Stepwise transition to higher degrees of coherence in a random network of phase oscillators

We consider a model system of phase oscillators which are connected in a random network. The network favors the connection of oscillators with close values of phases. We extend the order parameter used in the study of synchronization of phase oscillators and define generalized order parameters for the model system. We investigate the equilibrium properties of the model and reveal a phenomenon of stepwise transitions to higher degrees of coherence as the system goes through a series of second-order phase transitions for the order parameters. We also discuss a possible realization of the model in real physical systems. Copyright c © EPLA, 2012 Introduction. – The phenomenon of collective synchronization was first recognized to be ubiquitous in nature and studied mathematically by Wiener [1]. Since then the synchronization in nonlinear systems of coupled phase oscillators as a phenomenon that plays a key role in many processes taking place in nature and technology was systematically studied by Winfree [2] and Kuramoto [3]. Recent reviews on the development can be found in [4–6]. Among the systems where the phenomenon was revealed are biological clocks [2], chemical reactions [3], coupled map lattices [7–10], coupled random frequency oscillators [11,12], cardiorespiratory coupled system [13], etc. The direction was recently connected with the research on complex networks [14–16] and processes taking places in them [17,18]. Synchronization in networks such as the Erdős-Rényi random graph [19], small-world networks [20,21], scale-free networks [22–27] and the synchronization of chaotic systems on time-varying networks [28] are phenomena that were considered due to their importance in physical processes taking place in real-world networks. Influence of various noises and delay on synchronization of the stochastic Kuramoto model had also been studied ([29,30] and references therein). Another class of models consists of nonlinear systems in the presence of thermal (a)E-mail: [email protected] (b)E-mail: [email protected] noise. In this case the analog of synchronization phenomenon is a transition to coherence when particles move in unison being in a thermodynamically stable state. Among such nonlinear systems are coupled classical phase oscillators with inertia [31–33]. They have attracted increasing attention and have been the subject of extensive studies [34]. The system, called Hamiltonian mean-field (HMF) model, demonstrates a transition from an incoherent to a coherent state [31] and was shown to exhibit many interesting behaviors, including an oscillation of macroscopic variables [35], diffusive anomalies [36–38] and emergence of quasistationary states [39] and their loss in the presence of stochastic dynamics in the long time limit [40]. It was also shown that the model (also called as globally coupled rotors) possesses an inequivalence between microcanonical and canonical ensembles [41,42]. A logical step in the research area related to HMF model would be to study its properties on various complex networks, starting with one particular network setup. Thus in this paper we introduce a model of phase oscillators with inertia, a non-globally-coupled HMF model, connected in a network with a random number of links [43]. The phase oscillators are located at the nodes of the network that has its links favor connection of oscillators with close values of phases. We are interested in the equilibrium properties of the system. As the phase oscillators begin to interact, they tend to become coherent as the coupling strength