Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows

Aubin Arroyo; Federico Rodriguez Hertz

doi:10.1016/S0294-1449(03)00016-7

Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows

Aubin Arroyo
, Federico Rodriguez Hertz

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

In this paper we prove that any C¹ vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C¹-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.

Original language	English (US)
Pages (from-to)	805-841
Number of pages	37
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	20
Issue number	5
DOIs	https://doi.org/10.1016/S0294-1449(03)00016-7
State	Published - 2003

All Science Journal Classification (ASJC) codes

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/S0294-1449(03)00016-7

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title = "Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows",

abstract = "In this paper we prove that any C1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C1-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincar{\'e} flow.",

author = "Aubin Arroyo and \{Rodriguez Hertz\}, Federico",

note = "Funding Information: ∗Corresponding author. E-mail addresses: [email protected] (A. Arroyo), [email protected] (F. Rodriguez Hertz). 1Partially supported by CNPq-Brazil and CONACYT-M{\'e}xico. 2Partially supported by CNPq-Brazil.",

year = "2003",

doi = "10.1016/S0294-1449(03)00016-7",

language = "English (US)",

volume = "20",

pages = "805--841",

journal = "Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire",

issn = "0294-1449",

publisher = "European Mathematical Society Publishing House",

number = "5",

}

TY - JOUR

T1 - Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows

AU - Arroyo, Aubin

AU - Rodriguez Hertz, Federico

N1 - Funding Information: ∗Corresponding author. E-mail addresses: [email protected] (A. Arroyo), [email protected] (F. Rodriguez Hertz). 1Partially supported by CNPq-Brazil and CONACYT-México. 2Partially supported by CNPq-Brazil.

PY - 2003

Y1 - 2003

N2 - In this paper we prove that any C1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C1-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.

AB - In this paper we prove that any C1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C1-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.

UR - https://www.scopus.com/pages/publications/0346724412

UR - https://www.scopus.com/pages/publications/0346724412#tab=citedBy

U2 - 10.1016/S0294-1449(03)00016-7

DO - 10.1016/S0294-1449(03)00016-7

M3 - Article

AN - SCOPUS:0346724412

SN - 0294-1449

VL - 20

SP - 805

EP - 841

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 5

ER -

Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows

Abstract

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