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

For citation:

Vadivasova T. E., Malyaev V. S. 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, vol. 21, iss. 2, pp. 113-134. DOI: 10.18500/0869-6632-2013-21-2-113-134

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2013no2p113.pdf
(downloads: 188)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
537.86/87:530.182
DOI: 
10.18500/0869-6632-2013-21-2-113-134

Bifurcations in van der pol oscillator with a hard excitation in a presence of parametrical noise: quasi-harmonic analyzes and the numerical simulations

Autors: 
Vadivasova Tatjana Evgenevna, Saratov State University
Malyaev Vladimir Sergeevich, Saratov State University
Abstract: 

In the work the behavior of a van der Pol oscillator with a hard excitation is considered near the excitation threshold under parametrical (multiplicative) Gaussian white noise disturbances, and in a case of the two noise sources presence: parametrical one and additive noise. The evolution of probability distribution is studied when a control parameter and a noise intensity are changed. A comparison of the theoretical results, obtained in the quasi-harmonic approach with the results  of numerical solutions of the oscillator stochastic equations is fulfilled.

Key words: 
noise
fluctuations
self-sustained oscillator
stochastic bifurcations
Subcritical Andronov–Hopf bifurcation
Reference: 
  1. Stratonovich RL. Selected questions of fluctuation theory in radio engineering.Moscow: Sovetskoe radio; 1961. 558 p. (In Russian).
  2. Malakhov AN. Fluctuations in self-oscillating systems. Moscow: Nauka; 1968. 660 p. (In Russian).
  3. Ventzel AD, Freidlin MI. Fluctuations in dynamic systems under the influence of small random disturbances. Moscow: Nauka; 1979. 424 p.
  4. Gardiner KV. Stochastic methods in the natural sciences. Moscow: Mir; 1986. 526 p. (In Russian).
  5. Risken Z. The Fokker-Planck Equation. Berlin: Springer; 1989. 485 p.
  6. Van Kampen NG. Stochastic processes in physics and chemistry. Moscow: Vyshaya shkola; 1990. 376 p. (In Russian).
  7. Klyatskin VI. Statistical description of dynamic systems with fluctuating parameters. Moscow: Nauka; 1975. 239 p. (In Russian).
  8. Horstnemke B, Lefebvre R. Noise-induced transitions. Moscow: Mir; 1987. 400 p. (In Russian).
  9. Arnold L. Random dynamical systems. Berlin: Spriger; 2003. 586 p.
  10. Kabashima S, Kawakubo T. Observation of a noise induced phase transition in a parametric oscillator. Phys. Lett. A. 1979;70(5-6):375–376. DOI: 10.1016/0375-9601(79)90335-9.
  11. Wiesenfeld K. Noisy precursors of nonlinear instabilities. J. Stat. Phys. 1985;58(5):1071–1097.
  12. Ebeling W, Herzel H, Richert W, Schimansky-Geier L. Influence of noise on Duffing–van der Pol oscillator. Zeitschrift f. Angew. Math. U. Mechanik. 1986;66(3):141–146. DOI:10.1002/zamm.19860660303.
  13. Franzoni L, Mannella R, McClintock P, Moss F. Postponement of Hopf bifurcations by multiplicative colored noise. Phys. Rev. F. 1987;36(2):834–841. DOI:10.1103/PhysRevA.36.834.
  14. Namachshivaya NS. Stochastic bifurcation. Apl. Math. and Computation. 1990;38:101.
  15. Schenk-Yoppe KR. Bifurcation scenarious of the noisy Duffing–van der Pol oscillator. Nonlinear Dynamics. 1996;11:255–274.
  16. Landa PS, Zaikin AA. Noise-induced phase transitions in a pendulum with a randomly vibrating suspension axis. Phys. Rev. E. 1996;54(4):3535–3544. DOI: 10.1103/physreve.54.3535.
  17. Crauel H, Flandol F. Additive noise destroys a pitchfork bifurcation. Journal of Dynamics and Differential Equations. 1998;10(2):259–274. DOI:10.1023/A:1022665916629
  18. Vadivasova TE, Anishchenko VS. Stochastic bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(5):3-16. DOI: 10.18500/0869-6632-2009-17-5-3-16.
  19. 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.
  20. Arnold L, Sri Namachshivaya N, Schenk-Yoppe KR. Toward an understanding ofstochastic Hopf bifurcation: A base study. Int. J. Bifurcation and Chaos. 1996;6(11):1947–1975. DOI:10.1142/S0218127496001272.
  21. Olarrea J, de la Rubia FJ. Stochastic Hopf bifurcation: The effect of colored noise on the bifurcational interval. Phys. Rev. E. 1996;53(1):268–271. DOI: 10.1103/physreve.53.268.
  22. Zakharova A, Vadivasova TE, 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(1–6). DOI: 10.1103/PhysRevE.81.011106.
  23. 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(1–7). DOI: 10.1103/PhysRevE.83.056215.
  24. Bashkirtseva I, Ryashko L, Schurz H. Analysis of noise-induced transitions for Hopf system with additive and multiplicativt random disturbances. Chaos, Solitons, and Fractals. 2009;39(1):72–82. DOI:10.1016/j.chaos.2007.01.128.
  25. Medvedev SYU, Muzychuk OV. Statistical characteristics of a nonlinear resonant system parametrically excited by random force. Radiophysics and Quantum Electronics. 1981;24(1):49–58.
  26. Bashkirceva IA, Perevalova TV, Ryashko LB. Analysis of noise­induced bifurcations for the Hopf system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):37-50. DOI: 10.18500/0869-6632-2010-18-1-37-50.
  27. Tikhonov VI, Mironov MA. Markov processes. Moscow: Sovetskoe radio; 1977. 488 p. (In Russian).
  28. Ritov SM. Introduction to statistical radiophysics. Moscow: Nauka; 1976. 491 p. (In Russian).
  29. Stratonovich RL, Romanovsky YuM. Simultaneous parametric effect of harmonic and random force on oscillating systems. Nauchnie doklady vyshey scholi. Fizmat nauki. 1958;4:161–1969.
  30. Muzychuk OV. On probabilistic characteristics of resonant stochastic system. Radiophysics and Quantum Electronics. 1980;23(6):707–713.
  31. Leonov GA, Kuznetsov NV. Time-varing linearization and the Perron effects. Internat. Journal of Bifurcation and Chaos. 2007;17(4):1079–1107. DOI:10.1142/S0218127407017732.
  32. Nikitin NN, Razevig VD. Methods for the digital simulation of stochastic differential equations and an estimate of their errors. Zh. Vychisl. Mat. Mat. Fiz., 18:1 (1978), 106–117.
Received: 
27.09.2012
Accepted: 
26.01.2013
Published: 
31.07.2013
Short text (in English):
PDF icon a2013no2p113.pdf
(downloads: 67)
  • 1766 reads