PMID: 15244903Jul 13, 2004Paper

Topological aspects of the structure of chaotic attractors in R3

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Tsvetelin D Tsankov, Robert Gilmore

Abstract

Strange attractors with Lyapunov dimension d(L) <3 can be classified by branched manifolds. They can also be classified by the bounding tori that enclose them. Bounding tori organize branched manifolds (classes of strange attractors) in the same way as the branched manifolds organize the periodic orbits in a strange attractor. We describe how bounding tori are constructed and expressed in a useful canonical form. We present the properties of these canonical forms and show that they can be uniquely coded by analogs of periodic orbits of period g-1, where g is the genus. We describe the structure of the global Poincaré surface of section for an attractor enclosed by a genus- g torus and determine the transition matrix for flows between the g-1 components of the Poincaré surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.

References

Apr 15, 1988·Physical Review A: General Physics·H G Solari, R Gilmore
May 15, 1990·Physical Review. a·N B TufillaroR Gilmore
Dec 1, 1993·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·G B MindlinR Gilmore
Oct 3, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·D Sciamarella, G B Mindlin
Oct 4, 2003·Physical Review Letters·Tsvetelin D Tsankov, Robert Gilmore

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Citations

Apr 8, 2006·Chaos·Gleison F V AmaralLuis Antonio Aguirre
Dec 17, 2004·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Greg ByrneChristophe Letellier
Oct 4, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Christophe LetellierRobert Gilmore
Feb 1, 2008·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Christophe LetellierTimothy Jones
May 17, 2012·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Christophe Letellier, Luis A Aguirre
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Daniel J CrossRobert Gilmore
Sep 28, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Christophe Letellier, Luis A Aguirre
Jan 15, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Daniel J Cross, R Gilmore
Aug 7, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Nicola RomanazziRobert Gilmore

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