Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach
Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach
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DOI：
10.1137/21m1457321
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发表时间：
2021-11
期刊：
影响因子：
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通讯作者：
Tung T. Nguyen;Roberto C. Budzinski;Jacqueline Ðoàn;F. Pasini;J. Mináč;L. Muller
中科院分区：
文献类型：
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作者：
Tung T. Nguyen;Roberto C. Budzinski;Jacqueline Ðoàn;F. Pasini;J. Mináč;L. Muller
仓本网络构成了研究网络系统中集体行为的范例模型。尽管近年来取得了许多进展，但复杂网络上由耦合 Kuramoto 振荡器组成的系统的解决方案仍然存在许多悬而未决的问题。在本文中，我们描述了一种代数方法来找到此类系统的平衡点，而无需在无限系统大小的限制或连续体限制中使用标准近似。为此，我们使用最近引入的 Kuramoto 动力学代数方法，产生一个可显式求解的复值方程，捕获原始 Kuramoto 模型的动力学。使用这种新方法，我们获得了非线性原始 Kuramoto 系统和复值系统的平衡。然后，我们在仓本最初研究的完整图的情况下对所有平衡进行完全分类。最后，我们继续研究具有相位滞后的耦合振荡器网络、广义循环网络、多层网络以及随机网络中的平衡。
Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto oscillators on complex networks. In this article, we describe an algebraic method to find equilibrium points for this kind of system without using standard approximations in the limit of infinite system size or the continuum limit. To do this, we use a recently introduced algebraic approach to the Kuramoto dynamics, which results in an explicitly solvable complex-valued equation that captures the dynamics of the original Kuramoto model. Using this new approach, we obtain equilibria for both the nonlinear original Kuramoto and complex-valued systems. We then completely classify all equilibria in the case of complete graphs originally studied by Kuramoto. Finally, we go on to study equilibria in networks of coupled oscillators with phase lag, in generalized circulant networks, multi-layer networks, and also random networks.