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The European Physical Journal Special Topics

, Volume 223, Issue 8, pp 1699–1709 | Cite as

Noise can reduce disorder in chaotic dynamics

  • Denis S. GoldobinEmail author
Regular Article Other Applications for Communication
Part of the following topical collections:
  1. Chaos, Cryptography and Communications

Abstract

We evoke the idea of representation of the chaotic attractor by the set of unstable periodic orbits and disclose a novel noise-induced ordering phenomenon. For long unstable periodic orbits forming the strange attractor the weights (or natural measure) is generally highly inhomogeneous over the set, either diminishing or enhancing the contribution of these orbits into system dynamics. We show analytically and numerically a weak noise to reduce this inhomogeneity and, additionally to obvious perturbing impact, make a regularizing influence on the chaotic dynamics. This universal effect is rooted into the nature of deterministic chaos.

Keywords

Chaotic System European Physical Journal Special Topic Unstable Manifold Chaotic Attractor Strange Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. McNamara, K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989)ADSCrossRefGoogle Scholar
  2. 2.
    L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 775 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    V.S. Anishchenko, in Dynamical Chaos – Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems, edited by L.O. Chua (World Scientific, 1995), p. 302Google Scholar
  5. 5.
    Ch. Zhou, J. Kurths, I.Z. Kiss, J.L. Hudson, Phys. Rev. Lett. 89, 014101 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    J.N. Teramae, D. Tanaka, Phys. Rev. Lett. 93, 204103 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    D.S. Goldobin, A.S. Pikovsky, Physica A 351, 126 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    D.S. Goldobin, A. Pikovsky, Phys. Rev. E 71, 045201(R) (2005)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    D.S. Goldobin, J.-N. Teramae, H. Nakao, G.B. Ermentrout, Phys. Rev. Lett. 105, 154101 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    P.W. Anderson, Phys. Rev. 109, 1492 (1958)ADSCrossRefGoogle Scholar
  11. 11.
    J.D. Maynard, Rev. Mod. Phys. 73, 401 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    D.S. Goldobin, E.V. Shklyaeva, J. Stat. Mech.: Theory Exp., P01009 (2009)Google Scholar
  13. 13.
    D.S. Goldobin, E.V. Shklyaeva, J. Stat. Mech.: Theory Exp., P09027 (2013)Google Scholar
  14. 14.
    E.G. Altmann, A. Endler, Phys. Rev. Lett. 105, 244102 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  16. 16.
    D.V. Lyubimov, M.A. Zaks, Physica 9D, 52 (1983)ADSMathSciNetGoogle Scholar
  17. 17.
    C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. A 37, 1711 (1988)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Ott, Chaos in Dynamical Systems (University Press, Cambridge, 1993)Google Scholar
  19. 19.
    B. Eckhardt, G. Ott, Z. Phys. B 93, 259 (1994)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    M.A. Zaks, D.S. Goldobin, Phys. Rev. E 81, 018201 (2010)ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Cvitanovic, C.P. Dettmann, R. Mainieri, G. Vattay, J. Stat. Phys. 93(3/4), 981 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    C.P. Dettmann, T.B. Howard, Physica D 238, 2404 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Lippolis, P. Cvitanovic, Phys. Rev. Lett. 104, 014101 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    P. Gaspard, J. Stat. Phys. 106(1/2), 57 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    E.G. Altmann, J.C. Leitao, J.V. Lopes, Chaos 22, 026114 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    P. Cvitanovic, Chaos: Classical and Quantum, http://chaosbook.org, ver. 14, 2012
  27. 27.
    R. Bowen, On Axiom A Diffeomorphisms, CBMS Regional Conference Series in Mathematics, Vol. 35 (American Mathematical Society, Providence, 1978)Google Scholar
  28. 28.
    A.N. Oraevskii, Sov. J. Quantum Electron. 11, 71 (1981)ADSCrossRefGoogle Scholar
  29. 29.
    J.L. Kaplan, J. A. Yorke, Commun. Math. Phys. 67, 93 (1979)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    C. Sparrow, The Lorenz equations: Bifurcations, chaos, and strange attractors (Springer, 1982)Google Scholar
  31. 31.
    A.S. Pikovsky, M. Rosenblum, J. Kurths, Synchronization–A Unified Approach to Nonlinear Science (Cambridge University Press, Cambridge, UK, 2001)Google Scholar
  32. 32.
    D. Goldobin, M. Rosenblum, A. Pikovsky, Phys. Rev. E 67, 061119 (2003)ADSCrossRefGoogle Scholar
  33. 33.
    D.S. Goldobin, Phys. Rev. E 78, 060104(R) (2008)ADSCrossRefGoogle Scholar
  34. 34.
    P. Reimann, C. van den Broeck, H. Linke, P. Hänggi, J.M. Rubi, A. Pérez-Madrid, Phys. Rev. Lett. 87, 010602 (2001)ADSCrossRefGoogle Scholar
  35. 35.
    S. Boccaletti, E. Allaria, R. Meucci, Phys. Rev. E 69, 066211 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    S. Banerjee, L. Rondoni, S. Mukhopadhyay, A.P. Misra, Opt. Commun. 284(9), 2278 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    S. Banerjee, L. Rondoni, S. Mukhopadhyay, Opt. Commun. 284(19), 4623 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    M. Montminy, Annu. Rev. Biochem. 66, 807 (1997)CrossRefGoogle Scholar
  39. 39.
    D. Bell-Pedersen, et al., Nat. Rev. Genet. 6, 544 (2005)CrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2014

Authors and Affiliations

  1. 1.Institute of Continuous Media Mechanics, UB RASPermRussia
  2. 2.Department of MathematicsUniversity of Leicester, Leicester LE1 7RHLeicesterUK

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