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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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  1. 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 \(d=2\).

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

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

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

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

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


  1. Abrams, D.M., Strogatz, S.H.: Chimera states in a ring of nonlocally coupled oscillators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16(1), 21–37 (2006)

    MathSciNet  Article  Google Scholar 

  2. Andronov, A.A., Vitt, A.A., Khaĭkin, S.È.: Theory of Oscillators. Dover Publications, Inc., New York (1987). Translated from the Russian by F. Immirzi, Reprint of the 1966 translation

  3. Antoni, M., Ruffo, S.: Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E 52, 2361–2374 (1995)

    Article  ADS  Google Scholar 

  4. Ashwin, P., Burylko, O.: Weak chimeras in minimal networks of coupled phase oscillators. Chaos 25(1), (2015) 013106, 9

  5. Bachelard, R., Dauxois, T., De Ninno, G., Ruffo, S., Staniscia, F.: Vlasov equation for long-range interactions on a lattice. Phys. Rev. E 83, 061132 (2011)

    Article  ADS  Google Scholar 

  6. Barré, J., Métivier, D.: Bifurcations and singularities for coupled oscillators with inertia and frustration. Phys. Rev. Lett. 117, 214102 (2016)

    Article  ADS  Google Scholar 

  7. Belykh, I.V., Brister, B.N., Belykh, V.N.: Bistability of patterns of synchrony in Kuramoto oscillators with inertia. Chaos: Interdiscip. J. Nonlinear Sci. 26(9), 094822 (2016)

    MathSciNet  Article  Google Scholar 

  8. Chiba, H.: unpublished notes

  9. Chiba, H.: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergod. Theory Dynam. Syst. 35(3), 762–834 (2015)

    MathSciNet  Article  Google Scholar 

  10. Chiba, H., Medvedev, G.S.: The mean field analysis of the Kuramoto model on graphs I. The mean field equation and transition point formulas. Discret. Contin. Dyn. Syst. 39(1), 131–155 (2019)

    MathSciNet  Article  Google Scholar 

  11. Chiba, H., Medvedev, G.S., Mizuhara, M.S.: Instability of Mixing in the Kuramoto Model: From Bifurcations to Patterns. Pure and Applied Functional Analysis, accepted (2020)

  12. Dörfler, F., Bullo, F.: Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators. SIAM J. Control Optim. 50(3), 1616–1642 (2012)

    MathSciNet  Article  Google Scholar 

  13. Dupuis, P., Medvedev, G.S.: The large deviation principle for interacting dynamical systems on random graphs, arxiv preprint, 2007.13899 (2020)

  14. Jaros, P., Brezetsky, S., Levchenko, R., Dudkowski, D., Kapitaniak, T., Maistrenko, Yu: Solitary states for coupled oscillators with inertia. Chaos 28 (2018), no. 1, 011103, 7

  15. Kaliuzhnyi-Verbovetskyi, D., Medvedev, G.S.: The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit. SIAM J. Math. Anal. 50(3), 2441–2465 (2018)

    MathSciNet  Article  Google Scholar 

  16. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002)

    Google Scholar 

  17. Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34(7), 523–535 (2005)

    MathSciNet  Article  ADS  Google Scholar 

  18. Medvedev, G.S.: The continuum limit of the Kuramoto model on sparse random graphs. Commun. Math. Sci. 17(4), 883–898 (2019)

    MathSciNet  Article  Google Scholar 

  19. Neunzert, H.: Mathematical investigations on particle–in–cell methods. Fluid Dyn. Trans. 9, 229–254 (1978)

    Google Scholar 

  20. Oliveira, R.I., Reis, G.H.: Interacting diffusions on random graphs with diverging average degrees: Hydrodynamics and large deviations. J. Stat. Phys. 176, 1057–1087 (2019)

    MathSciNet  Article  ADS  Google Scholar 

  21. Olmi, S.: Chimera states in coupled Kuramoto oscillators with inertia. Chaos 25 (2015), no. 12, 123125, 13

  22. Olmi, S., Martens, E.A., Thutupalli, S., Torcini, A.: Intermittent chaotic chimeras for coupled rotators. Phys. Rev. E 92, 030901 (2015)

    Article  ADS  Google Scholar 

  23. Olmi, S., Navas, A., Boccaletti, S., Torcini, A.: Hysteretic transitions in the Kuramoto model with inertia. Phys. Rev. E 90, 042905 (2014)

    Article  ADS  Google Scholar 

  24. Olmi, S., Torcini, A.: Dynamics of Fully Coupled Rotators with Unimodal and Bimodal Frequency Distribution, pp. 25–45. Springer, Cham (2016)

    Google Scholar 

  25. Omel’chenko, O.E.: The mathematics behind chimera states. Nonlinearity 31(5), R121–R164 (2018)

    MathSciNet  Article  ADS  Google Scholar 

  26. Omelchenko, O.E.: Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26(9), 2469 (2013)

    MathSciNet  Article  ADS  Google Scholar 

  27. Salam, F., Marsden, J., Varaiya, P.: Arnold diffusion in the swing equations of a power system. IEEE Trans. Circ. Syst. 31(8), 673–688 (1984)

    MathSciNet  Article  Google Scholar 

  28. Strogatz, S.H., Mirollo, R.E.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63(3–4), 613–635 (1991)

    MathSciNet  Article  ADS  Google Scholar 

  29. Tanaka, H.-A., de Sousa Vieira, M., Lichtenberg, A.J., Lieberman, M.A., Oishi, S.: Stability of synchronized states in one-dimensional networks of second order PLLs. Int. J. Bifurc. Chaos Appl. Sci. Eng. 7(3), 681–690 (1997)

    MathSciNet  Article  Google Scholar 

  30. Tumash, L., Olmi, S., Scholl, E.: Effect of disorder and noise in shaping the dynamics of power grids. EPL 123(2), 20001 (2018)

  31. Tumash, L., Olmi, S., Scholl, E.: Stability and control of power grids with diluted network topology. Chaos: Interdiscip. J. Nonlinear Sci. 29(12), 123105 (2019)

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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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Correspondence to Georgi S. Medvedev.

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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