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

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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

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Language: 
Russian
Heading: 
Journal in the journal
Article type: 
Article
UDC: 
621.391.01
DOI: 
10.18500/0869-6632-2011-19-1-123-136

Nonlinear dynamics of a ring of three phase systems

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

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.

Key words: 
ensemble of oscillators
phase systems
dynamic regimes
synchronization
quasi-synchronization
beats
attractors
bifurcations
Reference: 
  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).  
Received: 
17.12.2010
Accepted: 
09.02.2011
Published: 
29.04.2011
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