Asymmetry in the Kuramoto model with nonidentical coupling

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 sigma. We give an analytical expression of the boundary smax of synchronization occurred when s > smax. 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 smax. 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.

Automated summary

In this paper, a method is proposed to characterize the synchronization of two oscillators without reconstructing the coupling structure. The method is based on the Kuramoto chain with three oscillators with constant but nonidentical coupling. The asymmetry A induced by the coupling disparity is proposed as a new characteristic of synchronization between the two oscillators. The asymmetry A is applied to describe the synchronization of solar magnetic field proxies and its potential use for the forecast of solar cycle anomalies is discussed.