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 Nelin. Dinam., 2014, Volume 10, Number 2, Pages 149–156 (Mi nd432)

Poincaré recurrences in a stroboscopic section of a nonautonomous van der Pol oscillator

International Research Institute of Nonlinear Dynamics Saratov State University, Astrakhanskaya 83, Saratov, 410026, Russia

Abstract: In the present work we analyze the statistics of a set that is obtained by calculating a stroboscopic section of phase trajectories in a harmonically driven van der Pol oscillator. It is shown that this set is similar to a linear shift on a circle with an irrational rotation number, which is defined as the detuning between the external and natural frequencies. The dependence of minimal return times on the size $\varepsilon$ of the return interval is studied experimentally for the golden ratio. Furthermore, it is also found that in this case, the value of the Afraimovich–Pesin dimension is $\alpha_c=1$.

Keywords: Poincaré recurrence, Afraimovich–Pesin dimension, Fibonacci stairs, circle map, van der Pol oscillator.

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UDC: 530.182
MSC: 37B20
Revised: 15.05.2014

Citation: Nadezhda I. Semenova, Vadim S. Anishchenko, “Poincaré recurrences in a stroboscopic section of a nonautonomous van der Pol oscillator”, Nelin. Dinam., 10:2 (2014), 149–156

Citation in format AMSBIB
\Bibitem{SemAni14} \by Nadezhda~I.~Semenova, Vadim~S.~Anishchenko \paper Poincar\'e recurrences in a stroboscopic section of a nonautonomous van der Pol oscillator \jour Nelin. Dinam. \yr 2014 \vol 10 \issue 2 \pages 149--156 \mathnet{http://mi.mathnet.ru/nd432}