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


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Kochanov A. A., Vadivasova T. E., Anishchenko V. S. Noise induced parametric instability and stochastic oscillations in the oscillator with nonlinear dissipation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 43-55. DOI: 10.18500/0869-6632-2011-19-2-43-55

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Russian
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Article
UDC: 
537.86:519.21

Noise induced parametric instability and stochastic oscillations in the oscillator with nonlinear dissipation

Autors: 
Vadivasova Tatjana Evgenevna, Saratov State University
Anishchenko Vadim Semenovich, Saratov State University
Abstract: 

The appearance of the instability of oscillator equilibrium state in a case of noisy modulation of the natural frequency is considered in the work. The threshold of instability and the properties of stochastic oscillations arising over the threshold are studied for the different noise characteristics.

Reference: 
  1. Stratonovich RL. Selected Questions of the Theory of Fluctuations in Radio Engineering. Moscow: Sovetskoe Radio; 1961. 560 p. (in Russian).
  2. Wentzell AD, Freidlin MI. Random Perturbations of Dynamical Systems. Moscow: Nauka; 1979. 424 p. (in Russian).
  3. Horsthemke W, Lefever R. Noise-Induced Transitions. Berlin: Springer; 1984. 322 p. DOI: 10.1007/3-540-36852-3.
  4. Gardiner KV. Stochastic Methods in Natural Sciences. Moscow: Mir; 1986. 538 p. (in Russian).
  5. Risken Z. The Fokker-Planck Equation. Berlin: Springer; 1989. 472 p. DOI: 10.1007/978-3-642-61544-3.
  6. Van Kampen NG. Stochastic Processes in Physics and Chemistry. North Holland Library; 1981. 419 p.
  7. Arnold L. Random Dynamical Systems. Berlin: Springer-Verlag; 1998. 586 p. DOI: 10.1007/978-3-662-12878-7.
  8. Anishchenko VS, Astakhov VV, Vadivasova TE, Neiman AB, Strelkova GI, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Berlin: Springer; 2007. 446 p. DOI: 10.1007/978-3-540-38168-6.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. Landa PS. Excitation of chaotic and stochastic oscillations in different systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):3–11 (in Russian). DOI: 10.18500/0869-6632-2010-18-1-3-11.
  14. Akhmanov SA, Dyakov YE , Chirkin AS. Introduction to Statistical Radiophysics and Optics. Moscow: Nauka; 1981. 640 p.
  15. Bobryk RV, Chrzeszczyk A. Colored-noise-induced parametric resonance. Physica A. 2002;316(1–4):225–232. DOI: 10.1016/S0378-4371(02)01312-2.
  16. Gitterman M. The Noisy Oscillator: The First Hundred Years, From Einstein Until Now. Singapore: World Scientific; 2005. 160 p. DOI: 10.1142/5949.
  17. Aumaitre S, Mallick K, Franсois P. Noise-induced bifurcations, multiscaling and on-off intermittency. J. Stat. Mech. 2007;(7):P07016. DOI: 10.1088/1742-5468/2007/07/P07016.
  18. Sirotkin OL. Features of the moment functions of an oscillator with parametric instability due to dichotomous noise with Erlang distribution functions. Radiophys. Quantum El. 2009;52(11):832–842. DOI: 10.1007/s11141-010-9190-3.
  19. Kapitsa PL. Dynamic stability of the pendulum at an oscillating suspension point. Sov. Phys. JETP. 1951;21(5):588–597 (in Russian).
  20. Anishchenko V, Vadivasova T, Strelkova G. Stochastic self-sustained oscillations of non-autonomous system. The European Physical Journal Special Topics. 2010;87(1):109–125. DOI: 10.1140/epjst/e2010-01276-1.
  21. Anishchenko VS, Vadivasova TE, Strelkova GI. Self-sustained oscillations of dynamical and stochastic systems and their mathematical image – an attractor. Russian Journal of Nonlinear Dynamics. 2010;6(2):107–126 (in Russian).
Received: 
02.12.2010
Accepted: 
02.12.2010
Published: 
31.05.2011
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