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The explosion of singular hyperbolic attractors

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C. A. Morales

Dynamical Systems · Submitted 20 Mar 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It is proved that any attractor obtained from such perturbations contains a singularity.

how it is stated · asserted 1.00 read the record →

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arXiv IDmath/0303253 (version 2)
Pith IDPYBVE27Z, Pith's permanent record of this paper and its review
CitationC. A. Morales. "The explosion of singular hyperbolic attractors." arXiv:math/0303253 (2003). https://pith.science/paper/PYBVE27Z
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,145 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.30 · compares with prior results 0.69 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains a singularity. So, there is an upper bound for the number of attractors obtained from such perturbations. Furthermore, every three-dimensional flow $C^r$ close to one exhibiting a singular hyperbolic attractor has a singularity non isolated in the non wandering set. We also give sufficient conditions for a singularity of a three-dimensional flow to be stably non isolated in the nonwandering set. These results generalize well known properties of the Lorenz attractor.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. background 1.00 0.05 0.03 asserted 1.00
s1 The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. background 1.00 0.05 0.02 asserted 1.00
s2 In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. contribution 0.99 0.71 0.03 asserted 0.89
s3 It is proved that any attractor obtained from such perturbations contains a singularity. result 1.00 0.89 0.54 asserted 1.00
s4 So, there is an upper bound for the number of attractors obtained from such perturbations. interpretation 0.64 0.79 0.17 asserted 0.99
s5 Furthermore, every three-dimensional flow $C^r$ close to one exhibiting a singular hyperbolic attractor has a singularity non isolated in the non wandering set. result 1.00 0.82 0.24 asserted 1.00
s6 We also give sufficient conditions for a singularity of a three-dimensional flow to be stably non isolated in the nonwandering set. result 0.59 0.91 0.07 asserted 0.99
s7 These results generalize well known properties of the Lorenz attractor. interpretation 0.99 0.67 0.06 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains a singularity. So, there is an upper bound for the number of attractors obtained from such perturbations. Furthermore, every three-dimensional flow $C^r$ close to one exhibiting a singular hyperbolic attractor has a singularity non isolated in the non wandering set. We also give sufficient conditions for a singularity of a three-dimensional flow to be stably non isolated in the nonwandering set. These results generalize well known properties of the Lorenz attractor.

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