ABSTRACT
We investigate the statistics of recurrences to finite size intervals for chaotic dynamical systems. We find that the typical distribution presents an exponential decay for almost all recurrence times except for a few short times affected by a kind of memory effect. We interpret this effect as being related to the unstable periodic orbits inside the interval. Although it is restricted to a few short times it changes the whole distribution of recurrences. We show that for systems with strong mixing properties the exponential decay converges to the Poissonian statistics when the width of the interval goes to zero. However, we alert that special attention to the size of the interval is required in order to guarantee that the short time memory effect is negligible when one is interested in numerically or experimentally calculated Poincaré recurrence time statistics.
- 1. J. L. Lebowitz, Rev. Mod. Phys. 71, 346 (1999). https://doi.org/10.1103/RevModPhys.71.S346, Google ScholarCrossref
- 2. G. Aquino, P. Grigolini, and N. Scafetta, Chaos, Solitons Fractals 12, 2023 (2001). https://doi.org/10.1016/S0960-0779(00)00162-4, Google ScholarCrossref
- 3. D. Ruelle, Physica A 263, 540 (1999). https://doi.org/10.1016/S0378-4371(98)00529-9, Google ScholarCrossref
- 4. H. L. Frisch, Phys. Rev. 104, 1 (1956). https://doi.org/10.1103/PhysRev.104.1, Google ScholarCrossref
- 5. M. Kac, Probability and Related Topics in Physical Sciences (Interscience, New York, 1959). Google Scholar
- 6. J. B. Gao, Phys. Rev. Lett. 83, 3178 (1999). https://doi.org/10.1103/PhysRevLett.83.3178, Google ScholarCrossref
- 7. M. S. Baptista and I. L. Caldas, Physica A 312, 539 (2002). Google ScholarCrossref
- 8. M. S. Baptista, I. L. Caldas, M. V. A. P. Heller, and A. A. Ferreira, Physica A10301, 150 (2001). https://doi.org/10.1016/S0378-4371(01)00395-8, Google ScholarCrossref
- 9. M. S. Baptista, I. L. Caldas, M. V. A. P. Heller, A. A. Ferreira, R. Bengtson, and J. Stöckel, Phys. Plasmas 8, 4455 (2001). https://doi.org/10.1063/1.1401117, Google ScholarScitation, ISI
- 10. B. V. Chirikov, Physica D 13, 395 (1984). https://doi.org/10.1016/S0167-2789(00)00016-6, Google ScholarCrossref
- 11. J. D. Meiss, Chaos 7, 139 (1997). https://doi.org/10.1063/1.166245, Google ScholarScitation
- 12. H. Hu, A. Rampioni, L. Rossi, and G. Turchetti, Chaos 14, 160 (2004). https://doi.org/10.1063/1.1629191, Google ScholarScitation
- 13. G. M. Zaslavsky and M. K. Tippett, Phys. Rev. Lett. 67, 3251 (1991). https://doi.org/10.1103/PhysRevLett.67.3251, Google ScholarCrossref
- 14. G. M. Zaslavsky, M. Edelman, and B. A. Niyazov, Chaos 7, 159 (1997). https://doi.org/10.1063/1.166252, Google ScholarScitation
- 15. V. Afraimovich, Chaos 7, 12 (1997). https://doi.org/10.1063/1.166237, Google ScholarScitation
- 16. V. Afraimovich and G. M. Zaslavsky, Phys. Rev. E 55, 5418 (1997). https://doi.org/10.1103/PhysRevE.55.5418, Google ScholarCrossref
- 17. N. Hadyn, J. Luevano, G. Mantica, and S. Vaienti, Phys. Rev. Lett. 88, 224502 (2002). https://doi.org/10.1103/PhysRevLett.88.224502, Google ScholarCrossref
- 18. B. Pitskel, Ergod. Theory Dyn. Syst. 11, 501 (1991). Google ScholarCrossref
- 19. M. Hirata, Ergod. Theory Dyn. Syst. 13, 533 (1993). https://doi.org/10.1016/0305-0548(86)90048-1, Google ScholarCrossref
- 20. M. Hirata, Poisson Law for the Dynamical Systems with the “Self-Mixing” Conditions, Vol. 1 (World Scientific, River Edge, NJ, 1995). Google Scholar
- 21. A. Galves and B. Schmitt, Random Comput. Dyn. 5, 337 (1997). Google Scholar
- 22. H. Bruin and S. Vaienti, Fundamenta Mathematicae 176, 77 (2003). Google ScholarCrossref
- 23. G. M. Zaslavsky, Phys. Rep. 371, 461 (2002). https://doi.org/10.1016/S0370-1573(02)00331-9, Google ScholarCrossref
- 24. M. Kac, Bull. Am. Math. Soc. 53, 1002 (1947). Google ScholarCrossref
- 25. V. Balakrishnan, G. Nicolis, and C. Nicolis, Phys. Rev. E 61, 2490 (2000). https://doi.org/10.1103/PhysRevE.61.2490, Google ScholarCrossref
- 26. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1950). Google Scholar
- 27. I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai, Ergodic Theory (Springer-Verlag, Berlin, 1982). Google Scholar
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
