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

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Kozlov A. K., Sushik M. M., Molkov J. I., Kuznetsov A. S. Symmetry breaking, multistability and chaos in the system of two coupled identical Van der Pol - Duffing oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 1999, vol. 7, iss. 1, pp. 68-80. DOI: 10.18500/0869-6632-1999-7-1-68-80

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Russian
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Journal in the journal
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Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1999-7-1-68-80

Symmetry breaking, multistability and chaos in the system of two coupled identical Van der Pol - Duffing oscillators

Autors: 
Kozlov Aleksandr Konstantinovich, Lobachevsky State University of Nizhny Novgorod
Sushik Mihail Mihajlovich, Lobachevsky State University of Nizhny Novgorod
Molkov Jaroslav Igorevich, Institute of Applied Physics of the Russian Academy of Sciences
Kuznetsov Aleksej Sergeevich, Saratov State University
Abstract: 

The phase bistability occurring at synchronization of two identical Van der Pol — Duffing oscillators with cubic nonlinear coupling is considered. Two scenarios of the transition to chaos were revealed for finite values of the coupling nonlinearity. The chaos in the system is observed in a wide range of coupling parameters.

Key words: 
-
Acknowledgments: 
The work was supported by the RFBR (projects 96-02-16559 and 97-02-17526), and Program for Support of Leading Scientific Schools of the Russian Federation (grant 96-02-96593).
Reference: 
  1. Haken Н. Advanced synergetics: Instability Hierarchies of Self-Organizing Systems and Devices. Berlin: Springer; 1983. 356 p. DOI: 10.1007/978-3-642-45553-7.
  2. Haken H. Information and Self—Organization: A Macroscopic Approach to Complex Systems. Berlin: Springer; 1988. 258 p. DOI: 10.1007/3-540-33023-2.
  3. Haken H, Kelso JAS, Bunz Н. A theoretical model оf phase transitions in human hand movements. Biol. Cybern. 1985;51(5):347-356. DOI: 10.1007/BF00336922.
  4. Kelso JAS, Scholz JP, Schoner G. Nonequilibrium phase transitions in coordinated biological motions: critical fluctuations. Phys. Lett. 1986;118(6):279-284. DOI: 10.1016/0375-9601(86)90359-2.
  5. Schoner G, Kelso JAS. Dynamic pattern generation in behavior and neural systems. Science. 1988;239(4847):1513-1520. DOI: 10.1126/science.3281253.
  6. Buchanan JJ, Kelso JAS, Fuchs A. Coordination dynamics оf trajectory formation. Biol. Cybern. 1995;74(1):41-54. DOI: 10.1007/BF00199136.
  7. Fuchs А, Jirsa VK, Haken Н, Kelso JАS. Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies. Biol. Cybern. 1996;74(1):21-30. DOI: 10.1007/BF00199134.
  8. Sternad D, Тurvey MT, Schmidt RС. Average phase difference theory and 1:1 phase entrainment in interlimb coordination. Biol. Cybern. 1992;67(3):223-231. DOI: 10.1007/BF00204395.
  9. Kelso JАS, Del Colle JD, Schoner G. Action—perception аs а pattern formation process. In: Jeannerod M, editor. Attention and Performance XIII: Motor representation and control. Hillsdale: Lawrence Erlbaum Associates; 1990. P. 139-169.
  10. Kelso JАS, Jeka JJ. Symmetry breaking dynamics of human multilimb coordination. J. Exp. Psychol. Hum. Percept. Perform. 1992;18(3):645-668. DOI: 10.1037//0096-1523.18.3.645.
  11. Molkov Yal, Sushchik MM, Kuznetsov АS, Kozlov АK, Zacharov DG. Dynamical model for locomotor-like movements of humans. In: Proc. 1998. Int. Symp. on Nonlinear Theory and its Applications(NOLTA’98). Vol. 3. 14-17 September. 1998. Crans — Montana Switzerland. 1998. P. 1325-1328.
  12. Gurafinkel VS, Levik YuS, Kazennikov OV, Selinov VА. Locomotor—like movements evoked by leg muscle vibration in humans. Eur. J. Neurosci. 1998;10(5):1608-1612. DOI: 10.1046/j.1460-9568.1998.00179.x.
  13. Kozlov AK, Huerta R, Rabinovitch MI, Abarbanel HD, Bazhenov MV. Neural ensembles with balanced communication as information receivers. Doklady Physics. 1997;357(6):752-757. (in Russian).
  14. Kozlov AK, Bazhenov MV, Huerta R, Rabinovitch MI. Multistability in neural ensembles with balance communication. Bulletin of the University of Nizhny Novgorod. 1998;1.
  15. Abarbanel HD, Rabinovitch MN, Selverston A, Bazhenov MV, Huerta R, Sushchik MM, Rrubchinskii LL. Synchronisation in neural networks. Phys. Usp. 1996;39(4):337-362. DOI: 10.1070/pu1996v039n04abeh000141.
  16. Arnold VI, Afraimovich VS, Ilyashenko YuS, Shilnikov LP. Dynamical Systems V: Bifurcation Theory and Catastrophe Theory. Berlin: Springer; 1994. 274 p. DOI: 10.1007/978-3-642-57884-7.
  17. Berge P, Pomeau Y, Vidal C. Order within Chaos. Weinheim: Wiley;1987. 329 p.
  18. Neimark YuI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Kluwer Academic Publishers; 1992. 500 p.
  19. Pastor-Diaz I, Lopez-Fraguas А. Dynamics оf two Van der Pol oscillators. Phys. Rev. В. 1995;52(2):1480-1489. DOI: 10.1103/PhysRevE.52.1480.
  20. Panter PF. Modulation, Noise, and Spectral Analysis, Applied to Information Tranmission. New York: McGraw—Hill; 1965. 759 p.
  21. Pikovsky AS, Rosenblum MG, Osipov GV, Kurths J. Phase synchronization of chaotic oscillators by external driving. Physica D. 1997;104(3-4):219-238. DOI: 10.1016/S0167-2789(96)00301-6.
  22. Pecora LM, Carroll TL. Sinchronization in chaotic system. Phys. Rev. Lett. 1990;64(8):821-824. DOI: 10.1103/PhysRevLett.64.821.
  23. Volkovskii AR, Rulkov NF. Sinchronous chaotic response of а nonlinear oscillator system as a principle for the detection of the information component of chaos. Tech. Phys. Lett. 1993;19(2):71–75.
  24. Kozlov AK, Shalfeev VD. Selective suppression оf deterministic chaotic signals. Tech. Phys. Lett. 1993;19(12):769-770.
Received: 
05.01.1999
Accepted: 
15.04.1999
Published: 
28.05.1999
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