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Mean-field approach to finite-size fluctuations in the Kuramoto-Sakaguchi model

Oleh E. Omel'chenko, Georg A. Gottwald

Abstract

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for solutions of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state. We corroborate our results with numerical simulations of the full Kuramoto-Sakaguchi model. The proposed methodology is sufficiently general such that it can be applied to other interacting particle systems.

Mean-field approach to finite-size fluctuations in the Kuramoto-Sakaguchi model

Abstract

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for solutions of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state. We corroborate our results with numerical simulations of the full Kuramoto-Sakaguchi model. The proposed methodology is sufficiently general such that it can be applied to other interacting particle systems.

Paper Structure

This paper contains 2 sections, 1 theorem, 92 equations, 8 figures.

Table of Contents

  1. Appendix: Derivations of equations and additional results
  2. Acknowledgements

Key Result

Proposition 1

If $|\Delta|\ne|H|$, the solution of the Adler equation (Eq:Adler) is of the form where $\xi$ is defined in Eq. Def:xi, and In addition, in the degenerate case $|\Delta|=|H|$, the solution of the Adler equation (Eq:Adler) is given by where $z_* = i \Delta / H$. (For brevity of notations, we do not specify that $\xi$, $\chi_1(t)$, $\chi_2(t)$ and $z_*$ depend on $\Delta$ and $H$.)

Figures (8)

  • Figure 1: Covariance $R(\tau)$ of fluctuations $\zeta(t)$ of the complex order parameter for the KS model \ref{['Eq:KS:original']}. (Top row) Partially synchronized state at $K = 6$ and $\lambda = \pi/4$, with theoretical prediction \ref{['Formula:R:ps']}. (Bottom row) Completely incoherent state at $K = 0.5$ and $\lambda = \pi/4$, with theoretical prediction \ref{['Formula:R:incoh']}.
  • Figure 2: (a) Averaged order parameter $\langle r(t) \rangle$, with theoretical predictions from classical mean-field theory \ref{['Eq:SC']} for $K>K_{\rm{crit}}$ and from our approach \ref{['Formula:incoh:r']} for $K<K_{\rm{crit}}$, and (b) its variance $V$, with theoretical predictions \ref{['Formula:V:ps']}, \ref{['Formula:incoh:r']}, versus coupling strength $K$ in the KS model \ref{['Eq:KS:original']} with $\lambda = \pi/4$.
  • Figure 3: Covariance $R(\tau)$ and pseudo-covariance $\tilde{R}(\tau)$ of the complex order parameter fluctuation $\zeta(t)$ for a partially synchronized state at $K = 5$ and $\lambda = \pi/4$ in the KS model \ref{['Eq:KS:original']}. Numerical simulations (solid curve) vs. theoretical prediction \ref{['Formula:R:ps']} (dashed curve).
  • Figure 4: The same fluctuation characteristics as in Fig. \ref{['Fig:Covariances:K5']} but for a partially synchronized state at $K = 4$ and $\lambda = \pi/4$.
  • Figure 5: The same fluctuation characteristics as in Fig. \ref{['Fig:Covariances:K5']} but for a partially synchronized state at $K = 3$ and $\lambda = \pi/4$.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3