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

## 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.

- Authors:

- Faculty of Information Studies in Novo mesto, 8000 Novo mesto (Slovenia)
- (United States)
- Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106 (United States)

- Publication Date:

- OSTI Identifier:
- 22402556

- Resource Type:
- Journal Article

- Journal Name:
- Chaos (Woodbury, N. Y.)

- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1054-1500

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CHAOS THEORY; DYNAMICS; FREQUENCY ANALYSIS; MAPS; MATHEMATICAL OPERATORS; PARTITION FUNCTIONS; PERIODICITY; PHASE SPACE

### Citation Formats

```
Levnajić, Zoran, Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, and Mezić, Igor.
```*Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets*. United States: N. p., 2015.
Web. doi:10.1063/1.4919767.

```
Levnajić, Zoran, Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, & Mezić, Igor.
```*Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets*. United States. doi:10.1063/1.4919767.

```
Levnajić, Zoran, Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, and Mezić, Igor. Fri .
"Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets". United States. doi:10.1063/1.4919767.
```

```
@article{osti_22402556,
```

title = {Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets},

author = {Levnajić, Zoran and Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106 and Mezić, Igor},

abstractNote = {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.},

doi = {10.1063/1.4919767},

journal = {Chaos (Woodbury, N. Y.)},

issn = {1054-1500},

number = 5,

volume = 25,

place = {United States},

year = {2015},

month = {5}

}