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

For citation:

Matrosov V. V., Shmelev A. V. Nonlinear dynamics of a ring of three phase systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 1, pp. 123-136. DOI: 10.18500/0869-6632-2011-19-1-123-136

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 122)
Article type: 

Nonlinear dynamics of a ring of three phase systems

Matrosov Valerij Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Shmelev Aleksej Vjacheslavovich, Lobachevsky State University of Nizhny Novgorod

Nonlinear dynamics of the ensemble consisting of three phase-locked generators, which are coupled in a ring, is discovered. By force of computational modeling, which is based on the theory of oscillations, the regimes of the generators collective behavior is examined; the districts of synchronous and quasi-synchronous regimes are distinguished in the parameter space; the restructuring of the dynamics behavior on the boards of the distinguished districts is analyzed.

  1. Barbashin EA, Tabueva VA. Dynamical Systems with a Cylindrical Phase Space. Moscow: Nauka; 1969. 300 p. (in Russian).
  2. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillators. Pergamon; 1966. 815 p. DOI: 10.1016/C2013-0-06631-5.
  3. Shakhgildyan VV, Lyakhovkin AA. Phase Locking Systems. Moscow: Svyaz; 1972. 446 p. (in Russian).
  4. Lindsey V. Synchronization Systems in Communication and Control. Prentice-Hall; 1972. 704 p.
  5. Barone A, Paterno J. Physics and Applications of the Josephson. Wiley; 1982. 529 p. DOI: 10.1002/352760278X.
  6. Likharev KK. Introduction to the Dynamics of Josephson Junctions. Moscow: Nauka; 1985. 320 p. (in Russian).
  7. Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Camridge University Press; 2003. 432 p.
  8. Anishchenko VS, Astakhov VV, Vadivasova TE, Strelkova GI. Synchronization of Regular, Chaotic and Stochastic Oscillations. Moscow-Izhevsk: Institute for Computer Research; 2008. 144 p. (in Russian).
  9. Kuznetsov AP, Emelyanova YP, Sataev IR, Tyuryukina LV. Synchronization in Tasks. Saratov: «Nauka»; 2010. 256 p. (in Russian).
  10. Afraimovich VS, Nekorkin VI, Osipov GV, Shalfeev VD. Stability, Structures and Chaos in Nonlinear Synchronization Networks. World Scientific; 1995. 260 p. DOI: 10.1142/2412.
  11. Matrosov VV, Kasatkin DV. Dynamic modes of coupled phase controlled oscillators. J. Commun. Technol. Electron. 2003;48(6):637–645 (in Russian).
  12. Krjukov AK, Osipov GV, Polovinkin AV. Variety of synchronous regimes in ensembles of nonidentical oscillators: two coupled elements. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(2):16–28 (in Russian). DOI: 10.18500/0869-6632-2009-17-2-16-28.
  13. Matrosov VV, Shmelev AV. Nonlinear dynamics of a ring of two coupled phase locked loops. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(4):67–80 (in Russian). DOI: 10.18500/0869-6632-2010-18-4-67-80.
  14. Shmelev AV, Matrosov VV. Peculiarities of dynamics of two ring-coupled phase-locked loops synchronized in anti-phase. Vestnik of Lobachevsky University of Nizhni Novgorod. 2011;2(1)55–61 (in Russian).
  15. Shmelev AV, Matrosov VV. Synchronous and quasi-synchronous modes of ring connection of three phase systems. In: Abstracts of the Conference of Young Scientists. Fundamental and Applied Problems of Nonlinear Physics. XV Scientific School «Nonlinear Waves – 2010». Nizhni Novgorod; 2010. P. 134 (in Russian).
  16. Shmelev AV, Matrosov VV. Bifurcation analysis of the dynamics of three phase systems united in a ring. In: Grach CM, Yakimov AV, editors. Proceedings of the 14th Scientific Conference on Radiophysics, Dedicated to the 80th Anniversary of Y.N. Babanova. Nizhni Novgorod: LU; 2010. P. 123 (in Russian).
  17. Shmelev AV. Modeling the dynamics of phase systems in the ADS package. In: Proceedings of the 9th International School «Chaotic Self-Oscillations and Formation of Structures», October 4-9, 2010. Saratov, Russia. Saratov; 2010. P. 72–75 (in Russian).
  18. Matrosov VV. Dynamics of Nonlinear Systems. A Software Package for the Study of Nonlinear Dynamic Systems With Continuous Time. Nizhni Novgorod: LU; 2002 (in Russian).
  19. Anosov DV, Aranson SK, Bronstein IU, Grines VV. Smooth dynamical systems. II. Modern Problems Of Mathematics. Fundamental Directions. Vol. 1. Results of Science and Technology. Moscow: All-Russian Institute of Scientific and Technical Information of the Academy of Sciences of the USSR; 1985. P. 151–240 (in Russian).
  20. Andronov AA, Leontovich EA, Gordon II, Maier AG. The theory of bifurcations of dynamical systems on the plane. Wiley; 1973. 482 p.
  21. Belyakov LA. On the structure of bifurcation sets in systems with a saddle-focus separatrix loop. In: Abstracts of the IX International Conference on Nonlinear Oscillations. Kiev; 1981. P. 57 (in Russian).
  22. Shilnikov LP. Bifurcation theory and Lorentz model. In: Marsden J. McCracken M. Cycle Birth Bifurcation and Its Applications. Moscow: Mir; 1980. P. 317 (in Russian).
  23. Belyakov LA. On a bifurcation set in a system with a homoclinic saddle curve. Mathematical Notes. 1980;28(6):911–922 (in Russian).  
Short text (in English):
(downloads: 71)