Skip to main content

Stochasticity and Chaos

  • Chapter

Part of the book series: Lecture Notes in Statistics ((LNS,volume 128))

Abstract

We discuss some results concerning stochastic perturbations of chaotic systems. In particular stochastic stability of SRB measures, asymptotic laws for entrance and exit times in small sets and rates of leaking due to noise.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeo morphisms. Lecture Notes in Mathematics 470. Springer, Berlin Heidelberg New York 1975.

    Google Scholar 

  2. V.Balady, L.-S.Young. On the spectra of randomly perturbed expanding maps. Commun. Math. Phys. 156, 355–385 (1993).

    Article  Google Scholar 

  3. M.Benedicks, L.-S.Young. Absolutely continuous invariant measures and random perturbations for someone dimensional maps. Ergod. Theor. & Dyn. Syst. 12, 13–37 (1992).

    MathSciNet  MATH  Google Scholar 

  4. M.Benedicks, L.-S.Young. SRB measures for certain Hénon maps. Invent. Math. 112, 541–576 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. M.Blank, G.Keller. Stochastic stability versus localization in chaotic dynamical systems. Preprint 1996.

    Google Scholar 

  6. V.Baladi, A.Kondah, B.Schmitt. Random correlations for small perturbations of expanding maps. Preprint 1995.

    Google Scholar 

  7. M.N.Bussac, R.B.White, L.Zuppiroli. Particle and heat transport in a partially stochastic magneticfiled. Physics Letters A, 190, 101–105 (1994).

    Article  Google Scholar 

  8. S.Childress, A.Gilbert. Stretch, twist, fold: the fast dynamo. Springer, Berlin, Heidelberg, New York 1995.

    MATH  Google Scholar 

  9. Z.Coelho, P.Collet. Asymptotic Limit Law for the Close Approach of Two Trajectories in Expanding Maps of the Circle. Prob. Theor. and Related Fields 99, 237–250 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. P.Collet Stochastic perturbations of the invariant measure of some hyperbolic dynamical systems. In Nonlinear Evolution and Chaotic Phenomena, G.Gallavotti and P.Zweifel editors, Plenum, New York London 1988.

    Google Scholar 

  11. P.Collet. Some Ergodic Properties of Maps of the Interval. In “Dynamical Systems & Frustrated Systems”, R.Bamon, J.-M.Gambaudo and S.Martinez editors, to appear.

    Google Scholar 

  12. Z.Coelho, E. de Faria. Limit laws of entrance times for homeomorphisms of the circle. To appear in the Israel Journal of Maths

    Google Scholar 

  13. P.Collet, A.Galves. Asymptotic distribution of entrance times for expanding maps of the interval. Dynamical Systems and Applications. R.P. Agar wal editor, World Scientific 1995.

    Google Scholar 

  14. P.Collet, A.Lesne. Renormalization group analysis of dynamical systems with noise. Journ. Stat. Phys. 57, 967 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  15. P.Collet, S.Martinez, B.Schmitt. The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7, 1437–1443 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  16. P.Collet, S.Martinez, B.Schmitt. In preparation.

    Google Scholar 

  17. J.-P. Eckmann, D.Ruelle. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985).

    Article  MathSciNet  Google Scholar 

  18. M.Freidlin, A.Wentzell. Random perturbations of dynamical systems. Springer, Berlin Heidelberg New York 1984.

    MATH  Google Scholar 

  19. M.Hirata. Poisson law for axiom A diffeomorphisms. Ergod. Theor. & Dyn. Syst. 13, 533–556 (1993).

    MathSciNet  MATH  Google Scholar 

  20. F.Hofbauer, G.Keller. Ergodic properties of invariant measures for piece- wise monotonic transformations. Math. Z. 180, 119–140 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  21. M.Hirsch, C.Pugh, M.Shub. Invariant Manifolds. Lecture Notes in Mathematics 583. Springer, Berlin Heidelberg New York 1977.

    Google Scholar 

  22. M.Kac. Bull. Amer. Math. Soc. 53, 1002 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  23. G.Keller. Stochastic stability in some chaotic dynamical systems. Mh. Math. 94, 313–333 (1982).

    Article  MATH  Google Scholar 

  24. Y.Kifer. Random perturbations of dynamical systems. Birkhauser, Boston 1988.

    MATH  Google Scholar 

  25. P.D.Liu, M.Qian. Smooth ergodic theory of random dynamical systems. Lecture Notes in Mathematics 1606, Springer Verlag 1995.

    Google Scholar 

  26. A.Lasota, J.Yorke. On the existence of invariant measures for piecewise monotone transformations. Trans. Amer. Math. Soc. 186, 481–488 (1973).

    Article  MathSciNet  Google Scholar 

  27. A.Lasota, M.Mackey. Probabilistic Properties of deterministic Systems. Cambridge University Press, Cambridge 1985.

    MATH  Google Scholar 

  28. J.Olarrea, F.J.Rubia. Stochastic Hopf Bifurcation. Phys. Rev. E 53, 268–271 (1996).

    Article  Google Scholar 

  29. B.Pitskel. Ergod. Th. k Dynam. Sys. 11, 501 (1991).

    MathSciNet  MATH  Google Scholar 

  30. D.Ruelle. Thermodynamic Formalism. Addison-Wesley, Reading 1978.

    MATH  Google Scholar 

  31. D.Ruelle, F.Takens. On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971) and 21, 64 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  32. S.Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  33. L.-S.Young. Stochastic stability of hyperbolic attractors. Ergod. Theor. & Dyn. Syst. 6, 311–319 (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Collet, P. (1998). Stochasticity and Chaos. In: Accardi, L., Heyde, C.C. (eds) Probability Towards 2000. Lecture Notes in Statistics, vol 128. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2224-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2224-8_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98458-2

  • Online ISBN: 978-1-4612-2224-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics