Schröder Malte, Timme Marc, Witthaut Dirk
Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany.
Forschungszentrum Jülich, Institute for Energy and Climate Research - Systems Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, Germany.
Chaos. 2017 Jul;27(7):073119. doi: 10.1063/1.4995963.
We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously prove that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids.
我们分析了用于测量复杂振荡器网络中同步和锁相的序参量的性质。首先，我们回顾了先前引入的网络序参量，并揭示了几个缺点：对于任意有限网络，所引入的序参量均无法捕捉从非相干到锁相再到完全同步的所有转变。然后，我们引入了一种替代的通用序参量，它能准确跟踪部分锁相和同步的程度，通过调整传统定义以考虑网络拓扑及其对振荡器相位相干性的影响。我们严格证明，在锁相状态下，该序参量随耦合强度严格单调增加，直接反映了网络的动态稳定性。此外，它在临界耦合强度处通过发散斜率表明完全锁相的开始。该序参量可能在不同类型同步可能出现的各种系统中找到应用，包括生物网络和电网。
