Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

Montserrat Corbera; Jaume Llibre; Ernesto Pérez-Chavela

doi:https://doi.org/10.1088/0305-4470/39/50/001

Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

Montserrat Corbera, Jaume Llibre, Ernesto Pérez-Chavela

Department of Mathematics

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

3 Citations (Scopus)

Abstract

In this paper we consider vector fields in ℝ3 that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' L formed by two singular points (e+ and e-) and their invariant manifolds: one of dimension 2 (a sphere minus the points e + and e-) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e-). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincaré map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in ℝ3, and the second one is the charged rhomboidal four-body problem. © 2006 IOP Publishing Ltd.

Original language	English
Article number	001
Pages (from-to)	15313-15326
Journal	Journal of Physics A: Mathematical and General
Volume	39
DOIs	https://doi.org/10.1088/0305-4470/39/50/001
Publication status	Published - 15 Dec 2006

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title = "Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2",

abstract = "In this paper we consider vector fields in ℝ3 that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' L formed by two singular points (e+ and e-) and their invariant manifolds: one of dimension 2 (a sphere minus the points e + and e-) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e-). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincar{\'e} map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in ℝ3, and the second one is the charged rhomboidal four-body problem. {\textcopyright} 2006 IOP Publishing Ltd.",

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Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2. / Corbera, Montserrat; Llibre, Jaume; Pérez-Chavela, Ernesto.

In: Journal of Physics A: Mathematical and General, Vol. 39, 001, 15.12.2006, p. 15313-15326.

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

TY - JOUR

T1 - Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

AU - Corbera, Montserrat

AU - Llibre, Jaume

AU - Pérez-Chavela, Ernesto

PY - 2006/12/15

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N2 - In this paper we consider vector fields in ℝ3 that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' L formed by two singular points (e+ and e-) and their invariant manifolds: one of dimension 2 (a sphere minus the points e + and e-) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e-). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincaré map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in ℝ3, and the second one is the charged rhomboidal four-body problem. © 2006 IOP Publishing Ltd.

AB - In this paper we consider vector fields in ℝ3 that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' L formed by two singular points (e+ and e-) and their invariant manifolds: one of dimension 2 (a sphere minus the points e + and e-) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e-). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincaré map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in ℝ3, and the second one is the charged rhomboidal four-body problem. © 2006 IOP Publishing Ltd.

U2 - https://doi.org/10.1088/0305-4470/39/50/001

DO - https://doi.org/10.1088/0305-4470/39/50/001

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EP - 15326

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