Architecture of chaotic attractors for flows in the absence of any singular point
Abstract
Some chaotic attractors produced by threedimensional 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 onedimensional 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 onedimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such onedimensional 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:
 CORIAUMR 6614 Normandie Université, CNRSUniversité et INSA de Rouen, Campus Universitaire du Madrillet, F76800 SaintEtienne du Rouvray (France)
 Université de Lyon, ENTPE, Laboratoire Génie Civil et Bâtiment, 3 Rue Maurice Audin, F69518 VaulxenVelin 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; ONEDIMENSIONAL CALCULATIONS; THREEDIMENSIONAL CALCULATIONS
Citation Formats
Letellier, Christophe, and Malasoma, JeanMarc. 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, JeanMarc. Architecture of chaotic attractors for flows in the absence of any singular point. United States. doi:10.1063/1.4954212.
Letellier, Christophe, and Malasoma, JeanMarc. 2016.
"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, JeanMarc},
abstractNote = {Some chaotic attractors produced by threedimensional 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 onedimensional 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 onedimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such onedimensional 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 = 2016,
month = 6
}

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