Comptes Rendus
no. 10
Complete classification of homoclinic cycles in ℝ 4 in the case of a symmetry group 𝐆⊂ SO (4)
[Classification complète des cycles homoclines de ℝ 4 dans le cas d'un groupe de symétrie 𝐆⊂ SO (4)]
Nicola Sottocornola1
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.

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.

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Révisé le :
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DOI : 10.1016/S1631-073X(02)02371-3
Nicola Sottocornola 1

1 Institut non-linéaire de Nice, UMR 6618 CNRS, 1361, route des Lucioles, 06560 Valbonne, France 
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     title = {Complete classification of homoclinic cycles in $ \mathbb{R}^{\mathbf{4}}$ in the case of a symmetry group $ \mathbf{G\subset }\mathrm{SO}\mathbf{(4)}$},
     journal = {Comptes Rendus. Math\'ematique},
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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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[7] M. Krupa; I. Melbourne Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynamical Systems, Volume 15 (1995) no. 1, pp. 121-147

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[10] N. Sottocornola Sur la classification des cycles homoclines, C. R. Acad. Sci. Paris, Serie I, Volume 332 (2001), pp. 695-698

[11] N. Sottocornola, Robust homoclinic cycles in ℝ 4 , Nonlinearity, submitted

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