Cite this article as:
Anishchenko V. S., Astakhov S. V., Boev Y. I., Biryukova N. I., Strelkova G. I. Statistics of Poincare Recurrence with Considering Effect of Fluctuations. Izvestiya of Saratov University. New series. Series Physics, 2013, vol. 13, iss. 2, pp. 5-15. DOI: https://doi.org/10.18500/1817-3020-2013-13-2-5-15
Statistics of Poincare Recurrence with Considering Effect of Fluctuations
The basic statistical characteristics of Poincare recurrence are obtained numerically for the logistic map in a chaotic regime. The mean values, variation and recurrence distribution density are calculated and their dependence on a return size is analysed. Afraimovich–Pesin dimension values are obtained. It is verified that the Afraimovich–Pesin dimension corresponds to the Lyapunov exponent. the peculiarities of the influence of noise on the recurrence statistics are studied in local and global approaches. It is shown that the obtained numerical data fully conform to the theoretical results. It is demonstrated that the Poincare recurrence theory can be applied to diagnose.
1. Немыцкий В. В., Степанов В. В. Качественная теория дифференциальных уравнений. М. : ОГИЗ, 1947.
2. Afraimovich V., Ugalde E., Urias J. Размерности для последовательности времен возвращений Пуанкаре. М. ; Ижевск : Изд-во РХД ; Ижевский ин-т комп. ис- следований, 2011.
3. Kac M. Lectures in Applied Mathematics. London : Interscience, 1957.
4. Ott E., Grebogi C., Yorke J. Controlling chaos // Phys. Rev. Lett. 1990. Vol. 64, iss. 11. P. 1196–1199.
5. Hirata M., Saussol B., Vaienti S. Statistics of Return Times : A General Framework and New Applications // Communications in Mathematical Physics. 1999. Vol. 206. P. 33.
6. Afraimovich V. Pesin’s dimension for Poincaré recurrences // Chaos. 1999. Vol. 206, iss. 1. P. 33–35.
7. Afraimovich V., Zaslavsky G. M. Fractal and multifractal properties of exit times and Poincarérecurrences // Phys. Rev. E. 1997. Vol. 55. P. 5418.
8. Pesin Y. B. Dimension Theory in Dynamical Systems : Contemporary Viewsand Applications // Chicago Lectures in Mathematics. Chicago University Press, 1997.
9. Adler R. L., Konheim A. G., McAndrew M. H. Topological entropy // Trans. Amer. Math. Soc. 1965. Vol. 114. P. 309.
10. Afraimovich V. S., Lin W. W., Rulkov N. F. Fractal dimension for Poincaré recurrences as an indicator of synchronized chaotic regimes // Intern. J. Bifurcat. Chaos. 2000. Vol. 10, № 10. P. 2323–2337.
11. Каток А. Б., Хассельблат Б. Введение в современную теорию динамических систем. М. : Факториал, 1999.
12. Шустер Г. Детерминированный хаос. Введение. М. : Мир, 1988.
13. Анищенко В. С., Хайрулин М. Е. Влияние индуцированного шумом кризиса аттракторов на характеристики времен возврата Пуанкаре // Письма в ЖТФ. 2011. Т. 37, вып. 12. С. 35–43.
14. Anishchenko V., Khairulin M., Strelkova G., Kurths J. Statistical characteristics of the Poincare return times for an one-dimensional nonhyperbolic map // Eur. Phys. J. B. 2011. Vol. 82. P. 219–225.
15. Анищенко В. С., Астахов С. В., Боев Я. И., Куртс Ю. Возвраты Пуанкаре в системе с хаотическим нестранным аттрактором // Нелинейная динамика. 2012. Т. 8, № 1. С. 29–41.
16. Saussol B., Troubetzkoy S., Vaienti S. Recurrence, Dimensions, and Lyapunov Exponents // J. Stat. Phys. 2002. Vol. 106, № 3–4. P. 623–634.
17. Astakhov S. V., Anishchenko V. S. Afraimovich–Pesin dimension for Poincaré recurrences in one- and twodimensional deterministic and noisy chaotic maps // Phys. Lett. A. 2012. Vol. 376, № 47–48. P. 3620–3624.
18. Anishchenko V. S., Astakhov S. Relative Kolmogorov Entropy of a Chaotic System in the Presence of Noise // Intern. J. Bifurcat. Chaos. 2008. Vol. 18, № 9. P. 2851–2855.
19. Benzi R., Sutera A., Vulpiani A. The mechanism of stochastic resonance // J. Phys. A: Mathematical and General. 1981. Vol. 14, № 11. P. L453–L457.
20. Anishchenko V. S., Astakhov V. V., Neiman A. B., Vadivasova T. E., Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems // Tutorial and Modern Development. 2nd ed. Berlin ; Heidelberg : Springer, 2007.
21. Анищенко В. С., Нейман А. Б., Мосс Ф., Шиманский- Гайер Л. Стохастический резонанс как индуцированный шумом эффект увеличения степени порядка // УФН. 1999. Т. 169, № 1. С. 7–38.
22. Anishchenko V. S., Neiman A. B., Safonova M. A. Stochastic Resonance in chaotic systems // J. Stat. Phys. 1993. Vol. 70, № 1–2. P. 183–196.
23. Anishchenko V. S., Boev Y. I. Diagnostics of stochastic resonance using Poincaré recurrence time distribution // Communications in Nonlinear Science and Numerical Simulation. 2013. Vol. 18, iss. 4. P. 953–958.
24. Pecora L. M., Carroll T. L. Synchronization in chaotic systems // Phys. Rev. Lett. 1990. Vol. 64. P. 821–824.
