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

Nadezhda I. Semenova, Vadim S. Anishchenko

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.

UDC: 530.182

MSC: 37B20

Received: 18.04.2014
Revised: 15.05.2014