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

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Shabunin A. V. Multistability in dynamical small world networks. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 3, pp. 63-76. DOI: 10.18500/0869-6632-2014-22-3-63-76

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Multistability in dynamical small world networks

Shabunin Aleksej Vladimirovich, Saratov State University

We explore phase multistability which takes place in an ensemble of periodic oscillators under the action of long-distance couplings, which appear randomly between the arbitrary cells. The  system under study is Kuromoto’s model with additional dynamical interconnections between phase oscillators. The sequence of bifurcations, which accompany increasing of the strength of the global coupling is determined. Regions of multistability existance are defined.

  1. Watt DJ, Strogatz SH. Collective dynamics of «small-world» networks. Nature. 1998; 393(4):440–442. DOI: 10.1038/30918.
  2. Milgram S. The small world problem. Psychology Today. 1967;2:60–67.
  3. Belykh IV, Hasler M, Belykh VN. Blinking model and synchronization in small-world networks with a time-varying coupling. Physica D. 2004;195(1–2):188–206. DOI 10.1016/j.physd.2004.03.013
  4. Li C, Chen G. Phase synchronization in small-world networks of chaotic oscillators. Physica A. 2004;341(1-4):73–79. DOI: 10.1016/j.physa.2004.04.112
  5. Percha B, Dzakpasu R, Zochowski M. Parent Transition from local to global phase synchrony in small world neural network and its possible implications for epilepsy. Physical Review E. 2005;72(3):1–3. DOI: 10.1103/PhysRevE.72.031909
  6. Wang Q, Duan Z, Perc M, Chen G. Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability. Euroipean Physical Letters. 2008;83(5):50008–50014. DOI: 10.1209/0295-5075/83/50008
  7. Yu H, Wang J, Deng B, Wei X, Wong YK, Chan WL, Tsang KM, Yu Z. Chaotic phase synchronization in small-world networks of bursting neurons. Chaos. 2011;21(1):013127. DOI: 10.1063/1.3565027
  8. Rothkegel A, Lehnertz K. Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrate-and-fire neurons. Chaos. 2009;19(1):015109. DOI: 10.1063/1.3087432.
  9. Gao Z, Hu B, Hu G. Stochastic resonance of small-world networks. Physical Review E. 2001;65:016209. DOI: 10.1103/PhysRevE.65.016209
  10. Hou Z, Xin H. Oscillator death on small-world networks. Physical Review E. 2003;68(2):551031. DOI: 10.1103/PhysRevE.68.055103
  11. Moukarzel CF. Percolation in networks with long-range connections. Physica A. 2006; 372(2):340–346. DOI: 10.1016/j.physa.2006.08.049
  12. Yevin IA. Introduction to the theory of complex networks. Computer Research and Modeling. 2010; 2(2):121–141.DOI: 10.20537/2076-7633-2010-2-2-121-141
  13. Dvornikov AA, Utkin GM, Chukov AM. On the mutual synchronization of the chain of resistively connected auto-generators. Radiophisics. 1984;27(11):1388–1394.
  14. Ermentrout GB. The behaviour of rings of coupled oscillators. J. of Math. Biol. 1985;23(1):55–74. DOI:10.1007/BF00276558
  15. Shabunin AV, Akopov AA, Astahov VV, Vadivasova TE. Running waves in a discrete anharmonic self-oscillating medium. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(4):37-55. DOI: 10.18500/0869-6632-2005-13-4-37-55
  16. Astakhov VV, Bezruchko BP, Gulyaev YV, Seleznev YP. Multistable states of dissipatively-connected feigenbaum system. Pisma v Zhurnal Tekhnicheskoi Fiziki. 1989;15(3):60–65.
  17. Astakhov BV, Bezruchko BP, Pudovochkin OB, Seleznev EP. Phase multi-stability and establishment of oscillations in nonlinear systems with period doubling. Journal of Communications Technology and Electronics. 1993;38(2):2.
  18. Shabunin AV, Astahov VV. Phase multistability in an array of period-doubling self-sustained oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(6):99-118. DOI: 10.18500/0869-6632-2009-17-6-99-118
  19. Shabunin AV. Random distant couplings influence to a system with phase multistability. Izvestiya VUZ. Applied Nonlinear Dynamics. 2013;21(2):20-33. DOI: 10.18500/0869-6632-2013-21-2-20-33
  20. Kuramoto Y. Chemical oscillators, waves and turbulence. New-York: Springer; 1984. 156 p.
  21. Gurtovnik AU, Neymark UI. Synchronisms in a cyclically loosely coupled oscillator system. Dynamic systems: Inter-university collection of scientific works. NNUP; 1991. p. 84. (In Russian).
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