ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Shabunin A. V. Multistability of traveling waves in an ensemble of harmonic oscillators with long-range couplings. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 1, pp. 48-63. DOI: 10.18500/0869-6632-2018-26-1- 48-63

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 133)
Article type: 
517.9, 621.372

Multistability of traveling waves in an ensemble of harmonic oscillators with long-range couplings

Shabunin Aleksej Vladimirovich, Saratov State University

The work is devoted to study of multistability of traveling waves in a ring of harmonic oscillators with a linear non-local couplings. It analyses the influence of the strength and radius of the couplings on stability of spatially periodic regimes with different values of their wavelengths. The system under study is an array of identical van der Pol generators in the approximation of quasi-harmonic oscillations. On the one hand, the chosen model is a very simple one, that allows analytical studies; on the other hand, it applicable to a wide range of oscillatory systems with almost harmonic behavior.The research of the multistability is carried out in the way of the constucting analytical solutions by means of the method of slowly-changing amplitudes and then, by the standard methods of the stability analysis of the linearization matrix eigenvalues. In some cases the analitycal solution are supported by numerical calculations.

The study has shown that the number of simultaneously coexisting regimes is bounded by the value of the phase shift between oscillations of the subsystems on the length of the links. In the contrary of the locally coupled oscillators, here the maximum value of the phase shift may exceed the value of 0.5π and can reach a value of 0.7π. The every coexisting wave is born from the equilibrium in the origin as a saddle limit cycle (excluding the in-phase oscillating mode), which then becomes stable further on the parameter. Regions of stability of spatially periodic regimes represent a set of cones, where regions of shorter wave locate inside of the regions with much longer ones.


  1. Blekhman I.I., Landa P.S., Rosenblum M.G. Synchronization and chaotization in interacting dynamical systems. Appl. Mech. Rev, 1995, Vol. 11, part 1, pp. 733–752.
  2. Anishchenko V.S., Vadivasova T.E. Synchronization of self-oscillations and noiseinduced oscillations. Journal of Communications Technology and Electronics, 2002, vol. 47, no.2, pp. 117–148.
  3. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2003.
  4. Malafeev V.M., Polyakova M.S., Romanovsky Yu.M. Izvestya VUZ. Radiofizika, 1970, vol. 13, no.6, pp. 936–940 (in Russian).
  5. Mynbaev D.K., Shilenkov M.I. Radiotekhnika i elektronika, 1981, vol. 26, no. 2, pp. 361–370 (in Russian).
  6. Maltzev A.A., Silaev A.M. Izvestya VUZ, Radiofizika, 1979, vol. 22, no.7, pp. 826– 833 (in Russian).
  7. Dvornikov A.A., Utkin G.M., Chukov A.M. Izvestya VUZ, Radiofizika, 1984, vol. 27, no.11, pp. 1388–1393 (in Russian).
  8. Ermentrout G.B. The behaviour of rings of coupled oscillators. J. of Math. Biol, 1985, vol. 23, pp. 55–74.
  9. Shabunin A.V., Akopov A.A., Astakhov V.V., Vadivasova T.E. Izvestya VUZ, Applied Nonlinear Dynamics, 2005, vol. 13, iss. 4, pp. 37–54 (in Russian).
  10. Matias M.A., Guemez J., Perez-Munuzuri V., Marino I.P., Lorenzo M.N., PerezVillar V. Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys. Rev. Lett., 1997, vol. 78, no. 2, pp. 219–222. 
  11. Marino I.P., Perez-Munuzuri V., Perez-Villar V., Sanchez E., Matias M.A. Interaction of chaotic rotating waves in coupled rings of chaotic cells. Physica D, 2000, vol. 128, pp. 224–235.
  12. Shabunin A., Astakhov V., Anishchenko V. Developing chaos on base of traveling waves in a chain of coupled oscillators with period-doubling: Synchronization and hierarchy of multistability formation. Int. J. of Bifurcation and Chaos, 2002, vol. 12, no. 8. pp. 1895–1908.
  13. Nekorkin V.I., Makarov V.A., Velarde M.G. Spatial disorder and waves in a ring chain of bistable oscillators. Int. J. Bifurcation and Chaos, 1996, vol. 6, pp. 1845– 1858.
  14. Kuramoto Y. Chemical Oscillations Waves and Turbulence. Berlin: Springer, 1984.
  15. Ermentrout G.B., Kopell N. Symmetry and phase locking in chains of weakly coupled oscillators. Comm. Pure Appl. Math., 1986, vol. 49, pp. 623–660.
  16. Ermentrout G.B., Kopell N. Phase transitions and other phenomena in chains of coupled oscillators. SIAM J. of Appl. Math., 1990, vol. 50, pp. 1014–1052.
  17. Ermentrout G.B. Synchronization in a pool of mutually coupled oscillators with random frequencies. J. of Math. Biol., 1985, vol. 22, pp. 1–9.
  18. Crawford J.D., Davies K.T.R. Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings. Physica D, 1990, vol. 125, pp. 1–46.
  19. Gurtovnik A.S., Neimark Yu.I. Dinamicheskie Sistemy: Mezhvuzovskii sbornik nauchnyh trudov, 1991, pp. 84–97 (in Russian).
  20. Ren L., Ermentrout G.B. Phase locking in chains of multiple-coupled oscillators. Physica D, 2000, vol. 143, pp. 56–73.
  21. Astakhov V., Shabunin A., Uhm W., Kim S. Multistability formation and synchronization loss in coupled Henon maps: Two sides of the single bifurcational mechanism. Phys. Rev. E., 2001, vol. 63, 056212.
  22. Bezruchko B.P., Prokhorov M.D., Seleznev E.P. Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems. Chaos, Solitons and Fractals, 2003, vol. 15, pp. 695–711. 
Short text (in English):
(downloads: 86)