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

For citation:

Shabunin A. V. Random distant couplings influence to a system with phase multistability. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 2, pp. 20-33. DOI: 10.18500/0869-6632-2013-21-2-20-33

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: 89)
Article type: 

Random distant couplings influence to a system with phase multistability

Shabunin Aleksej Vladimirovich, Saratov State University

We explore the destruction of phase multistability which takes place in an ensemble of period doubling oscillators under the action of long-distance couplings, which appear randomly between the arbitrary cells. The investigation is carried out on the example of a chain of Rossler’s oscillators with periodic boundary conditions, where alongside with local couplings between the elements exist long-range interconnections. The sequence of bifurcations, which accompany increasing of the strength of the global coupling is determined.

  1. Watts DJ, Strogatz SH. Collective dynamics of 'small-world' networks. Nature. 1998;393(6684):440–442. DOI: 10.1038/30918.
  2. Lago-Fernández LF, Huerta R, Corbacho F, Sigüenza JA. Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett. 2000;84(12):2758–2761. DOI: 10.1103/PhysRevLett.84.2758.
  3. Barahona M, Pecora LM. Synchronization in small-world systems. Phys Rev Lett. 2002;89(5):054101. DOI: 10.1103/PhysRevLett.89.054101.
  4. Mori F, Odagaki T. Synchronization of coupled oscillators on small-world networks. Physica D. 2009;238(14):1180–1185. DOI: 10.1016/J.PHYSD.2009.04.002.
  5. Wang X, Chen G. Synchronization in small-world dynamical networks. Int. J. Bifurcation and Chaos. 2002;12(1):187–192.
  6. Astakhov VV, Bezruchko BP, Pudovochkin OB, Seleznev EP. Phase multi-stability and establishment of oscillations in nonlinear systems with period doubling. Soviet Journal Of Communications Technology And Electronics. 1993;38(2):291–295.
  7. Landa PS. Self-oscillations in systems with a finite number of degrees of freedom. Moscow: Nauka; 1980. 360 p. (In Russian).
  8. Dvornikov AA, Utkin GM, Chukov AM. On the mutual synchronization of the chain of resistively connected auto-generators Radiophysics and Quantum Electronics. 1984;27(11):1388–1394.
  9. Ermentrout GB. The behavior of rings of coupled oscillators. J Math Biol. 1985;23(1):55–74. DOI: 10.1007/BF00276558.
  10. Ermentrout GB. Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. of Appl. Math. 1992;52(6):1664–1687. DOI: 10.1137/0152096.
  11. Shabunin AV, Akopov AA, Astakhov VV, Vadivasova TE. Izvestija VUZ, Applied Nonlinear Dynamics. 2005;13(4):37–54 (in Russian).
  12. Astakhov VV, Bezruchko BP, Erastova EN, Seleznev EP. Oscillation modes and their evolution in dissipatively coupled Feigenbaum systems. Tech. Phys. 1990;60(10):19–26 (in Russian).
  13. Astakhov VV, Bezruchko BP, Gulyaev YuV, Seleznev YP. Multistable States Of Dissipatively-Connected Feigenbaum System. Pisma v Zhurnal Tekhnicheskoi Fiziki. 1989;15(3):60–65.
  14. Astakhov VV, Bezruchko BP, Ponomarenko VI, Seleznev EP. Multi-stability in the system of radio-technical generators with capacitive communication. Soviet Journal of Communications Technology and Electronics. 1991;36(11):2167–2170.
  15. Astakhov VV, Bezruchko BP, Ponomarenko VI. Formation of multi-stability, classification of isomers and their evolution in related Feigenbaum systems. Radiophysics and Quantum Electronics. 1991;34(1):35–39.
  16. Anishchenko VS, Astakhov VV, Vadivasova TE, Sosnovtseva OV, Wu CW, Chua L. Dynamics of two coupled Chua’s curcuits. Int. J. of Bifurcation and Chaos. 1995;5(6):1677–1699.
  17. Bezruchko BP, Prokhorov MD, Seleznev EP. Oscillation types, multistability, and basins of attractors in symetrically coupled period-doubling systems. Chaos, Solitons and Fractals. 2003;15(4):695–711. DOI: 10.1016/S0960-0779(02)00171-6.
  18. Matias MA, Perez-Munuzuri V, Marino IP, Lorenzo MN, Perez-Villa V. Size instabilities in ring of chaotic synchronized systems. Europhys. Lett. 1997;37(6):379–384. DOI: 10.1209/epl/i1997-00159-8.
  19. Matias MA, Guemez J, Perez-Munuzuri V, Marino IP, Lorenzo MN, Perez-Villar V. Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys. Rev. Lett. 1997;78(2):219–222. DOI: 10.1103/PHYSREVLETT.78.219.
  20. Marino IP, Perez-Munuzuri V, Perez-Villar V, Sanchez E, Matias MA. Interaction of chaotic rotating waves in coupled rings of chaotic cells. Physica D. 2000;128(2-4):224–235. DOI: 10.1016/S0167-2789(98)00303-0.
  21. 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;12(8):1895–1907. DOI: 10.1142/S021812740200556X.
  22. Shabunin AV, Astahov VV. Phase multistability in an array of period-doubling self­sustained oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(6):99–118. DOI: 10.18500/0869-6632-2009-17-6-99-118.
  23. Pikovsky AS, Rosenblum MG, Kurths J. Synchronization: a universal concept in nonlinear sciences. Cambridge: University Press; 2001. 433 p.
  24. Gurtovnik AS, Neymark YuI. Synchronisms in the system of cyclically weakly coupled oscillators. Dynamic systems: Inter-university collection of scientific works. Nizhny Novgorod: UNN Press; 1991. P. 84.
Short text (in English):
(downloads: 47)