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Recurrence time statistics in chaotic dynamics. I. Discrete time maps


The dynamics of transitions between the cells of a finite-phase-space partition in a variety of systems giving rise to chaotic behavior is analyzed, with special emphasis on the statistics of recurrence times. In the case of one-dimensional piecewise Markow maps the recurrence problem is cast into a-renewal process. In the presence of intermittency, transitions between cells define a non-Markovian, non-renewal process reflected in the presence of power-law probability distributions and of divergent variances and mean values.

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  1. 1.

    M. Kac,Probability and Related Topics in Physical Sciences (Interscience, New York, 1969);Bull. Am. Math. Soc. 53:1002 (1947).

  2. 2.

    E. Ott,Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).

  3. 3.

    G. Nicolis,Introduction to Nonlinear Science (Cambridge University Press, Cambridge, 1995).

  4. 4.

    A. Lasota and M. Mackey,Probabilistic Properties of Deterministic Systems (Cambridge University Press, Cambridge, 1985).

  5. 5.

    J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).

  6. 6.

    D. MacKernan and G. Nicolis,Phys. Rev. E 50:988 (1994).

  7. 7.

    G. Nicolis and C. Nicolis,Phys. Rev. A 38:427 (1988).

  8. 8.

    G. Nicolis, S. Martinez, and E. Tirapegui,Chaos Solitons Fractals 1:25 (1991).

  9. 9.

    D. R. Cox and H. D. Miller,The Theory of Stochastic Processes (Chapman and Hall, London, 1994).

  10. 10.

    N. G. de Bruijn,Asymptotic Methods in Analysis (Dover, New York, 1981).

  11. 11.

    P. Manneville and Y. Pomeau,Physica 1D:219 (1980).

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Balakrishnan, V., Nicolis, G. & Nicolis, C. Recurrence time statistics in chaotic dynamics. I. Discrete time maps. J Stat Phys 86, 191–212 (1997).

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Key Words

  • Recurrence time
  • escape time
  • Markov partition
  • fully developed chaos
  • intermittent chaos