Self-consistent autocorrelation of a disordered Kuramoto model in the asynchronous state

Abstract

The Kuramoto model has provided deep insights into synchronization phenomena and remains an important paradigm to study the dynamics of coupled oscillators. Yet, despite its success, the asynchronous regime in the Kuramoto model has received limited attention. Here, we adapt and enhance the mean-field approach originally proposed by Stiller and Radons [Phys. Rev. E 58 (1998)] to study the asynchronous state in the Kuramoto model with a finite number of oscillators and with disordered connectivity. By employing an iterative stochastic mean field (IMF) approximation, the complex N-oscillator system can effectively be reduced to a one-dimensional dynamics, both for homogeneous and heterogeneous networks. This method allows us to investigate the power spectra of individual oscillators as well as of the multiplicative "network noise" in the Kuramoto model in the asynchronous regime. By taking into account the finite system size and disorder in the connectivity, our findings become relevant for the dynamics of coupled oscillators that appear in the context of biological or technical systems.

Figures (6)

• Figure 1: Synchronous and asynchronous regimes of the model. Variation of the order parameter $r$ as a function of coupling strength $K$ (solid lines). There are three scenarios: disorder in connectivity (blue), disorder in frequencies (red), and concurrent disorder in both connectivity and frequencies (green). This paper exclusively focuses on the asynchronous regime (shaded areas). Parameters used: $N=10^4, T=10^3, R=20$.
• Figure 2: Convergence of the iterative mean field (IMF) approach for the self-consistent noise spectrum. Noise spectra $S_\zeta$ labeled by the iterative step; we start with a Lorentzian spectrum i.e. $2/(\pi^2 + 20 \omega^2)$ (labeled as 0) and quickly approach a self-consistent shape (spectra for 15 and 30 steps agree within line thickness). Parameters: $k=1$, $K=0$, $\sigma_\omega=0$, $T=10^3$, $dt=10^{-3}$, with $N_{trial}=10^4$ number of realizations.
• Figure 3: Single-oscillator dynamics with (a) and without (b) connectivity disorder. Power spectra for three distinct oscillators with $\omega_6=-2.6, \omega_{32}=0.02, \omega_{79}=2.3$ and with Gaussian disorder in the connectivity ($k=1$ in (a)) or without disorder in the connectivity ($k=0$ in (b)). Red solid lines (IMF) and black dashed lines (ND) show matching results for both network types. The analysis is conducted with a frequency variability of $\sigma_\omega=1$, coupling strength $|K|=1$, in a system of size $N=10^4$ and over a time window $T=10^4$. All results were obtained from a single realization of frequency and connectivity disorder: $R=1, M=1$.
• Figure 4: Dependence of single-oscillator power spectrum on network disorder with (a) and without (b) frequency variability. Power spectrum of 32nd oscillator $S_{x_\ell}$ for increasing values of the Gaussian network disorder $k$ as indicated (a) with natural frequency $\omega_{32}=0.02$ (same as the middle curve in Fig. \ref{['fig:fig3']}) and with a variability in natural frequencies of the network oscillators of $\sigma_\omega=1$. In (b), the same spectra but without frequency variability (all oscillators $\omega_\ell=0$). Dashed lines represent the full network dynamics (ND) for a single realization ($R=1$) with $N=10000$. Solid lines depict the IMF approach after iterations within $I=20$ in (a) and $I=50$ in (b), both with $M=1, N=10^4, T=10^5$. All data correspond to $D=0$, except the cyan line (IMF) and its corresponding black dashed line (ND) for $D=0.5$ in both panels and cyan circles (Eq. \ref{['eq:d']}) in (b). In (a), $|K|=1$; in (b), $|K|=0$.
• Figure 5: Average power spectra $S_z=\langle S_{x_\ell} \rangle_\ell$ with (a) and without (b) connectivity disorder. Spectra averaged over all oscillators in the network with Gaussian connectivity ($k=1$ in (a)) or without ($k=0$ in (b)); in both panels we indicate increasing disorder in frequencies, $\sigma_\omega$. Dashed lines indicate ND with $N=10^3$. We used $R=10^3$ realizations with time window $T=10^5$ (a) and $R=200$ realizations with time window $T=10^4$ (b). Solid lines illustrate the IMF method with $I=20$ iterations, employing $M=100$ and $T=10^5$. We set the parameter $K$ equal to the value of $|\sigma_\omega|$.