Journal of Statistical Physics

, Volume 121, Issue 5–6, pp 759–778 | Cite as

Weak Noise Approach to the Logistic Map

Article

Abstract

Using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1D maps. We replace the noise-driven 1D map by an area-preserving 2D map modelling the Poincare sections of a conserved dynamical system with unbounded energy manifolds. We analyze the properties of the 2D map and draw conclusions concerning the interference of noise on the nonlinear time evolution. We apply this technique to the standard period-doubling sequence in the logistic map. From the 2D area-preserving analogue we, in addition to the usual period-doubling sequence, obtain a series of period doubled cycles which are elliptic in nature. These cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events.

Keywords

Weak noise random maps Hamiltonian systems elliptic fixed points period doubling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Kramers, H.A. 1940Kramers Physica7284MATHMathSciNetGoogle Scholar
  2. Hänggi, P., Talkner, P., Borkovec, M. 1990Rev. Mod. Phys.62251CrossRefADSGoogle Scholar
  3. Graham, R., Tél, T. 1984J. Stat. Phys35729MATHADSGoogle Scholar
  4. Graham, R., Tél, T. 1984Phys. Rev. Lett529ADSMathSciNetGoogle Scholar
  5. Graham, R., Roekaerts, D., Tél, T. 1985Phys. Rev. A313364ADSMathSciNetGoogle Scholar
  6. Graham, R., Tél, T. 1986Phys. Rev. A331322ADSMathSciNetGoogle Scholar
  7. R. Graham, Noise in nonlinear dynamical systems, Vol. 1, in Theory of Continuous Fokker-Planck Systems, F. Moss and P. E. V. McClintock, eds. (Cambridge University Press, Cambridge, 1989).Google Scholar
  8. Noise in Nonlinear Dynamical Systems, F. Moss and P. V. E. McClintock, eds., (Cambridge University Press, Cambridge, 1989).Google Scholar
  9. Collet, P., Eckmann, J. P. 1980Iterated Maps on the Interval as Dynamical SystemBirkhauserBostonGoogle Scholar
  10. Grossmann, S., Thomae, S. 1977Z.Naturforsch321353MathSciNetADSGoogle Scholar
  11. Feigenbaum, M. J. 1978J. Stat. Phys925MathSciNetADSGoogle Scholar
  12. Feigenbaum, M. J. 1979Phys. Lett. A74375CrossRefADSMathSciNetGoogle Scholar
  13. Haken, H., Mayer-Kress, G. 1981Z. Phys. B43185CrossRefMathSciNetADSGoogle Scholar
  14. Linz, S. J., Lücke, M. 1986Phys. Rev. A332694CrossRefADSGoogle Scholar
  15. Crutchfield, J.R., Nauenberg, M., Rudnick, J. 1981Phys. Rev. Lett.46933CrossRefADSMathSciNetGoogle Scholar
  16. Shraiman, B., Wayne, C. E., Martin, P. C. 1981Phys. Rev. Lett.46935CrossRefADSMathSciNetGoogle Scholar
  17. Reimann, P., Talkner, P. 1991Hel. Phys. Acta64946Google Scholar
  18. Graham, R., Hamm, A., Tél, T. 1991Phys. Rev. Lett.663089CrossRefADSMathSciNetMATHGoogle Scholar
  19. Hirsch, J. E., Nauenberg, M., Scalapino, D. J. 1982Phys. Lett. A87391CrossRefADSMathSciNetGoogle Scholar
  20. Arecchi, F. T., Badii, R., Politi, A. 1984Phys. Lett. A1033CrossRefADSGoogle Scholar
  21. Grassberger, P. 1989J. Phys. A423283ADSMathSciNetGoogle Scholar
  22. Talkner, P., Haenggi, P., Freidkin, E., Trautmann, D. 1987J. Stat. Phys.48231CrossRefADSGoogle Scholar
  23. Crutchfield, J. R., Farmer, J. D., Huberman, B. A. 1982Phys. Rep.9245CrossRefADSMathSciNetGoogle Scholar
  24. Reimann, P., Talkner, P. 1991Phys. Rev. A446348CrossRefADSMathSciNetGoogle Scholar
  25. Reimann, P., Müller, R., Talkner, P. 1994Phys. Rev. E493670ADSMathSciNetGoogle Scholar
  26. Reimann, P., Talkner, P. 1995Phys. Rev. E514105CrossRefADSGoogle Scholar
  27. Cvitanovic, P., Dettmann, C. P., Mainieri, R., Vattay, G. 1998J. Stat. Phys.93981MathSciNetMATHADSGoogle Scholar
  28. Cvitanovic, P., Dettmann, C. P., Palla, G., Søndergaard, N., Vattay, G. 1999Phys. Rev. E603936ADSCrossRefGoogle Scholar
  29. Cvitanovic, P., Dettmann, C. P., Mainieri, R, Vattay, G. 1999Nonlinearity12939CrossRefADSMathSciNetMATHGoogle Scholar
  30. Ott, E. 1993Chaos in Dynamical SystemsCambridge University PressCambridgeMATHGoogle Scholar
  31. H. G. Schuster, Deterministic Chaos, An Introduction, 2nd ed. (VCH Verlagsgesellschaft mbH, Weinheim, 1989).Google Scholar
  32. Strogatz, S. H. 1994Nonlinear Dynamics and ChaosPerseus BooksReadingGoogle Scholar
  33. vanKampen, N. G. 1992Stochastic Processes in Physics and ChemistryNorth-HollandAmsterdamGoogle Scholar
  34. Freidlin, M.I., Wentzel, A.D. 1998Random Perturbations of Dynamical Systems2SpringerNew YorkMATHGoogle Scholar
  35. Martin, P. C., Siggia, E. D., Rose, H. A. 1973Phys. Rev. A8423ADSGoogle Scholar
  36. Baussch, R., Janssen, H. K., Wagner, H. 1976Z. Phys. B24113ADSGoogle Scholar
  37. Medina, E., Hwa, T., Kardar, M., Zhang, Y. C. 1989Phys. Rev. A393053CrossRefADSMathSciNetGoogle Scholar
  38. Kardar, M., Parisi, G., Zhang, Y. C. 1986Phys. Rev. Lett.56889CrossRefADSMATHGoogle Scholar
  39. Fogedby, H. C., Eriksson, A. B., Mikheev, L. V. 1995Phys. Rev. Lett.751883CrossRefADSMathSciNetMATHGoogle Scholar
  40. Fogedby, H. C. 1998Phys. Rev. E574943ADSMathSciNetGoogle Scholar
  41. Fogedby, H. C. 1999Phys. Rev. E595065CrossRefADSMathSciNetGoogle Scholar
  42. Fogedby, H. C., Brandenburg, A. 2002Phys. Rev. E66016604ADSMathSciNetGoogle Scholar
  43. Fogedby, H. C. 2003Phys. Rev. E68026132ADSMathSciNetGoogle Scholar
  44. Landau, L., Lifshitz, E. 1959Quantum MechanicsPergamon PressOxfordGoogle Scholar
  45. Reichl, L. E. 1987The Transition to ChaosSpringer-VerlagNew YorkGoogle Scholar
  46. Wenzel, W., Biham, O., Jayaprakash, C. 1991Phys. Rev. A436550CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of AarhusAarhus CDenmark
  2. 2.NORDITACopenhagen ØDenmark
  3. 3.Niels Bohr Institute for Astronomy, Physics, and GeophysicsCopenhagen ØDenmark

Personalised recommendations