Eur. Phys. J. B 82, 219-225 (2011)
https://doi.org/10.1140/epjb/e2011-20184-0

Statistical characteristics of the Poincaré return times for a one-dimensional nonhyperbolic map^*

¹ International Institute of Nonlinear Dynamics, Department of Physics, Saratov State University, Astrakhanskaya street, 83, 410012 Saratov, Russia
² Potsdam Institute for Climate Impact Research, Potsdam, Germany

Corresponding author: ^a wadim@info.sgu.ru

Received: 10 March 2011
Revised: 31 May 2011
Published online: 12 August 2011

Abstract

Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac's theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with the form of probability measure and the complete correspondence of the obtained results with the mathematical theory is shown. The influence of the attractor crisis on the return time characteristics is also analyzed. The obtained results have a methodical and educational significance and can be used for solving a number of applied tasks.