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Vadivasova T. E., Anishchenko V. S. Stochastic bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 5, pp. 3-16. DOI: 10.18500/0869-6632-2009-17-5-3-16

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Stochastic bifurcations

Vadivasova Tatjana Evgenevna, Saratov State University
Anishchenko Vadim Semenovich, Saratov State University

The modern knowledges of bifurcations of dynamical systems in the presence of noise are presenred. The main definitions are given and certain typical examples of the bifurcations in the presence of additive and multiplicative noise are considered.

  1. Arnol'd VI, Afraimovich VS, Ilyashenko YuS, Shilnikov LP. Bifurcation theory. Dynamical systems – 5, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 5. Moscow: VINITI; 1986. 5–218 p. (In Russian).
  2. Anishchenko V, Astakhov V, Neiman A, Vadivasova T, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Second Edition. Berlin: Springer; 2007. 446 p.
  3. Andronov AA, Witt AA, Khaikin SE. Theory of Oscillators. Oxford: Pergamon Press; 1966. 916 p.
  4. Horsthemke W, Lefever R. Noise-induced Transitions. Berlin: Springer; 1984. 322 p.
  5. Arnold L. Random dynamical systems. Chapter 9. Bifurcation theory. Berlin: Spriger; 2003. 586 p.
  6. 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.
  7. Fronzoni L, Mannella R, McClintock PV, Moss F. Postponement of Hopf bifurcations by multiplicative colored noise. Phys Rev A Gen Phys. 1987;36(2):834-841. DOI: 10.1103/physreva.36.834.
  8. Sri Namachshivaya N. Stochastic bifurcation. Appl. Math. and Computation. 1990;38(2):101–159. DOI: 10.1016/0096-3003(90)90051-4.
  9. Arnold L, Sri Namachshivaya N, Schenk-Yoppe’ KR. Toward an understanding of stochastic Hopf bifurcation: a base study. Int. J. Bifurcation and Chaos. 1996;6(11):1947–1975. DOI: 10.1142/S0218127496001272.
  10. Schenk-Yoppe’ KR. Bifurcation scenarious of the noisy Duffing–van der Pol oscillator. Nonlinear Dynamics. 1996;11:255–274. DOI: 10.1007/BF00120720.
  11. 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.
  12. Landa PS, Zaikin AA. Noise-induced phase transitions in a pendulum with a randomly vibrating suspension axis. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1996;54(4):3535-3544. DOI: 10.1103/physreve.54.3535.
  13. 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.
  14. Leung HK. Stochastic Hopf bifurcation in a based van der Pol model. Physica A. 1998;254(1):146–155. DOI: 10.1016/S0378-4371(98)00017-X.
  15. 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.
  16. Ebeling W, Herzel H, Richert W, Schimansky-Geier L. Influence of noise on Duffing–van der Pol oscillator. Zeitschrift f. Angew. Math. und Mechanik. 1986;66(3):141–146. DOI: 10.1002/zamm.19860660303.
  17. Schimansky-Geier L, Herzel H. Positive Lyapunov exponents in the Kramers oscillator. J. of Stat. Physics. 1993;70:141–147. DOI: 10.1007/BF01053959.
  18. Matsumoto K, Tsuda I. Noise-induced order. J. Stat. Phys. 1983;31(1):87–106. DOI: 10.1007/BF01010923.
  19. Hramov AE, Koronovskii AA, Moskalenko OI. Are generalized synchronization and noise-induced synchronization identical types of synchronous behavior of chaotic oscillators? Phys. Lett. A. 2006;354(5-6):423–427. DOI: 10.1016/j.physleta.2006.01.079.
  20. Stratonovich RL. Selected Questions of the Theory of Fluctuations in Radio Engineering. Moscow: Sovetskoe Radio; 1961. 560 p. (in Russian).
  21. Malakhov AN. Fluctuations in self-oscillating systems. Moscow: Nauka; 1968. 660 p. (In Russian).
  22. Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance. J. Phys. A: Math. Gen. 1981;14(11):L453—L457. DOI: 10.1088/0305-4470/14/11/006.
  23. Gammaitoni L, Marchesoni F, Menichella-Saetta E, Santucci S. Stochastic resonance in bistable systems. Phys Rev Lett. 198923;62(4):349-352. DOI: 10.1103/PhysRevLett.62.349.
  24. Pikovsky A, Kurths J. Coherence resonance in a noisy driven excitable system. Phys. Rev. Lett. 1997;78(5):775–778. DOI: 10.1103/PHYSREVLETT.78.775.
  25. Lindner B, Schimansky-Geier L. Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1999;60(6):7270–7276. DOI: 10.1103/physreve.60.7270.
  26. Shulgin B, Neiman A, Anishchenko V. Mean switching frequency locking in stochastic bistable system driven by a periodic force. Phys. Rev. Lett. 1995;75(23):4157–4160. DOI: 10.1103/PHYSREVLETT.75.4157.
  27. Anishchenko V, Neiman B. Stochastic synchronization. Stochastic Dynamics. Eds. L. Schimansky-Geier and T. Poschel. Berlin: Springer, 1997. P. 155.
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