Elaeva M, Blanter E, Shnirman M, Shapoval A
Department of Higher Mathematics, HSE University, Moscow 109028, Russia.
Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, Moscow 117997, Russia.
Phys Rev E. 2023 Jun;107(6-1):064201. doi: 10.1103/PhysRevE.107.064201.
Synchronization and desynchronization of coupled oscillators appear to be the key property of many physical systems. It is believed that to predict a synchronization (or desynchronization) event, the knowledge on the exact structure of the oscillatory network is required. However, natural sciences often deal with observations where the coupling coefficients are not available. In the present paper we suggest a way to characterize synchronization of two oscillators without the reconstruction of coupling. Our method is based on the Kuramoto chain with three oscillators with constant but nonidentical coupling. We characterize coupling in this chain by two parameters: the coupling strength s and disparity σ. We give an analytical expression of the boundary s_{max} of synchronization occurred when s>s_{max}. We propose asymmetry A of the generalized order parameter induced by the coupling disparity as a new characteristic of the synchronization between two oscillators. For the chain model with three oscillators we present the self-consistent inverse problem. We explore scaling properties of the asymmetry A constructed for the inverse problem. We demonstrate that the asymmetry A in the chain model is maximal when the coupling strength in the model reaches the boundary of synchronization s_{max}. We suggest that the asymmetry A may be derived from the phase difference of any two oscillators if one pretends that they are edges of an abstract chain with three oscillators. Performing such a derivation with the general three-oscillator Kuramoto model, we show that the crossover from the chain to general network of oscillators keeps the interrelation between the asymmetry A and synchronization. Finally, we apply the asymmetry A to describe synchronization of the solar magnetic field proxies and discuss its potential use for the forecast of solar cycle anomalies.
耦合振子的同步与去同步似乎是许多物理系统的关键特性。据信，要预测同步（或去同步）事件，需要了解振荡网络的确切结构。然而，自然科学常常处理耦合系数不可得的观测情况。在本文中，我们提出了一种无需重构耦合来表征两个振子同步的方法。我们的方法基于具有三个振子且耦合恒定但不相同的Kuramoto链。我们用两个参数来表征该链中的耦合：耦合强度(s)和差异(\sigma)。我们给出了(s > s_{max})时发生同步的边界(s_{max})的解析表达式。我们提出由耦合差异引起的广义序参量的不对称性(A)作为两个振子之间同步的一个新特征。对于具有三个振子的链模型，我们提出了自洽逆问题。我们探索为逆问题构建的不对称性(A)的标度性质。我们证明，当模型中的耦合强度达到同步边界(s_{max})时，链模型中的不对称性(A)最大。我们认为，如果将任意两个振子当作具有三个振子的抽象链的边，那么不对称性(A)可以从它们的相位差推导得出。对一般的三振子Kuramoto模型进行这样的推导，我们表明从链到一般振子网络的转变保持了不对称性(A)与同步之间的相互关系。最后，我们应用不对称性(A)来描述太阳磁场代理的同步，并讨论其在预测太阳周期异常方面的潜在用途。
