PHYSICAL REVIEW E 94, 052216 (2016) Generalized synchrony of coupled stochastic processes with multiplicative noise Haider Hasan Jafri, 1 R. K. Brojen Singh, 2 and Ramakrishna Ramaswamy 2, 3 1 Department of Physics, Aligarh Muslim University, Aligarh 202 002, India 2 School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi 110 067, India 3 School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India (Received 6 August 2016; published 18 November 2016) We study the effect of multiplicative noise in dynamical flows arising from the coupling of stochastic processes with intrinsic noise. Situations wherein such systems arise naturally are in chemical or biological oscillators that are coupled to each other in a drive-response configuration. Above a coupling threshold we find that there is a strong correlation between the drive and the response: This is a stochastic analog of the phenomenon of generalised synchronization. Since the dynamical fluctuations are large when there is intrinsic noise, it is necessary to employ measures that are sensitive to correlations between the variables of drive and the response, the permutation entropy, or the mutual information in order to detect the transition to generalized synchrony in such systems. DOI: 10.1103/PhysRevE.94.052216 I. INTRODUCTION The synchronization of rhythms is an important and extensively studied phenomenon that is ubiquitous in nature [1] and can arise in linear as well as nonlinear systems, for weak as well as strong coupling, and when the dynamics is periodic as well as when it is chaotic or otherwise complex [2–4]. This makes synchrony one of the most widely observed instances of emergent and cooperative behavior. Technological applications of synchronization are numerous, ranging from signal processing and communication protocols [5,6] to information transfer in multiprocessors [7]. Over the past decades the notion of synchrony has expanded to include the situation when distinct signals become strongly correlated without becoming identical. This is the case of the so-called generalized synchronization (GS), which was first described in the context of a skew-product (or one-way coupled) system [8]: A response system is coupled to the output of the drive, and, for sufficiently strong coupling, the output of the response is uniquely defined by the drive. Consider, for instance, the following system: ˙ y = F y (P y ,y), (1) ˙ x = F x (P x ,x) + ε(y − x), (2) where the subscripts y and x denote the drive and the response respectively, the dynamical variables of either system are denoted by y(t ) ∈ R n and x(t ) ∈ R m , F y and F x specify the respective n- and m-dimensional flows, and P y,x are the sets of parameters in the two systems. The coupling considered above is linear (diffusive), but the discussion below applies to other forms of coupling as well. GS is said to occur in the system when there is a unique functional relationship, x = [y], (3) between the drive and response variables. Depending on whether  is differentiable, the generalized synchronization is termed as strong or weak [9,10], and numerous studies have examined the nature and characteristics of GS in a variety of systems, including those with nonlinear coupling. In a number of situations of practical importance, however, the microscopic dynamics is governed by a set of coupled stochastic processes [11–14], and it is therefore of interest to examine how these ideas of synchrony can be extended to dynamical systems in which fluctuations cannot be suppressed. The present work addresses this issue in the context of coupled microscopic chemical reactions, wherein the dynamics is subject to both intrinsic and extrinsic noise and the variables therefore can undergo large fluctuations. Such a situation is common, for instance, in biological reactions at the cellular and subcellular levels. The methods of analysis that are applicable to deterministic dynamical systems [1] cannot be easily adapted in a straightforward manner to stochastic systems. In the present work, we consider two sets of coupled chemical reactions that interact with each other. Our interest is in the manner in which the two subsystems become correlated, in a manner that is analogous to synchrony. We also approximate the master equation that describes this system by the Langevin equation [15] and study the dynamics of this system in which the noise that appears is multiplicative. One consequence is that the noise cannot be “switched off” except in the thermodynamic limit, and thus we seek measures that can assess the degree of synchrony accurately in systems that are dominated by noise. The basic framework of our study is outlined in Sec. II, and application is made to model coupled stochastic processes. The examples we consider are chosen to correspond to well-known dynamical systems such as the Brusselator and the Lorenz system, primarily so the dynamics is well understood in the thermodynamic limit. We calculate correlations and information-theoretic measures to study the transition. The transition to generalized synchrony in case of chemical oscillators is described in Sec. III. We apply the above-discussed order parameters to these coupled systems in order to detect the transition in finite systems. We conclude with a discussion and summary in Sec. IV. II. STOCHASTIC DRIVE-RESPONSE SYSTEMS WITH MULTIPLICATIVE NOISE The effect of external additive noise has been studied in detail in the context of generalized synchrony [16,17]. Our interest here is on the nature of intrinsic noise, such as might arise in systems far from the thermodynamic limit. 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