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Bashkirtseva I. A., Perevalova T. V., Ryashko L. B. Analysis of noise­induced bifurcations for the Hopf system. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 1, pp. 37-50. DOI: 10.18500/0869-6632-2010-18-1-37-50

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Analysis of noise­induced bifurcations for the Hopf system

Bashkirtseva Irina Adolfovna, Ural Federal University named after the first President of Russia B.N.Yeltsin
Perevalova Tatjana Vladimirovna, Ural Federal University named after the first President of Russia B.N.Yeltsin
Ryashko Lev Borisovich, Ural Federal University named after the first President of Russia B.N.Yeltsin

We consider the Hopf system as a classical model of a stiff birth of a cycle. In the presence of parametrical and additive random disturbances, various types of the stochastic attractors are observed. The solution of the corresponding Fokker–Planck–Kolmogorov equation is found. The qualitative changes of the form for stochastic attractors under multiplicative noise are shown. The phenomenon of backward stochastic bifurcations is described in details.

  1. Stratonovich RL, Landa PS. The effect of noise on a generator with rigid excitation. Radiophysics and Quantum Electronics. 1959;2(1):3744 (in Russian).
  2. Horsthemke W, Lefever R. Noise-Induced Transitions. Moscow: Mir; 1987. 400 p. (in Russian).
  3. Neimark YuI, Landa PS. Stochastic and Chaotic Oscillations. Moscow: Nauka; 1987. 424 p. (in Russian).
  4. Landa PS, McClintock PVE. Changes in the dynamical behavior of nonlinear systems induced by noise. Physics Reports. 2000;323:180. DOI: 10.1016/S0370-1573(99)00043-5.
  5. Fedotov S, Bashkirtseva I, Ryashko L. Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition. Phys. Rev. E. 2002;66:066310. DOI: 10.1103/PhysRevE.66.066310.
  6. Fedotov S, Bashkirtseva I, Ryashko L. Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo. Phys. A. 2004;342(3-4):491506. DOI: 10.1016/j.physa.2004.05.084.
  7. Pontryagin LS, Andronov AA, Witt AA. On the statistical investigation of a dynamical system. J. Exp. and Theoretical Phys. 1933;3:165180 (in Russian).
  8. Stratonovich RL. Selected Problems of the Theory of Fluctuations in Radio Engineering, Moscow: Sovetskoe Radio; 1961. (in Russian).
  9. Ibrahim RA. Parametric Random Vibration. New York: John Wiley and Sons; 1985.
  10. Soong TT, Grigoriu M. Random Vibration of Mechanical and Structural Systems. New Jersey: RTL Prentice-Hall, Englewood Cliffs; 1993.
  11. Venttsel’ AD, Freidlin MI. Fluctuations in Dynamical Systems under the Action of Small Random Perturbations. Moscow: Nauka; 1979. (in Russian).
  12. Dembo M, Zeitouni O. Large Deviations Techniques and Applications. Boston: Jones and Bartlett Publishers; 1995.
  13. Naeh T, Klosek MM, Matkowsky BJ, Schuss Z. A direct approach to the exit problem. SIAM J. Appl. Math. 1990;50(2):595627. DOI: 10.1137/0150036.
  14. Roy RV. Asymptotic analysis of first passage problem. Int. J. Non-Linear Mechanics. 1997;32:173186.
  15. Smelyanskiy VN, Dykman MI, Maier RS. Topological features of large fluctuations to the interior of a limit cycle. Physical Review E. 1997;55(3):23692391. DOI: 10.1103/PhysRevE.55.2369.
  16. Ryashko LB, Bashkirtseva IA, Stichin PV. Stochastical sensitivity of cycles of Ressler system in transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):3247 (in Russian).
  17. Bashkirtseva IA, Ryashko LB. Stochastic sensitivity of 3D-cycles. Mathematics and Computers in Simulation. 2004;66(1):5567. DOI: 10.1016/j.matcom.2004.02.021.
  18. Ryagin M, Ryashko L. The analysis of the stochastically forced periodic attractors for Chua’s circuit. Int. J. Bifurcation Chaos. 2004;14(11):39813987. DOI: 10.1142/S0218127404011600.
  19. Bashkirtseva IA, Ryashko LB. Sensitivity and chaos control for the forced nonlinear oscillations. Chaos, Solitons and Fractals. 2005;26(5):14371451. DOI: 10.1016/j.chaos.2005.03.029.
  20. Lefever R, Turner J. Sensitivity of a Hopf bifurcation to external multiplicative noise. In: Fluctuations and Sensitivity in Nonequilibrium Systems. Ed. by Horsthemke W, Kondepudi DK. Berlin: Springer. 1984:143149.
  21. Lefever R, Turner J. Sensitivity of a Hopf bifurcation to multiplicative colored noise. Phys. Rev. Lett. 1986;56(16):16311634. DOI: 10.1103/PhysRevLett.56.1631.
  22. Fronzoni L, Mannella R, McClintock P, Moss F. Postponement of Hopf bifurcations by multiplicative colored noise. Phys. Rev. A. 1987;36:834841. DOI: 10.1103/physreva.36.834.
  23. Arnold L, Bleckert G, Schenk-Hoppe K. The stochastic Brusselator: Parametric noise destroys Hopf bifurcation. In: Stochastic Dynamics. Bremen; 1997. New York: Springer; 1999:7192. 
  24. Malick K, Marcq P. Stability analysis of noise-induced Hopf bifurcation. Eur. Phys. J. 2003;36:119128. DOI: 10.1140/epjb/e2003-00324-y.
  25. Leung HK. Stochastic Hopf bifurcation in a biased van der Pol model. Physica A. 1998;254(1-2):146155. DOI: 10.1016/S0378-4371(98)00017-X.
  26. Namachchivaya NSri. Hopf bifurcation in the presence of both parametric and external stochastic excitations. J. Appl. Mech. 1988;55(4):923930. DOI: 10.1115/1.3173743.
  27. Schenk-Hoppe KR. Bifurcation scenarios of the noisy Duffing–van der Pol oscillator. Nonlinear dynamics. 1996;11:255274. DOI: 10.1007/BF00120720.
  28. 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;39:7282. DOI: 10.1016/j.chaos.2007.01.128.
  29. Gardiner KV. Stochastic Methods in the Natural Sciences. Moscow: Mir; 1986. (in Russian).
  30. Arnold L. Random Dynamical Systems. Berlin: Springer; 1998.
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