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Spatiotemporal Regimes in the Kuramoto–Battogtokh System of Nonidentical Oscillators

  • STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS
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Abstract

We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of the long-term evolution of such a system. We have obtained homogeneous, twisted, and nonhomogeneous stationary solutions to the Ott–Antonsen equations corresponding to key variants of the realized collective rotational motion of elements of the medium in question with nonzero mesoscopic characteristics determining the degree of coherence of the dynamics of neighboring particles. We have described the procedures of the search for the class of nonhomogeneous solutions as stationary points of the auxiliary point map and of determining the stability based on analysis of the eigenvalue spectrum of the composite operator. Static and breather cluster regimes have been demonstrated and described, as well as the regimes with an irregular behavior of averaged complex fields including, in particular, the local order parameter.

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REFERENCES

  1. G. V. Osipov, J. Kurths, and Ch. Zhou, Synchronization in Oscillatory Networks (Springer, Berlin, 2007).

    Book  MATH  Google Scholar 

  2. V. S. Afraimovich, V. I. Nekorkin, G. V. Osipov, and V. D. Shalfeev, Stability, Structures and Chaosin Nonlinear Synchronization Networks (World Scientific, Singapore, 1994).

    MATH  Google Scholar 

  3. J. Pantaleone, Am. J. Phys. 70, 992 (2002).

    Article  ADS  Google Scholar 

  4. S. Chhabria, K. A. Blaha, F. D. Rossa, and F. Sorrentino, Chaos 28, 111102 (2018).

    Article  MathSciNet  Google Scholar 

  5. J. Machowski, J. W. Bialek, and J. R. Bumby, Power System Dynamics: Stability and Control (Wiley, New York, 2002).

    Google Scholar 

  6. P. J. Menck, J. Heitzig, J. Kurths, and H. J. Schellnhuber, Nat. Commun. 5, 3969 (2014).

    Article  ADS  Google Scholar 

  7. S. Ryu, W. Yu, and D. Stroud, Phys. Rev. E 53, 2190 (1996).

    Article  ADS  Google Scholar 

  8. Z. Zheng, B. Hu, and G. Hu, Phys. Rev. E 57, 1139 (1998).

    Article  ADS  Google Scholar 

  9. S. Homma and S. Takeno, Prog. Theor. Phys. 72, 4 (1984).

    Google Scholar 

  10. S. Takeno and S. Homma, Prog. Theor. Phys. 77, 3 (1987).

    Article  Google Scholar 

  11. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge Univ. Press, Cambridge, 2001).

    Book  MATH  Google Scholar 

  12. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).

    Book  MATH  Google Scholar 

  13. A. Pikovsky and M. Rosenblum, Chaos 25, 097616 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  14. E. Brown, J. Moehlis, and P. Holmes, Neural Comp. 16, 4 (2004).

    Article  Google Scholar 

  15. C. R. Laing, J. Math. Neurosci. 8, 4 (2018).

    Article  ADS  Google Scholar 

  16. J. Acebron, L. Bonilla, C. P. Vicente, F. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 (2005).

    Article  ADS  Google Scholar 

  17. F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths, Phys. Rep. 610, 1 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  18. S. Gupta, A. Campa, and S. Ruffo, Statistical Physics of Synchronization (Springer Int., Switzerland, 2018).

    Book  MATH  Google Scholar 

  19. S. Watanabe and S. H. Strogatz, Phys. Rev. Lett. 70, 2391 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  20. E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  21. E. Ott and T. M. Antonsen, Chaos 19, 023117 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  22. B. Pietras and A. Daffertshofer, Chaos 26, 103101 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  23. D. Chowdhury and M. C. Cross, Phys. Rev. E 82, 016205 (2010).

    Article  ADS  Google Scholar 

  24. M. J. Panaggio and D. M. Abrams, Nonlinearity 28, R67 (2015).

    Article  ADS  Google Scholar 

  25. N. Yao and Z. Zheng, Int. J. Mod. Phys. B 30, 1630002 (2016).

    Article  ADS  Google Scholar 

  26. O. E. Omel’chenko, Nonlinearity 31, R121 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  27. S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Phys. Life Rev. 28, 100 (2019).

    Article  ADS  Google Scholar 

  28. E. Schöll, A. Zakharova, and R. G. Andrzejak, Chimera States in Complex Networks (Frontiers Media SA, Lausanne, 2020).

    Book  Google Scholar 

  29. E. Schöll, Eur. Phys. J. Spec. Top. 225, 891 (2016).

    Article  Google Scholar 

  30. A. E. Motter, Nat. Phys. 6, 164 (2010).

    Article  Google Scholar 

  31. L. Smirnov, G. Osipov, and A. Pikovsky, J. Phys. A: Math. Theor. 50, 08LT01 (2017).

    Article  Google Scholar 

  32. M. I. Bolotov, L. A. Smirnov, G. V. Osipov, and A. S. Pikovsky, JETP Lett. 106, 393 (2017).

    Article  ADS  Google Scholar 

  33. M. I. Bolotov, L. A. Smirnov, G. V. Osipov, and A. Pikovsky, Chaos 28, 045101 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  34. O. E. Omel’chenko, Nonlinearity 26, 9 (2013).

    MathSciNet  Google Scholar 

  35. M. Wolfrum, O. E. Omel’chenko, S. Yanchuk, and Y. L. Maistrenko, Chaos 21, 013112 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  36. G. B. Ermentrout, SIAM J. Appl. Math. 52, 1665 (1992).

    Article  MathSciNet  Google Scholar 

  37. C. R. Laing, Phys. D (Amsterdam, Neth.) 238, 1569 (2009).

  38. C. R. Laing, Chaos 52, 013113 (2009).

    Article  ADS  Google Scholar 

  39. W. S. Lee, J. G. Restrepo, E. Ott, and T. M. Antonsen, Chaos 21, 023122 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  40. G. S. Medvedev and X. Tang, J. Nonlin. Sci. 25, 1169 (2015).

    Google Scholar 

  41. O. E. Omel’chenko, M. Wolfrum, and C. R. Laing, Chaos 24, 023102 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  42. M. Wolfrum, S. V. Gurevich, and O. E. Omel’chenko, Nonlinearity 29, 257 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  43. O. E. Omel’chenko, Chaos 30, 043103 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  44. D. Wiley, S. Strogatz, and M. Girvan, Chaos 16, 015103 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  45. T. Girnyk, M. Hasler, and Y. Maistrenko, Chaos 22, 013114 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  46. D. Bolotov, M. Bolotov, L. Smirnov, G. Osipov, and A. Pikovsky, Regular Chaotic Dyn. 24, 717 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  47. Y. Kuramoto and D. Battogtokh, Nonlin. Phenom. Complex Syst. 5, 380 (2002).

    Google Scholar 

  48. I. N. Pesin, Classical and Modern Integration Theories, Vol. 8 of Probability and Mathematical Satistics (Academic, New York, 1970).

  49. W. Eckhaus, Studies in Nonlinear Stability Theory (Springer, Berlin, 1965).

    Book  MATH  Google Scholar 

  50. F. C. Moon, Chaotic Vibrations: An Introduction for Applied Scientists and Engineers (Wiley-VCH, 2004).

    Book  MATH  Google Scholar 

  51. Y. Suda and K. Okuda, Phys. Rev. E 97, 042212 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  52. J. Xie, E. Knobloch, and H.-C. Kao, Phys. Rev. E 92, 042921 (2015).

    Article  ADS  Google Scholar 

  53. O. E. Omel’chenko, M. Wolfrum, and E. Knobloch, SIAM J. Appl. Dyn. Syst. 17, 97 (2018).

    Article  MathSciNet  Google Scholar 

  54. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1995).

    Book  MATH  Google Scholar 

Download references

Funding

This study was supported by the Russian Science Foundation (project no. 19-12-00367) (Sections 1 and 2), the Ministry of Science and Higher Education of the Russian Federation (project no. 0729-2020-0036) (Section 3), and by the Russian Foundation for Basic Research (project no. 19-52-12053) (Sections 4 and 5). AP was supported by German Science Foundation (grant PI 220/22-1).

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Authors and Affiliations

  1. Research and Education Mathematics Center “Mathematics of Future Technologies,” Institute of Information Technologies, Mathematics, and Mechanics, Lobachevsky State University of Nizhny Novgorod, 603950, Nizhny Novgorod, Russia

    M. I. Bolotov, L. A. Smirnov, E. S. Bubnova, G. V. Osipov & A. S. Pikovsky

  2. Institute of Applied Physics, Russian Academy of Sciences, 603950, Nizhny Novgorod, Russia

    L. A. Smirnov

  3. Institute for Physics and Astronomy, University of Potsdam, D-14476, Potsdam-Golm, Germany

    A. S. Pikovsky

Authors
  1. M. I. Bolotov
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  2. L. A. Smirnov
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  3. E. S. Bubnova
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  4. G. V. Osipov
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  5. A. S. Pikovsky
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Corresponding authors

Correspondence to M. I. Bolotov, L. A. Smirnov, G. V. Osipov or A. S. Pikovsky.

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Translated by N. Wadhwa

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Bolotov, M.I., Smirnov, L.A., Bubnova, E.S. et al. Spatiotemporal Regimes in the Kuramoto–Battogtokh System of Nonidentical Oscillators. J. Exp. Theor. Phys. 132, 127–147 (2021). https://doi.org/10.1134/S1063776121010106

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  • DOI: https://doi.org/10.1134/S1063776121010106

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