Effects of inertia on the asynchronous state of a disordered Kuramoto model

TL;DR

This work extends the iterative mean-field (IMF) framework to the Kuramoto model with inertia to quantify self-consistent fluctuation statistics in the asynchronous state. It demonstrates that IMF reproduces full network spectra with high accuracy while identifying a nonmonotonic effect: at an intermediate oscillator mass $m$, temporal correlations are minimized and spectral broadening is maximized, accompanied by a peak in the network's Kolmogorov-Sinai entropy. The analysis links spectral properties to the Lyapunov spectrum, revealing a transition from monotonic to damped-oscillatory correlations and emergent chaotic activity for a substantial fraction of oscillators. These findings have potential implications for how inertia-driven temporal structure in disordered oscillator networks influences information processing and stimulus responsiveness, and suggest avenues for analytical development via fluctuation-dissipation relations in driven, non-equilibrium settings.

Abstract

We investigate the role of inertia in the asynchronous state of a disordered Kuramoto model. We extend an iterative simulation scheme to the case of the Kuramoto model with inertia in order to determine the self-consistent fluctuation statistics, specifically, the power spectra of network noise and single oscillators. Comparison with network simulations demonstrates that this works well whenever the system is in an asynchronous state. We also find an unexpected effect when varying the degree of inertia: the correlation time of the oscillators becomes minimal at an intermediate mass of the oscillators; correspondingly, the power spectra appear flatter and thus more similar to white noise around the same value of mass. We also find a similar effect for the Lyapunov spectra of the oscillators when the mass is varied.

Figures (6)

• Figure 1: Defining the asynchronous regime. The order parameter $r$ vs $K$ for a system with a random network ($k=1$) or a random frequency ($\sigma_\omega=0.25$) distribution for $m=0, 1, 10, 100$. Other parameters: $N=10^4, T=2000, t_d=200, R=1$. For all masses shown and $K<1$, the network is in the asynchronous regime with $r\ll 1$.
• Figure 2: IMF method works for the Kuramoto model with inertia. A comparison between the IMF and ND methods is presented for masses $m=0, 1, 10, 100$. (a) Single-oscillator spectra for the selected oscillator with $\omega_3=0.0165$ and (b) mean power spectra are shown. The IMF results are represented by solid lines, while the ND results are shown with black dashed lines. Fixed parameters: $\sigma_\omega=0.25, k=1, N=10^4, T=10^5, t_d=1000, R=1$.
• Figure 3: Maximized spectral flatness and minimized temporal correlation for an intermediate oscillator mass. (a) Mean power spectrum for $m=0, 4, 7, 100$: solid lines (IMF method), black dashed lines (ND method). (b) Autocorrelation function for $m=0, 4, 7, 100$. Dashed lines, matching the colors of the corresponding solid lines, represent $|C(\tau)|$ and highlight its use in (d). (c) Spectrum width, Eq. (\ref{['eq:delta_omega']}) at 80% maximum of $S_z(\omega)$ vs mass with a maximum at $m=7$, marked by a black dashed line. (d) Correlation time vs mass: $\tau_C$ according to Eq. (\ref{['eq:tauc']}) has a minimum at $m= 4$. Dashed red line represents $\hat{\tau}_C=S_z(0)/2$, according to Eq. (\ref{['eq:tauc_absval']}), with a minimum at $m=7$, marked by a black dashed line. Fixed parameters: $k=1, \sigma_\omega=0, N=10^4, I=20, R=1, T=10^4, t_d=1000$.
• Figure 4: Dependence on the coupling coefficient can be captured by scaling arguments. (a) Spectrum width at 80% maximum of $S_z(\omega)$ for $k=0.5, 1, 2$. Solid lines represent $\Delta\omega$, and dashed lines represent the rescaled $\overline{\Delta\omega}$ using Eq. (\ref{['eq:rescale_k']}). (b) Correlation time as a function of mass for $k=0.5,1,2$. Dashed lines show the rescaled $\tau_c$ according to Eq. (\ref{['eq:rescale_k']}). Fixed parameters: $\sigma_\omega=0, N=10^4, R=1, T=10^4, t_d=1000$.
• Figure 5: Maximizing flatness and minimizing correlation time at intermediate mass is robust against moderate disorder in natural frequencies. For $\sigma_{\omega} = 0, 0.15, 0.25$, the lines are black, red, and green, respectively, while the insets show $\sigma_{\omega} = 1$ (purple) and 2 (blue). (a) Spectral width $\Delta\omega$ at 80% maximum of $S_z(\omega)$ vs mass $m$. (b) Correlation time $\tau_{C}$ vs mass $m$. Dashed lines represent $S_z(0)/2$, with colors matching the solid lines for each corresponding $\sigma_\omega$. Fixed parameters: $k=1, N=10^4, R=1, T=10^4, t_d=1000$.