Desynchronization and on-off intermittency in complex networks

Most existing works on synchronization in complex networks concern the synchronizability and its dependence on network topology. While there has also been work on desynchronization wave patterns in networks that are regular or nearly regular, little is known about the dynamics of synchronous patterns in complex networks. We find that, when a complex network becomes desynchronized, a giant cluster of a vast majority of synchronous nodes can form. A striking phenomenon is that the size of the giant cluster can exhibit an extreme type of intermittent behavior: on-off intermittency. We articulate a physical theory to explain this behavior. This phenomenon may have implications to the evolution of real-world systems. Copyright c © EPLA, 2009 When a regular oscillator network becomes desynchronized, either wave patterns are generated [1–8] or stationary synchronization clusters are formed [9]. Consider, for instance, a one-dimensional ring of identical oscillators, each coupled with its nearest neighbors. For nonlinear oscillator dynamics, it is typical that synchronization occurs only when the coupling parameter, say ε, lies in a finite range: ε1 ε ε2 [10]. When ε is decreased through ε1, long-wave bifurcation occurs in the sense that the desynchronization wave patterns generated for ε ε1 have wavelengths of the order of the system size [1,2]. As ε is increased through ε2, wave patterns with wavelength much smaller than the system size are generated, henceforth the term short-wave bifurcation [2–4]. For ε ε2, even when the actual coupling strengths are randomized (in this case ε is a nominal coupling parameter), robust regular wave patterns can arise [7]. Considering that in the synchronization regime, noise and small system mismatch can induce desynchronization bursts, the occurrence of stable wave patterns in the desynchronization parameter regime is quite remarkable. A more recent work reveals that regular wave patterns can persist in small-world (a)E-mail: [email protected] networks that deviate slightly in topology from a regular network, but the patterns can be destroyed if there are too many random links in the network [8]. Previous works have also revealed the situation where, when desynchronization occurs, a regular network breaks into a finite number of synchronous clusters. That is, oscillators in each cluster are synchronized but the synchronized dynamics differ from cluster to cluster. Such clusters are usually stationary due to the regularity of the underlying network. While wave patterns associated with desynchronization in regular networks, e.g., lattice and globally coupled systems, have been relatively well understood, little has been done to address the problem in complex networks, for example random [11] and scale-free networks [12]. Intuitively, since the underlying network does not possess a regular topology, no wave patterns can be formed. However, we find that, in the desynchronization regime, synchronization clusters can occur commonly. The remarkable phenomenon that we wish to report in this paper is that the evolution of the clusters can exhibit an extreme form of intermittency. In particular, there can be a giant cluster containing a substantial fraction of synchronized nodes most of the time, but there can