Abstract

It is of interest in many natural and artificial networks to find simpler systems to understand spatio-temporal dynamics. For this, we consider an extended fragmentary networks by considering a system of phase oscillators with time delay and degree heterogeneity on Cayley trees to address it. The coupled oscillators on Cayley trees exhibit various dynamical regimes such as clusters synchronization, desynchronization, and phase-flip between synchronized clusters. The dynamical regimes and transitions of oscillators on the Cayley tree are found to be similar to the star network’s oscillator dynamics. We confirm the Cayley tree and star network’s functional equivalence by finding the collective frequencies of phase oscillators in both systems. Analytical estimates of linearly stable common frequencies of oscillators on star networks agree with oscillator’s numerical solutions on Cayley trees. Our results hold for identical and non-identical oscillators, and they provide a theoretical framework to understand the collective dynamics of delayed phase oscillators in extended systems.

Graphic abstract

中文翻译：

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星网络粗化Cayley树上延迟相位振荡器的动力学。

摘要

在许多自然和人工网络中，寻找更简单的系统来理解时空动力学是很重要的。为此，我们通过考虑在Cayley树上具有时间延迟和度异质性的相位振荡器系统来解决扩展的碎片网络。Cayley树上的耦合振荡器表现出各种动态机制，例如集群同步，去同步和同步集群之间的相位翻转。发现Cayley树上的振荡器的动力学状态和跃迁与星形网络的振荡器动力学相似。通过找到两个系统中相位振荡器的集体频率，我们确定了Cayley树和星形网络的功能等效性。星形网络上振荡器线性稳定公共频率的分析估计与Cayley树上振荡器的数值解一致。我们的结果适用于相同和不同的振荡器，它们为理解扩展系统中延迟相位振荡器的集体动力学提供了理论框架。

图形摘要