Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs


The Kuramoto model of coupled phase oscillators with inertia on Erdős–Rényi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

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    It is easy to generalize the equations determining stability of d-cluster stattes for \(d\ge 2\), but the analysis of this system is already challenging for \(d=2\).

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    In fact, the same condition guarantees that the sequence of ER graphs satisfies the large deviation principle [13].

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    See Remark 2.1 for more details on the validity of the mean field limit.

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    We maintain \({\bar{\omega }}_1<0<{\bar{\omega }}_2\) in all numerical experiments.

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    Note that \(\arcsin \left( \frac{1}{K}\right) <\frac{\pi }{2}\) while \(\arcsin \left( 1\right) + \arccos (K^{-1})>\frac{\pi }{2}\).

  6. 6.

    Note that we are not using the analytic equation for \(\alpha ^*\).


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This work was supported in part by NSF grants DMS 1715161 and 2009233 (to GSM). Numerical simulations were completed using the high performance computing cluster (ELSA) at the School of Science, The College of New Jersey. Funding of ELSA is provided in part by National Science Foundation OAC-1828163. MSM was additionally supported by a Support of Scholarly Activities Grant at The College of New Jersey.

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Communicated by Shin-ichi Sasa.

Appendix A: \(K_c\) is an Increasing Function of \(\sigma \)

Appendix A: \(K_c\) is an Increasing Function of \(\sigma \)

Suppose g zero mean Gaussian density with standard deviation \(\sigma \)

$$\begin{aligned} g(s) = \frac{1}{\sigma \sqrt{2\pi }}{\text {exp}}\left( -\frac{s^2}{2\sigma ^2}\right) . \end{aligned}$$

By plugging (A.1) into (4.5) we have

$$\begin{aligned} K_c(\sigma ) = 2\sqrt{2\pi }\sigma \left( \pi -\displaystyle \int _{\mathbb {R}} \frac{{\mathrm{exp}}\left( -\frac{s^2}{2\sigma ^2}\right) }{\gamma ^2+(s/\gamma )^2}ds\right) ^{-1}, \end{aligned}$$

which can be further rewritten as

$$\begin{aligned} K_c(\sigma ) = \frac{2\sqrt{2}\sigma }{\sqrt{\pi }}\left( 1-{\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right) \right) ^{-1}, \end{aligned}$$

where \({\text {Erfc}}(x) = 1- \frac{2}{\sqrt{\pi }}\int _0^x e^{-s^2}ds\) is the function.

Lemma A.1

\(K_c\) in (A.2) is an increasing function of \(\sigma >0\).


Let \(A = {\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right) \) and note that

$$\begin{aligned} \frac{dK_c}{d\sigma } = \frac{2\sqrt{2}}{\sqrt{\pi }}\left( (1-A)+\sigma (1-A)^{-2}\frac{dA}{d\sigma }\right) . \end{aligned}$$

Below we show that \(A<1\) and \(\frac{dA}{d\sigma }>0\) for all \(\sigma >0\). Assume first that \(\sigma < \gamma ^2\). Then, using the well known bound

$$\begin{aligned} {\text {Erfc}}(z) < \frac{{\text {exp}}(-z^2)}{\sqrt{\pi }z}, \end{aligned}$$

it follows that \(A< \frac{\sqrt{2}{\sigma }}{\sqrt{\pi }\gamma ^2} <1\).

Next consider when \(\sigma \ge \gamma ^2.\) Since the Taylor series expansion of \({\text {Erfc}}(z)\) is an alternating series

$$\begin{aligned} {\text {Erfc}}(z) = 1-\frac{2}{\sqrt{\pi }} \left( z-\frac{z^3}{3}+\frac{z^5}{10}-\cdots \right) , \end{aligned}$$

we have

$$\begin{aligned} {\text {Erfc}}(z) < 1 -\frac{2}{\sqrt{\pi }} z + \frac{2}{3\sqrt{\pi }} z^3. \end{aligned}$$


$$\begin{aligned} A < {\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) \left( 1-\frac{\sqrt{2}}{\sqrt{\pi }}\cdot \frac{\gamma ^2}{\sigma } + \frac{1}{3\sqrt{2\pi }} \cdot \frac{\gamma ^6}{\sigma ^3}\right) . \end{aligned}$$

In this case we have

$$\begin{aligned} A< e^{1/2}\left( 1-\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{1}{3\sqrt{2\pi }}\right) <1. \end{aligned}$$

Finally, we show that \(\frac{dA}{d\sigma }>0\). Indeed, by direct calculation,

$$\begin{aligned} \frac{dA}{d\sigma } = \frac{\sqrt{2}\gamma ^2}{\sqrt{\pi }\sigma ^2 }- \frac{\gamma ^4}{\sigma ^3}{\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right) . \end{aligned}$$

Again use (A.3) to see that

$$\begin{aligned} \frac{dA}{d\sigma } > \frac{\sqrt{2}\gamma ^2}{\sqrt{\pi }\sigma ^2}-\frac{\gamma ^4}{\sigma ^3}\cdot \frac{\sqrt{2}\sigma }{\sqrt{\pi }\gamma ^2} =0 . \end{aligned}$$

\(\square \)

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Medvedev, G.S., Mizuhara, M.S. Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs. J Stat Phys 182, 30 (2021).

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  • Collective dynamics
  • Synchronization
  • Continuum limit
  • Chimera