Complete classification of homoclinic cycles in $ \mathbb{R}^{\mathbf{4}}$ in the case of a symmetry group $ \mathbf{G\subset }\mathrm{SO}\mathbf{(4)}$

Nicola Sottocornola

doi:10.1016/S1631-073X(02)02371-3

Complete classification of homoclinic cycles in

ℝ^{4}

in the case of a symmetry group

𝐆 \subset SO (4)

[Classification complète des cycles homoclines de

ℝ^{4}

dans le cas d'un groupe de symétrie

𝐆 \subset SO (4)

]

Nicola Sottocornola ¹

¹ Institut non-linéaire de Nice, UMR 6618 CNRS, 1361, route des Lucioles, 06560 Valbonne, France

Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 859-864.

Résumé
Abstract

Des cycles homoclines avec groupes de symétrie contenus dans SO(4) sont déjà apparus dans la littérature. Ces cycles ont 2, 3, 6, 8, 12 ou 24 points d'équilibre. Dans cette Note, on montre que cette classification est complète en utilisant un résultat sur les équations diophantiennes à angles rationnels.

Some homoclinic cycles in $ℝ^{4}$ with symmetry groups contained in SO(4) have already appeared in the literature. These cycles have 2, 3, 6, 8, 12, or 24 equilibria. In this Note we show that this classification is complete using a result in diophantine trigonometric equations with rational angles.

Reçu le : 2002-01-28
Révisé le : 2002-03-18
Publié le : 2002-04-15

DOI : 10.1016/S1631-073X(02)02371-3

Affiliations des auteurs :

Nicola Sottocornola ¹

¹ Institut non-linéaire de Nice, UMR 6618 CNRS, 1361, route des Lucioles, 06560 Valbonne, France

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Nicola Sottocornola. Complete classification of homoclinic cycles in $ \mathbb{R}^{\mathbf{4}}$ in the case of a symmetry group $ \mathbf{G\subset }\mathrm{SO}\mathbf{(4)}$. Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 859-864. doi : 10.1016/S1631-073X(02)02371-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02371-3/

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Ale Jan Homburg; Björn Sandstede Homoclinic and Heteroclinic Bifurcations in Vector Fields, Volume 3 (2010), p. 379 | DOI:10.1016/s1874-575x(10)00316-4
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Nicola Sottocornola Simple homoclinic cycles in low-dimensional spaces, Journal of Differential Equations, Volume 210 (2005) no. 1, pp. 135-154 | DOI:10.1016/j.jde.2004.10.023 | Zbl:1066.34042

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