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


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Bashkirtseva I. A., Ryashko L. B., Fedotov S. P., Tsvetkov I. N. Backward stochastic bifurcations of the henon map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 31-42. DOI: 10.18500/0869-6632-2011-19-2-31-42

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Language: 
Russian
Article type: 
Article
UDC: 
531.36

Backward stochastic bifurcations of the henon map

Autors: 
Bashkirtseva Irina Adolfovna, 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
Fedotov Sergej Petrovich, School of Mathematics, University of Manchester
Tsvetkov Ivan Nikolaevich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Abstract: 

We study the stochastically forced limit cycles of discrete dynamical systems in a period-doubling bifurcation zone. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation. In this paper, for such a bifurcation analysis we suggest a stochastic sensitivity function technique. The constructive possibilities of this method are demonstrated for analysis of the two-dimensional Henon model.

Reference: 
  1. Stratonovich RL. Topics in the Theory of Random Noise. New York: Gordon and Breach; 1963. 306 p.
  2. Horsthemke W, Lefever R. Noise-Induced Transitions. Berlin: Springer; 1984. 322 p. DOI: 10.1007/3-540-36852-3.
  3. Landa PS, McClintock PVE. Changes in the dynamical behavior of nonlinear systems induced by noise. Physics Reports. 2000;323(1):1–80. DOI: 10.1016/S0370-1573(99)00043-5.
  4. Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Physics Reports. 2004;392(6):321–424. DOI: 10.1016/j.physrep.2003.10.015.
  5. Gammaitoni L, et al. Stochastic resonance. Rev. Mod. Phys. 1998;70(1):223–287. DOI: 10.1103/RevModPhys.70.223.
  6. McDonnell MD, Stocks NG, Pearce CEM, Abbott D. Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization. Cambridge University Press; 2008. 425 p. DOI: 10.1017/CBO9780511535239.
  7. Matsumoto K, Tsuda I. Noise-induced order. J. Stat. Phys. 1983;31(1):87–106. DOI: 10.1007/BF01010923.
  8. Gassmann F. Noise-induced chaos-order transitions. Phys. Rev. E. 1997;55(3):2215–2221. DOI: 10.1103/PhysRevE.55.2215.
  9. Gao JB, Hwang SK, Liu JM. When can noise induce chaos? Phys. Rev. Lett. 1999;82(6):1132–1135. DOI: 10.1103/PhysRevLett.82.1132.
  10. Mayer-Kress G, Haken H. The influence of noise on the logistic model. J. Stat. Phys. 1981;26(1):149–171. DOI: 10.1007/BF01106791.
  11. Anishchenko VS. Complex Vibrations in Simple Systems. Moscow: Nauka; 1990. 312 p. (in Russian).
  12. 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.
  13. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically and periodically forced Brusselator. Physica A. 2000;278(1–2):126–139. DOI: 10.1016/S0378-4371(99)00453-7.
  14. Fedotov S, Bashkirtseva I, Ryashko L. Stochastic dynamo model for subcritical transition. Phys. Rev. E. 2006;73(6):066307. DOI: 10.1103/PhysRevE.73.066307.
  15. Arnold L. Random Dynamical Systems. Berlin: Springer-Verlag; 1998. 586 p. DOI: 10.1007/978-3-662-12878-7.
  16. Vadivasova TE, Anishchenko VS. Stochastic bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(5):3–16 (in Russian). DOI: 10.18500/0869-6632-2009-17-5-3-16.
  17. Lefever R, Turner J. Sensitivity of a Hopf bifurcation to external multiplicative noise. In: Horsthemke W, Kondepudi DK. Fluctuations and Sensitivity in Equilibrium Systems. Berlin: Springer; 1984. P. 143–149. DOI: 10.1007/978-3-642-46508-6_15.
  18. 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.
  19. 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.
  20. Arnold L, Bleckert G, Schenk-Hoppe K. The stochastic Brusselator: Parametric noise destroys Hopf bifurcation. In: Stochastic Dynamics. New-York: Springer; 1999. P. 71–92. DOI: 10.1007/0-387-22655-9_4.
  21. Namachchivaya NS. Hopf bifurcation in the presence of both parametric and external stochastic excitations. J. Appl. Mech. 1988;55(4):923–930. DOI: 10.1115/1.3173743.
  22. Schenk-Hoppe KR. Bifurcation scenarios of the noisy Duffing–van der Pol oscillator. Nonlinear Dynamics. 1996;11(3):255–274. DOI: 10.1007/BF00120720.
  23. Leung HK. Stochastic Hopf bifurcation in a biased van der Pol model. Physica A. 1998;254(1):146–155. DOI: 10.1016/S0378-4371(98)00017-X.
  24. 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.
  25. 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.
  26. Crutchfield JP, Nauenberg M, Rudnick J. Scaling for external noise at the onset of chaos. Phys. Rev. Lett. 1981;46(14):933–935. DOI: 10.1103/PhysRevLett.46.933.
  27. Crutchfield JP, Farmer J, Huberman BA. Fluctuation and simple chaotic dynamics. Phys. Rep. 1982;92(2):45–82. DOI: 10.1016/0370-1573(82)90089-8.
  28. Kuznetsov AP, Kapustina YV. Scaling property in the transition to chaos in model mappings with noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):78–87 (in Russian).
  29. Kuznetsov AP, Kuznetsov SP, Sedova YV. About scaling properties in the noisy circle map at the golden-mean winding number. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(5–6):56–76 (in Russian). DOI: 10.18500/0869-6632-2005-13-5-56-76.
  30. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of stochastically forced Lorenz model cycles under period-doubling bifurcations. Dynamic Systems and Applications. 2002;11(2):293–309.
  31. Ryagin M, Ryashko L. The analysis of the stochastically forced periodic attractors for Chua’s circuit. Int. J. Bifurcation Chaos. 2004;14(11):3981–3987. DOI: 10.1142/S0218127404011600.
  32. Bashkirtseva IA, Ryashko LB. Sensitivity and chaos control for the forced nonlinear oscillations. Chaos, Solitons and Fractals, 2005;26(5):1437–1451. DOI: 10.1016/j.chaos.2005.03.029.
  33. Bashkirtseva I, Ryashko L, Tsvetkov I. Sensitivity analysis of stochastic equilibria and cycles for the discrete dynamic systems. Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis. 2010;17(4):501–515.
  34. Bashkirtseva I, Ryashko L, Stikhin P. Noise-induced backward bifurcations of stochastic 3D-cycles. Fluctuation and Noise Letters. 2010;9(1):89–106. DOI: 10.1142/S0219477510000095.
  35. Bashkirtseva IA, Ryashko LB, Fedotov SP, Tsvetkov IN. Backward stochastic bifurcations of the discrete system cycles. Russian Journal of Nonlinear Dynamics. 2010;6(4):737–753 (in Russian).
  36. Elaydi SN. An Introduction to Difference Equations. Springer: Berlin; 1999. 540 p. DOI: 10.1007/0-387-27602-5.
Received: 
14.07.2010
Accepted: 
29.03.2011
Published: 
31.05.2011
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