Nadezhda Semenova

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 ε 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 $\alpha_c = 1$.

The statistics of Poincaré recurrences is studied numerically in a one-dimensional cubic map in the presence of harmonic and noisy excitations. It is shown that the distribution density of Poincare recurrences is periodically modulated by the harmonic forcing. It is substantiated that the theory of the Afraimovich–Pesin dimension can be applied to a nonautonomous map for both harmonic and noisy forcings. It is demonstrated that the relationship between the AP-dimension and Lyapunov exponents is violated in the nonautonomous system.

The concept of a local fractal dimension has been introduced in the framework of the average Poincaré recurrence time numerical analysis in an $\varepsilon$-vicinity of a certain point. Lozi and Hénon maps have been considered. It has been shown that in case of Lozi map the local dimension weakly depends on the point on the attractor and its value is close to the fractal dimension of the attractor. In case of a quasi attractor observed in both Hénon and Feugenbaum systems the local dimension significantly depends on both the diameter and the location of the $\varepsilon$-vicinity. The reason of this strong dependency is high non-homogenity of a quasi-attractor which is typical for non-hyperbolic chaotic attractors.