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Semenov V. V., Listov A. S., Vadivasova T. E. Experimental study of stochastic phenomena in a self­sustained oscillator with subcritical andronov–hopf bifurcation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 5, pp. 43-57. DOI: 10.18500/0869-6632-2014-22-5-43-57

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537.86; 519.21

Experimental study of stochastic phenomena in a self­sustained oscillator with subcritical andronov–hopf bifurcation

Semenov V. V., Saratov State University
Listov Aleksandr Serafimovich, Saratov State University
Vadivasova Tatjana Evgenevna, Saratov State University

The effect of noise on the self­sustained oscillator near subcritical Andronov–Hopf bifurcation is studied in numerical and full­scale experiments. Van der Pol oscillator is chosen as base model for investigation. The influence of both additive and multiplicative Gaussian white noise is considered. The regularities of evolution of the probability distribution in the self­sustained oscillator are analyzed with increase of the noise intensity for the cases of additive and parametric noise. The existence of a bifurcation interval is established experimentally for subcritical Andronov–Hopf bifurcation in the presence of additive noise. Besides of this, the  existence of a bifurcation interval is shown for the tangent bifurcation. The postponed character of the Andronov–Hopf bifurcation is confirmed for a multiplicative (parametric) noise excitation. The results of the full­scale modeling are compared with the numerical data.

  1. Horsthemke W, Lefever R. Noise Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Berlin: Springer; 1984. 322 p. DOI: 10.1007/3-540-36852-3.
  2. Arnold L. Random Dynamical Systems. Berlin: Springer; 1998. 586 p. DOI: 10.1007/978-3-662-12878-7.
  3. Ebeling W, Herzel H, Richert W, Schimansky-Geier L. Influence of noise on Duffing–van der Pol oscillators. Zeischrift angewandte Mathematik und Mechanik (ZAMM). 1986;66(3):141—146. DOI: 10.1002/zamm.19860660303.
  4. Lefever R, Turner J. Sensitivity of a Hopf bifurcation to multiplicative colored noise. Phys. Rev. Lett. 1986;56(16):1631—1634. DOI: 10.1103/PhysRevLett.56.1631.
  5. Franzoni L, Mannella R, McClintock P, Moss F. Postponement of Hopf bifurcations by multiplicative colored noise. Phys. Rev. A. 1987;36(2):834—841. DOI: 10.1103/PhysRevA.36.834.
  6. Namachchivaya NS. Stochastic bifurcation. Applied Mathematics and Computation. 1990;39(3):37s—95s. DOI: 10.1016/0096-3003(90)90003-L.
  7. Arnold L, Namachchivaya NS, Schenk-Yoppe KR. Toward an understanding of stochastic Hopf bifurcation: A base study. International Journal of Bifurcation and Chaos. 1996;6(11):1947—1975. DOI: 10.1142/S0218127496001272.
  8. Olarrea J, de la Rubia FJ. Stochastic Hopf bifurcation: The effect of colored noise on the bifurcation interval. Phys. Rev. E. 1996;53(1):268—271. DOI: 10.1103/PhysRevE.53.268.
  9. Schenk-Yoppe KR. Bifurcation scenarious of the noisy Duffing–van der Pol oscillator. Nonlinear Dynamics. 1996;11(3):255—271. DOI: 10.1007/BF00120720.
  10. Bashkirtseva I, Ryashko L, Schurz H. Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances. Chaos, Solitons, and Fractals. 2009;39(1):72—82. DOI: 10.1016/j.chaos.2007.01.128.
  11. Semenov VV, Vadivasova TE, Anishchenko VS. Experimental investigation of probability distribution in self-sustained oscillators with additive noise. Tech. Phys. Lett. 2013;39(7):632—635. DOI: 10.1134/S1063785013070213.
  12. Semenov VV, Zakoretskii KV, Vadivasova TE. Experimental investigation of stochastic Andronov–Hopf bifurcation in self-sustained oscillators with additive and parametric noise. Russian Journal of Nonlinear Dynamics. 2013;9(3):421—434. DOI: 10.20537/nd1303003.
  13. Bashkirceva IA, Perevalova TV, Ryashko LB. Analysis of noise­induced bifurcations for the Hopf system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):37—50 (in Russian). DOI: 10.18500/0869-6632-2010-18-1-37-50.
  14. Stratonovich RL. Random Processes in Dynamic Systems. Moscow; Izhevsk: Izhevsk Institute of Computer Research; 2009. 529 p. (in Russian).
  15. Ushakov OV, Wunsche HJ, Henneberger F, Khovanov IA, Schimansky-Geier L, Zaks MA. Coherence resonance near a Hopf bifurcation. Phys. Rev. Lett. 2005;95(12):123903. DOI: 10.1103/PhysRevLett.95.123903.
  16. Vadivasova TE, Zaharova AS, Anishchenko VS. Noise-induced bifurcations in bistable oscillator. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(2):114—122 (in Russian). DOI: 10.18500/0869-6632-2009-17-2-114-122.
  17. Zakharova A, Vadivasova T, Anishchenko V, Koseska A, Kurths J. Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator. Phys. Rev. E. 2010;81(1):011106. DOI: 10.1103/PhysRevE.81.011106.
  18. Xu Y, Gu R, Zhang H, Xu W, Duan J. Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys. Rev. E. 2011;83(5):056215. DOI: 10.1103/PhysRevE.83.056215.
  19. Vadivasova TE, Maljaev VS. Bifurcations in van der Pol oscillator with a hard excitation in a presence of parametrical noise: quasi-harmonic analyzes and the numerical simulations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2013;21(2):113—134 (in Russian). DOI: 10.18500/0869-6632-2013-21-2-113-134
  20. Andronov AA, Witt AA, Khaikin SE. Theory of Oscillators. Oxford: Pergamon Press; 1966. 916 p.
  21. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
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