Driven synchronization in random networks of oscillators

Abstract

Synchronization is a universal phenomenon found in many non-equilibrium systems. Much recent interest in this area has overlapped with the study of complex networks, where a major focus is determining how a system's connectivity patterns affect the types of behavior that it can produce. Thus far, modeling efforts have focused on the tendency of networks of oscillators to mutually synchronize themselves, with less emphasis on the effects of external driving. In this work we discuss the interplay between mutual and driven synchronization in networks of phase oscillators of the Kuramoto type, and explore how the structure and emergence of such states depends on the underlying network topology for simple random networks with a given degree distribution. We find a variety of interesting dynamical behaviors, including bifurcations and bistability patterns that are qualitatively different for heterogeneous and homogeneous networks, and which are separated by a Takens-Bogdanov-Cusp singularity in the parameter region where the coupling strength between oscillators is weak. Our analysis is connected to the underlying dynamics of oscillator clusters for important states and transitions.Collective behavior of complex networks is a very active field of theoretical and practical research.In particular, models of oscillator networks have drawn much attention due to their numerous applications across diverse fields, with a particular emphasis on synchronization phenomena. Here we study the dynamics of coupled oscillators, subject to periodic forcing, on random networks with different degrees of connectivity, and uncover many dynamical behaviors as a few parameters are varied. We find that the unfolding of synchronized states, and the possibility of bistability among them, differs for networks depending on how heterogeneous the degree of local connectivity is. This is explained through a combination of analytic and numerical results.

Periodic coupling suppresses synchronization in coupled phase oscillators

“…As the dynamical basis for the function and operation of many realistic systems, synchronization behaviors have been extensively studied by researchers from different fields over the past decades [9][10][11]. In exploring oscillator synchronization, one of the focusing issues is about the robustness of synchronization to external perturbations, including environmental noises and periodic drivings [12][13][14][15][16][17][18][19][20][21]. Due to the nonlinear feature of the system dynamics, intriguing phenomena could be generated in the presence of external perturbations, e.g.…”

“…Due to the nonlinear feature of the system dynamics, intriguing phenomena could be generated in the presence of external perturbations, e.g. the enhancement of synchronization by random noise [19,20], the control of synchronization by periodic signals [14,15], the manipulation of synchronization by periodic drivings [16], and the inducing of synchronization by configuring initial conditions [21]. It is noted that in these studies, the perturbation signals are normally added on the system variables.…”

Hybrid dynamics in delay-coupled swarms with mothership networks

“…For example, for bimodal networks, high and low-degree nodes can split into a state that is a composite of ring and rotating motions -mixing the behaviors in Fig.1 (b) and (d), respectively. For example, we find that each degree class's order-parameter (e.g., its centroid) has dynamics analogous to the distinct states [24]. Therefore, we call this a hybrid state, which is shown in Fig.1(c).…”

“…In addition to generalizing the patterns from homogeneous networks to heterogeneous networks with a specified degree distribution, we show that heterogeneity can produce novel hybrid motions, where different parts of the network have different collective dynamics depending on the degree of local connectivity. The production of new states that are mixes of distinct behaviors for homogeneous networks is an interesting feature of nonlinear processes occurring on heterogeneous networks and is seen in other systems, e.g., coupled oscillators [24], though other mechanisms for swarm splitting are known [26,27]. Hybrid behaviors are practically interesting in this context as well because they offer the possibility for synthetic swarms to perform multiple tasks simultaneously.…”

“…In general, the effects of complex network structure on swarm behavior are much less explored, particularly with time-delayed interactions, even though topology is known to strongly influence many processes and produce interesting new dynamics [1,[22][23][24]. Here, we focus on delay-coupled swarms interacting through heterogeneous networks that have a relatively small fraction of highly connected nodes, or "motherships."…”