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

For citation:

Shabunin A. V., Litvinenko A. N., Astakhov V. V. Controll of multistability by means of bi­phase resonance force. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 1, pp. 25-39. DOI: 10.18500/0869-6632-2011-19-1-25-39

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2011-19-1-25-39

Controll of multistability by means of bi­phase resonance force

Autors: 
Shabunin Aleksej Vladimirovich, Saratov State University
Litvinenko Artem Nikolaevich, Saratov State University
Astakhov Vladimir Vladimirovich, Yuri Gagarin State Technical University of Saratov
Abstract: 

We propose a new method of control of phase multistability in two coupled selfsustained oscillators. The method is based on the «pulling» of phases of oscillations to the target mode under two external harmonic forces, which influence the first and the second sub-systems simultaneosly. Varying the phase shift between the external signals results in control of switching between coexisting oscillating modes. Effectiveness of the method is demonstrated on the example of switching between periodic and chaotic regimes in two Chua’s oscillatotrs.

Key words: 
Phase multistability
synchronization
control of multistability
Reference: 
  1. Arecchi FT, Meucci R, Puccioni G, Tredicce J. Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser. Phys. Rev. Lett. 1982;49(17):1217–1220. DOI: 10.1103/PhysRevLett.49.1217.
  2. Prengel F, Wacker A, Scholl E. Simple model for multistability and domain formation in semiconductor superlattices. Phys. Rev. B. 1994;50(3):1705–1712. DOI: 10.1103/physrevb.50.1705.
  3. Sun NG, Tsironis GP. Multistability of conductance in doped semiconductor superlattices. Phys. Rev. B. 1995;51(16):11221–11224. DOI: 10.1103/PhysRevB.51.11221.
  4. Foss J, Longtin A, Mensour B, Milton J. Multistability and delayed recurrent loops. Phys. Rev. Lett. 1996;76(4):708–711. DOI: 10.1103/physrevlett.76.708.
  5. Astakhov VV, Bezruchko BP, Gulyaev YP, Seleznev EP. Multistable states in dissipatively coupled Feigenbaum systems. Tech. Phys. Lett. 1989;15(3):60–65 (in Russian).
  6. 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).
  7. Dvornikov AA, Utkin GM, Chukov AM. Mutual synchronization of a chain of resistance-coupled self-excited oscillators. Radiophys. Quantum Electron. 1984;27(11):967–972. DOI: 10.1007/BF01037390.
  8. Ermentrout GB. The behavior of rings of coupled oscillators. J. Math. Biol. 1985;23(1):55–74. DOI: 10.1007/bf00276558.
  9. Ermentrout GB. Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 1992;52(6):1664–1687. DOI: 10.1137/0152096.
  10. 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.
  11. Balanov AG, Janson NB, Astakhov VV, McClintock PVE. Role of saddle tori in the mutual synchronization of periodic oscillations. Phys. Rev. E. 2005;72(2):026214. DOI: 10.1103/PhysRevE.72.026214.
  12. Astakhov VV, Shabunin AV, Anishchenko VS. Spectral patterns in the formation of multistability in coupled oscillators with a doubling of the period. J. Commun. Technol. Electron. 1997;42(8):974–981 (in Russian).
  13. Shabunin A, Feudel U, Astakhov V. Phase multistability and phase synchronization in an array of locally coupled period-doubling oscillators. Phys. Rev. E. 2009;80(2):026211. DOI: 10.1103/PhysRevE.80.026211.
  14. Bezruchko BP, Prokhorov MD, Seleznev EP. Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems. Chaos, Solitons and Fractals. 2003;15(4):695–711. DOI: 10.1016/S0960-0779(02)00171-6.
  15. Lai YC. Driving trajectories to a desirable attractor by using small control. Phys. Lett. A. 1996;221(6):375–383. DOI: 10.1016/0375-9601(96)00609-3.
  16. Macau EEN, Grebogi C. Driving trajectories in complex systems. Phys. Rev. E. 1999;59(4):4062–4070. DOI: 10.1103/PhysRevE.59.4062.
  17. Pisarchik AN, Goswami BK. Annihilation of one of the coexisting attractors in a bistable system. Phys. Rev. Lett. 2000;84(7):1423–1426. DOI: 10.1103/physrevlett.84.1423.
  18. Egorov EN, Koronovskii AA. Dynamical control in multistable systems. Tech. Phys. Lett. 2004;30(3):186–189. DOI: 10.1134/1.1707162.
  19. Goswami BK, Euzzor S, Naimee KA, Geltrude A, Meucci R, Arecchi FT. Control of stochastic multistable systems: Experiment demonstration. Phys. Rev. E. 2009;80(1):016211. DOI: 10.1103/PhysRevE.80.016211.
  20. Goswami BK. Control of multistate hopping intermittency. Phys. Rev. E. 2008;78(6):066208. DOI: 10.1103/PhysRevE.78.066208.
  21. Astahov VV, Sherbakov PA, Kobljanskij SA, Shabunin AV. Synchronization of spatial-periodic modes in the ring of oscillators with phase multystability. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(4):65–73 (in Russian). DOI: 10.18500/0869-6632-2008-16-4-65-73.
  22. Komuro M, Tokunaga R, Matsumoto T, Chua LO, Hotta A. Global bifurcation analysis of the double-scroll circuit. Int. J. Bifurcation Chaos. 1991;1(1):139–182. DOI: 10.1142/S0218127491000105.
  23. Khibnik AI, Roose D, Chua L. On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity. In: Madan RN, editor. Chua’s Сircuit: A Paradigm for Chaos. Singapore: World Scientific; 1993. P. 145–178.
  24. Anishchenko VS, Astakhov VV, Vadivasova TE, Sosnovtseva OV, Wu CW, Chua L. Dynamics of two coupled Chua’s curcuits. Int. J. Bifurcation Chaos. 1995;5(6):1677–1699. DOI: 10.1142/S0218127495001241.
  25. Blekhman II. Synchronization in Nature and Technology. Moscow: Nauka; 1981. 440 p. (in Russian).
Received: 
05.07.2010
Accepted: 
12.12.2010
Published: 
29.04.2011
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