Return time in dynamical systems

Author(s)

Eduardo Goldani Altmann

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2004-02-13

Examining board members

Ibere Luiz Caldas; Marcus Aloizio Martinez de Aguiar; Ricardo Luiz Viana

• Advisor: Ibere Luiz Caldas

Abstract

We study the recurrence time in dynamical systems. The statistics of the recurrence time to a specific region of the phase space of chaotic dynamical systems were obtained numerically and compared with the binomial-like distribution, deduced for a random process. The main results are: the presence of the so called short time memory effect, typical for deterministic systems and related to the distribution of the unstable periodic orbits; the return time distribution captures the main temporal properties of intermittent systems. The possible connections of the recurrence time statistics to the anomalous transport were presented, with special attention to their limitations. The return time statistics was applied to analyze time series obtained from an Hamiltonian model and from magnetically confined plasma. In the first case we noticed that the recurrences of the series were similar to the recurrences obtained in the phase space of the Hamiltonian dynamical system: the standard map with a random phase. We analyze the dependence of the power-law tails of the distributions with the non-linearity and with the randomness of the system. One effect that appears only in the time series case is the multiple power law tails. We apply the log-normal cascade model to explain the probability density function of the series obtained in Tokamaks. The recurrence time statistics of the series is closely related to the short and long time correlation present on the series. (AU)