Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets

Zoran Levnajić; Igor Mezić

doi:10.1063/1.4919767

Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets

Chaos. 2015 May;25(5):053105. doi: 10.1063/1.4919767.

Authors

Zoran Levnajić¹, Igor Mezić²

Affiliations

¹ Faculty of Information Studies in Novo mesto, 8000 Novo mesto, Slovenia.
² Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, USA.

PMID: 26026317
DOI: 10.1063/1.4919767

Abstract

We present an application and analysis of a visualization method for measure-preserving dynamical systems introduced by I. Mezić and A. Banaszuk [Physica D 197, 101 (2004)], based on frequency analysis and Koopman operator theory. This extends our earlier work on visualization of ergodic partition [Z. Levnajić and I. Mezić, Chaos 20, 033114 (2010)]. Our method employs the concept of Fourier time average [I. Mezić and A. Banaszuk, Physica D 197, 101 (2004)], and is realized as a computational algorithms for visualization of periodic and quasi-periodic sets in the phase space. The complement of periodic phase space partition contains chaotic zone, and we show how to identify it. The range of method's applicability is illustrated using well-known Chirikov standard map, while its potential in illuminating higher-dimensional dynamics is presented by studying the Froeschlé map and the Extended Standard Map.