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 $s_{\text{max}}$ of synchronization occurred when $s>s_{\text{max}}$. 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 $s_{\text{max}}$. 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.

Figure 1

Schematic representation of three nonidentical Kuramoto oscillators coupled in a chain (top) and all-to-all (bottom) network.

Figure 2

The order parameter $r$ as the function of the disparity $\sigma$ (left) and coupling strength $s$ (right) in the chain model. Vertical dashed lines indicate two phase transitions corresponding to the boundaries of the partial synchronization ${\sigma }^{+}$ and ${\sigma }^{-}$ (left), $s_{\text{min}}$ and $s_{\text{max}}$ (right) (see Appendix pp1). The natural frequencies are the same as in Fig. 10 of Appendix pp1. The free coupling parameters are $s=\left|\mathrm{\Delta }k\right|=1$ (left panel), $\sigma =0.8$ (right panel).

Figure 3

Observed evolution of the solar data in the northern (blue) and southern (red) hemispheres. Left: 2-year averaged proxies of the toroidal (top) and poloidal (bottom) solar field component; top right: the phase evolution of the toroidal component; bottom right: the phase evolution of the poloidal component for MWO/WSO (solid) and Pulkovo (dashed) data. Desynchronization event of 1960 in the southern hemisphere exists only on the MWO/WSO data (solid red line in the right bottom panel).

Figure 4

Reconstructed asymmetry of the order parameter vs coupling strength $\widehat{s}$ of the reconstruction for the chain model with constant (right) and variable (left) natural frequencies. The power-law decay (left) is a general-case invariant, the exponential decay appears only for the synchronized oscillators with constant natural frequencies (right). Reconstruction is performed for ${\widehat{\omega }}_2$=0, $\widehat{\sigma }=0.8$. Right: The slope of the exponent depends on the coupling disparity $\sigma =0.8$ (blue), $\sigma =0.5$ (red) for fixed natural frequencies ${\omega }_1=1.07$, ${\omega }_2=0.29$, ${\omega }_3=0.39$ and coupling strength $s=1$. Left: Partial synchronization (red) and desynchronization (green) correspond to the scaling exponent $g=1$. Synchronization (blue) correspond to the scaling exponent $g=2$. Natural frequencies are ${\omega }_1=1.07+0.2(1+sin0.1t)$, ${\omega }_2=0.29$, ${\omega }_3=0.39$, coupling corresponds to the zone of synchronization $\sigma =0.8$, $s=2$ (blue), partial synchronization $\sigma =0.8$, $s=0.5$ (red), and desynchronization $\sigma =0.8$, $s=0.2$ (green).

Figure 5

Reconstructed asymmetry $\widehat{A}$ as a function of the original coupling strength $s$ for constant (left) and evolving (right) natural frequencies. Vertical dashed lines indicate boundaries between partial synchronization and desynchronization (red) or synchronization (blue). Parameters of the reconstruction are $\widehat{s}=2$, $\widehat{\sigma }=\sigma =0.8$. Natural frequencies are the same as on Fig. 4. Maximum of the reconstructed asymmetry is attained at the boundary of synchronization (vertical blue line).

Figure 6

Self consistent reconstruction with $\widehat{\sigma }=0.5$ in the general model with constant natural frequencies (${\omega }_1=0.84$, ${\omega }_2=0.45$, ${\omega }_3=0.47$) and coupling (${\kappa }_{12}={\kappa }_{21}=0.25$, ${\kappa }_{13}={\kappa }_{31}=-0.15$, ${\kappa }_{23}={\kappa }_{32}=-0.05$). The model presents a partial synchronization: oscillators 1 and 2 are synchronized, the third oscillator drifts causing desynchronization events. Reconstruction is performed for three pairs of oscillators: (1,2), (1,3) and (2,3) (from left to right). Top row: $\mathrm{\Delta }\widehat{\omega }/\widehat{s}$ reconstructed for weak $\widehat{s}=2$ and strong $\widehat{s}=128$ coupling. Desynchronization events for $\widehat{s}=128$ correspond to extremes of $\mathrm{\Delta }\widehat{\omega }/\widehat{s}$ equal to $\pm (1-|\widehat{\sigma }\left|\right)=\pm 0.5$. Middle row: self-consistency of the reconstruction, the original phase difference $2\alpha ={\theta }_i-{\theta }_j$ (yellow) taken modulus $2\pi$ is close to the phase difference reconstructed for $\widehat{s}=2$ (dashed blue) and $\widehat{s}=128$ (solid blue). The phase differences deviate near the desynchronization event. Bottom row: the reconstructed asymmetry $\widehat{A}$ vs coupling strength $\widehat{s}$ for the whole time interval (blue) and for three consecutive 15-year time intervals shown by the same colors on the top panels. The asymmetry curve is higher when the time interval is closer to the desynchronization event.

Figure 7

The self-consistent reconstruction of solar data. Reconstruction is performed independently in the northern (left) and southern (right) hemispheres. Top row: normalized difference between natural frequencies $\mathrm{\Delta }\widehat{\omega }/\widehat{s}$ reconstructed with $\widehat{s}=128$ and $\widehat{\sigma }=0.5$ from the phase difference ${\alpha }_m\left(t\right)$ (blue) and ${\alpha }_p\left(t\right)$ (red). The desynchronization event in the southern hemisphere (right panel, blue line) corresponds to the $\mathrm{\Delta }\widehat{\omega }/\widehat{s}=1-\widehat{\sigma }=0.5$. Middle row: comparison of the simulated phase difference (dashed) with the original one (solid) taken in $[0,2\pi ]$ for ${\alpha }_m\left(t\right)$ (blue) and ${\alpha }_p\left(t\right)$ (red). Apart from the start point and desynchronization event in the southern hemisphere, the simulated and the original phase differences coincide proving the self-consistence. Bottom row: Asymmetry $\widehat{A}$ derived from ${\alpha }_m\left(t\right)$ (blue full circle) and ${\alpha }_p\left(t\right)$ (red open circles) on the whole time interval of observations (blue), and 15-year time intervals $T_1=\left[1925\text{–}1940\right]$ (green) and $T_2=\left[1940\text{–}1955\right]$ (yellow). In the southern hemisphere (right) asymmetry curve is higher when the time interval is closer to the desynchronization event.

Figure 8

Influence of the finite interval of observations to the variability of the scaling exponent $g$. Asymmetry $A$ for each $s$ is reconstructed from the 110 consecutive time series of phase differences $\alpha \left(t\right)$ where $t\in [T_n,T_n+T],n=1,\cdots ,110$ simulated by the chain model with coupling strength $s$ and disparity $\sigma =0.8$. Reconstruction is performed with $\widehat{\sigma }=0.2$. Natural frequencies are the same as in Fig. 4, left. Vertical lines show theoretical boundaries of the coupling strength between desynchronization and partial synchronization (red) and between partial synchronization and synchronization (blue). The difference between minimal and maximal boundaries (dashed and solid line, respectively) appears due to variable natural frequencies. Near the boundary of the synchronization both exponents $g=1$ and $g=2$ are present.

Figure 9

Evolution of the phase differences (top row) and the order parameter (bottom row) in a chain of three Kuramoto oscillators. From left to right: phase-locking when coupling coefficients have different signs, partial synchronization, desynchronization. Colors of the top row indicate phase difference between oscillators ${\theta }_1-{\theta }_2$ (blue), ${\theta }_2-{\theta }_3$ (red), ${\theta }_1-{\theta }_3$ (yellow). Simulation is performed for ${\omega }_1=1.07$, ${\omega }_2=0.29$, ${\omega }_3=0.39$, $\sigma =0.8$, and $s=\left|\mathrm{\Delta }k\right|$ taking value $s=1$ (left), $s=0.8$ (middle), and $s=0.2$ (right).

Figure 10

Phase diagram of synchronization for a chain of three Kuramoto oscillators. Coupling domains, where three oscillators are phase-locked (red and blue zones), none of oscillators are phase locked (yellow rectangle), two of three oscillators are phase-locked (white zones outside the central rectangle). Synchronization corresponds to the stable phase difference ${\theta }_1-{\theta }_3=2\alpha$ taken from $-\pi$ to $\pi$ where $2\left|\alpha \right|<\pi /2$ (red zones) and $2\left|\alpha \right|>\pi /2$ (blue zones). Natural frequencies are ${\omega }_1=1.07$, ${\omega }_2=0.29$, ${\omega }_3=0.39$.