# Architecture of chaotic attractors for flows in the absence of any singular point

## Abstract

Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.

- Authors:

- CORIA-UMR 6614 Normandie Université, CNRS-Université et INSA de Rouen, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray (France)
- Université de Lyon, ENTPE, Laboratoire Génie Civil et Bâtiment, 3 Rue Maurice Audin, F-69518 Vaulx-en-Velin Cedex (France)

- Publication Date:

- OSTI Identifier:
- 22596653

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ATTRACTORS; CHAOS THEORY; EIGENVALUES; ONE-DIMENSIONAL CALCULATIONS; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Letellier, Christophe, and Malasoma, Jean-Marc.
```*Architecture of chaotic attractors for flows in the absence of any singular point*. United States: N. p., 2016.
Web. doi:10.1063/1.4954212.

```
Letellier, Christophe, & Malasoma, Jean-Marc.
```*Architecture of chaotic attractors for flows in the absence of any singular point*. United States. doi:10.1063/1.4954212.

```
Letellier, Christophe, and Malasoma, Jean-Marc. Wed .
"Architecture of chaotic attractors for flows in the absence of any singular point". United States.
doi:10.1063/1.4954212.
```

```
@article{osti_22596653,
```

title = {Architecture of chaotic attractors for flows in the absence of any singular point},

author = {Letellier, Christophe and Malasoma, Jean-Marc},

abstractNote = {Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.},

doi = {10.1063/1.4954212},

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

number = 6,

volume = 26,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}