Yaroslav Boev

The local theory of Poincaré recurrences is applied to estimate pointwise and information dimensions of chaotic attractors in two-dimensional nonhyperbolic and hyperbolic maps. It is shown that the local pointwise dimension can be defined by calculating the mean recurrence times depending on the return vicinity size. The values of pointwise, information, capacity, and Lyapunov dimensions are compared. It is also analyzed how the structure of attractors can affect the calculation of the dimensions.

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.

Statistical properties of Poincaré recurrences in a two-dimensional map with chaotic non-strange attractor have been studied in numerical simulations. A local and a global approaches were analyzed in the framework of the considered problem. It has been shown that the local approach corresponds to Kac’s theorem including the case of a noisy system in certain conditions which have been established. Numerical proof of theoretical results for a global approach as well as the Afraimovich–Pesin dimension calculation are presented.