Analytic invariants for the 1:-1 resonance
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, p. 1367-1426
Associated to analytic Hamiltonian vector fields in 4 having an equilibrium point satisfying a non semisimple 1:-1 resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.
Etant donnés des champs de vecteurs Hamiltoniens analytiques dans 4 ayant un point d’équilibre satisfaisant une résonance 1:-1 non semisimple, nous construisons deux constantes qui sont invariantes relativement aux changements de coordonnées symplectiques analytiques. Ces invariants sont égaux à zéro lorsque l’Hamiltonien est intégrable. Nous montrons également que ces invariants sont différents de zéro dans un ensemble ouvert et dense.
DOI : https://doi.org/10.5802/aif.2806
Classification:  37J20,  34M40,  34M30
Keywords: analytic classification, Stokes phenomenon, splitting of separatrices
@article{AIF_2013__63_4_1367_0,
     author = {Gaiv\~ao, Jos\'e Pedro},
     title = {Analytic invariants for the $1:-1$ resonance},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     pages = {1367-1426},
     doi = {10.5802/aif.2806},
     mrnumber = {3137358},
     zbl = {06359592},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_4_1367_0}
}
Gaivão, José Pedro. Analytic invariants for the $1:-1$ resonance. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1367-1426. doi : 10.5802/aif.2806. http://www.numdam.org/item/AIF_2013__63_4_1367_0/

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