Cite this article as:

Galaktionova T. I., Semenova N. I., Anishchenko V. S. Poincare Recurrences and Afraimovich–Pesin Dimension in a Nonautonomous Conservative Oscillator . Izvestiya of Saratov University. New series. Series Physics, 2016, vol. 16, iss. 4, pp. 195-203. DOI: https://doi.org/10.18500/1817-3020-2016-16-4-195-203

# Poincare Recurrences and Afraimovich–Pesin Dimension in a Nonautonomous Conservative Oscillator

**Background and Objectives: **One of the fundamental features of the temporal dynamics is Poincare recurrence. It have been shown that statistics of return time in global approach depends on topological entropy h. The case of h > 0 (set with mixing) has been already studied theoretically. The theoretical conclusions have been confirmed by numerical simulation. The case of the sets without mixing (h = 0) has been studied theoretically, but recent numerical results shows some special aspects which are absent in theory. In particular, it has been obtained that the dependence of the mean minimal return time on the size of ε-vicinity in a circle map is a step function («Fibonacci stairs»).** Materials and Methods: **In the present paper the Poincare recurrences are studied numerically for invariant curves in the stroboscopic section of trajectories of a nonautonomous conservative oscillator. **Results:** It is shown that the dependence of the mean minimal return time on the size of ε-vicinity is a step function («Fibonacci stairs»). In addition, the Afraimovich–Pesin dimension has been calculated in cases of rational and irrational natural and external frequency ratios.** Conclusion:** We have found the conditions of occurrence of «Fibonacci stairs» and the impact of the amplitude of the external harmonic force. Is is shown that this dependence exists only in some interval of ε. The size of the interval depends on amplitude of external force.

1. Poincaré H. Izbrannyye trudy: v 3 t. T. 2: Nebesnaya mekhanika. Topologiya. Teoriya chisel [Selected works: in 3 vols. Vol. 2: Celestial mechanics. Topology. Number theory]. Moscow : Nauka, 1972. 1000 p. (in Russian).

2. Hirata M., Saussol B., Vaienti S. Statistics of Return Times: A General Framework and New Applications. Commun. Math. Phys., 1999, vol. 206, iss. 1, pp. 33–55.

3. Afraimovich V., Ugalde E., Urias J. Fractal dimension for Poincaré recurrences. Elsevier, 2006. 245 p.

4. Anishchenko V. S., Astakhov S. V. Poincaré recurrence theory and its applications to nonlinear physics. Phys. Usp., 2013, vol. 56, pp. 955–972.

5. Afraimovich V. Pesin’s dimension for Poincaré recurrences. Chaos, 1997, vol. 7, iss. 1, pp. 12–20.

6. Afraimovich V., Zaslavsky G. M. Fractal and multifractal properties of exit times and Poincaré recurrences. Phys. Rev. E., 1997, vol. 55, pp. 5418–5426.

7. Penné V., Saussol B., Vaienti S. Fractal and statistical characteristics of recurrence times. Journal de Physique, 1998, vol. 8, iss. 6, pp. 163–171.

8. Anishchenko V. S., Astakhov S. V., Boev Ya. I., Biryukova N. I., Strelkova G. I. Statistics of Poincaré recurrences in local and global approaches. Commun. Nonlinear Sci. Numer. Simul., 2013, vol. 18, iss. 12, pp. 3423–3435.

9. Anishchenko V. S., Boev Ya. I., Semenova N. I., Strelkova G. I. Local and global approaches to the problem of Poincaré recurrences. Applications in nonlinear dynamics. Phys. Rep., 2015, vol. 587, pp. 1–39.

10. Semenova N. I., Vadivasova T. E., Strelkova G. I., Anishchenko V. S. Statistical properties of Poincaré recurrences and Afraimovich–Pesin dimension for the circle map. Commun. Nonlinear Sci. Numer. Simul., 2015, vol. 22, pp. 1050–1061.

11. Slater N. B. Gaps and steps for the sequence nθ mod 1. Proc. Camb. Philos. Soc., 1967, vol. 63, iss. 4, pp. 1115–1123.

12. Chirikov B. V., Shepelyansky D. L. Asymptotic Statistics of Poincaré Recurrences in Hamiltonian Systems with Divided Phase Space. Phys. Rev. Lett., 1999, vol. 82, pp. 528–531.

13. Shepelyansky D. L. Poincaré recurrences in Hamiltonian systems with a few degrees of freedom. Phys. Rev. E., 2010, vol. 82, pp. 055202.

14. Lichtenberg A. J., Liberman M. A. Regular and Stochastic Motion. Applied mathematical sciences. SpringerVerlag, 1982. 499 p.

15. Zaslavsky G. M., Kirichenko N. A. Khaos dinamicheskiy [Dynamic Chaos]. Fizicheskaya entsiklopediya [Physical encyclopedia: in 5 vols]. Ans. ed. A. M. Prokhorov. Moscow, Great Soviet Encyclopedia Publ., 1988, vol. 5, pp. 397–402 (in Russian).

16. Srivastava N., Kaufman C., Müller G. Hamiltonian chaos. Computers in Physics., 1990, vol. 4, pp. 549– 553.