[Classification complète des cycles homoclines de dans le cas d'un groupe de symétrie ]
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 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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@article{CRMATH_2002__334_10_859_0,
author = {Sottocornola, Nicola},
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},
pages = {859--864},
publisher = {Elsevier},
volume = {334},
number = {10},
year = {2002},
doi = {10.1016/S1631-073X(02)02371-3},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02371-3/}
}
TY - JOUR
AU - Sottocornola, Nicola
TI - Complete classification of homoclinic cycles in $ \mathbb{R}^{\mathbf{4}}$ in the case of a symmetry group $ \mathbf{G\subset }\mathrm{SO}\mathbf{(4)}$
JO - Comptes Rendus. Mathématique
PY - 2002
SP - 859
EP - 864
VL - 334
IS - 10
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02371-3/
DO - 10.1016/S1631-073X(02)02371-3
LA - en
ID - CRMATH_2002__334_10_859_0
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Sottocornola, Nicola. 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, Tome 334 (2002) no. 10, pp. 859-864. doi : 10.1016/S1631-073X(02)02371-3. http://www.numdam.org/articles/10.1016/S1631-073X(02)02371-3/
[1] P. Ashwin, J. Montaldi, Conditions for existence of robust relative homoclinic trajectories, Math. Proc. Cambridge Philos. Soc., 2002, to appear
[2] Transverse bifurcation of homoclinic cycles, Phys. D, Volume 100 (1997), pp. 85-100
[3] Methods in Equivariant Bifurcation and Dynamical Systems, Adv. Ser. Nonlinear Dynam., 15, World Scientific, 2000
[4] Homographies Quaternions and Rotations, Oxford University Press, 1964
[5] Structurally stable heteroclinic cycles, Math. Proc. Cambridge Philos. Soc., Volume 103 (1988), pp. 189-192
[6] Robust heteroclinic cycles, J. Nonlinear Sci., Volume 7 (1997), pp. 129-176
[7] Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynamical Systems, Volume 15 (1995) no. 1, pp. 121-147
[8] Rational products of sines of rational angles, Aequationes Math., Volume 45 (1993), pp. 70-82
[9] Some results on roots of unity, with an application to a diophantine problem, Aequationes Math., Volume 2 (1969), pp. 163-166
[10] 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 , Nonlinearity, submitted
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