled oscillators, say A and B , the drive and driven system, respectively. One can imagine an auxiliary response system B ′, which is identical to B and subject to the same driving signal, and asks whether there is synchronization between B and B ′. Abarbanel et al. showed that an * Corresponding author. E-mail address: [email protected] (S. Guan). 0375-9601/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physleta.2005.11.067 affirmative answer would imply a generalized synchronization between A and B . Note that the pioneering work on chaotic synchronization by Pecora and Carroll [5] focused on synchronization between identical subsystems under a common forcing. The auxiliary response-system approach is equivalent to treating B ′ as the replica of subsystem B in a single dynamical system that comprises A and B . Whether subsystems can be synchronized is determined by the sign of the conditional Lyapunov exponents evaluated for typical trajectories in any of the subsystems under the forcing. Another interesting synchronization phenomenon is the chaotic phase synchronization [12]. It occurs in certain chaotic systems where suitable phases can be defined. In phase synchronization, the phases between two chaotic oscillators can be locked while their amplitudes remain chaotic and uncorrelated. Compared with generalized synchronization, phase synchronization is a weaker form since there is no functional relation between the amplitudes of the two coupled oscillators. Parallel to the chaotic synchronization mentioned above, the phenomenon of noise-induced synchronization, i.e., synchronization among uncoupled nonlinear oscillators under “comS. Guan et al. / Physics Letters A 353 (2006) 30–33 31 mon (or identical) noise”, has also been intensively studied. The first work along this line was carried out by Maritan and Banavar over a decade ago [6]. This seemingly counter-intuitive phenomenon has since attracted a continuous interest [7–9], partly because it is another powerful demonstration of “noiseinduced order” as a result of the interplay between nonlinear dynamics and stochastic processes, in addition to stochastic resonance [10] and coherence resonance [11]. More recently, the phenomenon has been extended [9] to chaotic phase synchronization [12] by Zhou et al. who demonstrated numerically and experimentally that phase coherence between uncoupled chaotic oscillators in a statistical sense can be established by “common noise”. They further proposed that the phenomenon is due to the existence of distinct phase-space regions where infinitesimal vectors experience expansion and contraction, respectively, as a result of the common noisy forcing. Very recently, the noise effect on the fully synchronous regime of globally coupled chaotic systems has been investigated [13]. So far, chaotic synchronization and synchronization induced by “common noise” have appeared to be two almost independent research domains. Whether or not these two types of synchronization in chaotic systems can be understood in an unified framework is of particular interest. In this Letter, we argue that noise-induced synchronization can be understood naturally as a manifestation of generalized synchronization. Therefore, these two types synchronization phenomena can be unified conceptually. To state our result, we use the representative setting where two nonlinear oscillators are driven by a common random or chaotic forcing,