Abstract

Anti-phase synchronization is the spontaneous formation of 2 clusters of oscillators synchronized between themselves within a cluster but opposite in phase with the other cluster. Neuronal networks in human and animal brains, ecological networks, climactic networks, and lasers are all systems that exhibit anti-phase synchronization although the phenomenon is encountered less frequently than the celebrated in-phase synchronization. We show that this disparity in occurrence is due to fundamental limits on the size of networks that can sustain anti-phase synchronization. We study the influence of network structure and coupling conditions on anti-phase synchronization in networks composed of coupled Stuart-Landau oscillators. The dependence of probability of anti-phase synchronization on connectivity of the network, strength of interaction over distance, and symmetry of the network is illustrated. Regardless of favourable network conditions, we show that anti-phase synchronization is limited to small networks, typically smaller than 20 nodes.

Highlights

  • Coupled oscillators and phenomena associated with them have attracted considerable attention recently due to the behavioural richness they exhibit and due to their generality in capturing the essential dynamics of multiple real-world systems (e.g. Lasers[1], Josephson junctions[2], neuronal networks[3], ecological systems[4], among others[5])

  • This paper presents the first comprehensive analysis of the effects of network topology on AP synchronization of repulsively coupled oscillator networks

  • The generalization from phase oscillators to Stuart-Landau oscillators allows an expansion of the regime of probable AP synchronization from 4 to 20+ with higher sizes possible under favourable network conditions

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Summary

Introduction

Coupled oscillators and phenomena associated with them have attracted considerable attention recently due to the behavioural richness they exhibit and due to their generality in capturing the essential dynamics of multiple real-world systems (e.g. Lasers[1], Josephson junctions[2], neuronal networks[3], ecological systems[4], among others[5]). Some form of attractive coupling is essential for the oscillators to ‘pull’ others in the network to a common phase. From the simple case of two oscillators, it can be deduced that some form of repulsive coupling would be required in order to achieve anti-phase synchronization[19]. This has been shown in real networks.

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