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 $\epsilon$-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 $\epsilon$-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 $\epsilon$. The size of the interval depends on amplitude of external force.