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


For citation:

Bashkirtseva I. A., Ryashko L. B., Tsvetkov I. N. Stochastic sensitivity of equilibrium and cycles for 1D discrete maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 6, pp. 74-85. DOI: 10.18500/0869-6632-2009-17-6-74-85

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):
(downloads: 201)
Language: 
Russian
Article type: 
Article
UDC: 
531.36

Stochastic sensitivity of equilibrium and cycles for 1D discrete maps

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
Tsvetkov Ivan Nikolaevich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Abstract: 

The response problem of equilibrium and cycles for stochastically forced Verhulst population model is considered. Theoretical and empirical approaches are used for stochastically sensitivity analysis. The theoretical approach is based on the firth approximation method and the empirical approach is based on direct numerical simulation. The correspondence between the two approaches for Verhulst population model is demonstrated. The increase of discrete system sensitivity to external noise in the period­doubling bifurcation zone under transition to chaos is shown.

Reference: 
  1. Polak LS, Mikhailov AS. Self-organization in non-equilibrium physical and chemical systems. Moscow: Nauka; 1983. 285 p. (In Russian).
  2. Stanley H. Introduction to phase transitions and critical phenomena. United Kingdom: Clarendon Press; 1973. 419 p.
  3. Haken H. Synergetics – a field beyond irreversible thermodynamics. Lect. Notes in Phys. Berlin: Springer; 1978. Vol. 84. P. 140.
  4. Klimontovich YuL. Statistical physics. Moscow: Nauka; 1983. 608 p. (In Russian).
  5. Sinai JG. Theory of phase transitions. Moscow: Nauka; 1980. 207 p. (In Russian).
  6. Wilson KG. The renormalization group and critical phenomena. UFN. 1983;141(2):193–220. DOI: 10.3367/UFNr.0141.198310a.0193.
  7. Hu B. Intoduction to real-space renormalizatin-group methods in critical and chaotic phenomen. Phys. Rep. 1982;91(5):233–295. DOI: 10.1016/0370-1573(82)90057-6.
  8. Elaydi SN. An introduction to difference equations. New York: Springer; 1999. 540 p.
  9. Schuster G. Deterministic chaos: Introduction. Moscow: Mir; 1988. P. 115. (In Russian).
  10. Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 1978;19(1):25–52. DOI: 10.1007/BF01020332
  11. Feigenbaum MJ. The universal metric properties of nonlinear transformations. J. Stat. Phys. 1979;21(6):669–706. DOI: 10.1007/BF01107909.
  12. Feigenbaum MJ. The transition to aperiodic behavior in turbulent systems. Comm. Math. Phys. 1980;77(1):65–86. DOI: 10.1007/BF01205039.
  13. Hauser PR, Tsallis C, Curado MF. Criticallity of the routes to chaos of the 1 − α|x| z map. Phys. Rev. A. 1984;30(4):2074.
  14. Derida B, Gervois A, Pomeau Y. Universal metric properties of bifurcations of endomorphisms. J. Phys. A. 1979;12(3):269–296. DOI: 10.1088/0305-4470/12/3/004.
  15. Pikovsky AS. On the stochastic properties of the simplest model of stochastic self-oscillations. Radiophysics and Quantum Electronics. 1980;23(7):883–884.
  16. Huberman BA, Rudnick J. Scaling behavior of chaotic flows. Phys. Rev. Lett. 1980;45(3):154–156. DOI: 10.1103/PhysRevLett.45.154.
  17. Huberman BA, Zisook AB. Power spectra of strange attractors. Phys. Rev. Lett. 1981;46(10):626–628. DOI: 10.1103/PhysRevLett.46.626.
  18. Huberman BA, Hirisch JE, Scalapino DJ. Theory of intermittency. Phys. Rev. A. 1982;25(1):519–532. DOI: 10.1103/PHYSREVA.25.519.
  19. Anishchenko VS. Stochastic oscillations in radiophysical systems. Vol.1,2. Saratov: Saratov University Publishing; 1986. 197 p. (In Russian).
  20. Neimark YuI. On the occurrence of stochasticity in dynamic systemsRadiophysics and Quantum Electronics. 1974;17(4):602–607.
  21. Neymark YuI, Landa PS. Stochastic and chaotic fluctuations. Moscow: Nauka; 1987. 422 p. (In Russian).
  22. Crutchfield JP, Farmer J, Huberman BA. Fluctation and simple chaotic dynamics. Phys. Rep. 1982;92(2):45–82. DOI: 10.1016/0370-1573(82)90089-8.
  23. Crutchfield JP, Packard NH. Symbolic dynamics of noisy chaos. Physica D. 1983;7(1-3):201–223. DOI: 10.1016/0167-2789(83)90127-6.
  24. Gutierrez J, Iglesias A, Rodiguez MA. Logistic map driven by dichotomous noise. Phys. Rev. E. 1993;48(4):2507–2513. DOI: 10.1103/PhysRevE.48.2507.
  25. Hall P, Wolf RCL. Properties of invariant distributions and Lyapunov exponents for chaotic logistic maps. Journal of the Royal Statistical Society. 1995;57():439–452. DOI: 10.1111/J.2517-6161.1995.TB02038.X.
  26. Linz SJ, Lucke M. Parametric modulation of instabilities of a nonlinear discrete system. Phys. Rev. A. 1986;33:2694.
  27. Kuznetsov AP, Kapustina JV. Scaling properties at transition to chaos in model maps in the presence of noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):78–87.
  28. 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(6):56–76. DOI: 10.18500/0869-6632-2005-13-5-56-76.
  29. Crutchfield JP, Nauenberg M, Rudnick J. Scaling for external noise at the onset of chaos. Phys. Rev. Lett. 1981;46(14):933–935.
  30. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically and periodically forced brusselator. Phys. A. 2000;278(1):126--139. DOI: 10.1016/S0378-4371(99)00453-7.
  31. 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.
  32. Ryashko LB, Bashkirtseva IA, Stichin PV. Stochastical sensitivity of cycles of roessler system in transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):32–47.
  33. Bashkirtseva I, Ryashko L. Stochastic sensitivity of 3D-cycles. Mathematics and Computers in Simulation. 2004;66(1):55–67. DOI: 10.1016/j.matcom.2004.02.021.
  34. Bashkirtseva I, Ryashko L. Sensitivity and chaos control for the forced nonlinear oscillations. Chaos, Solitons & Fractals. 2005;26(5):1437–1451. DOI: 10.1016/j.chaos.2005.03.029.
Received: 
04.02.2008
Accepted: 
06.04.2009
Published: 
31.12.2009
Short text (in English):
(downloads: 117)