Analytic invariants for the 1:-1 resonance
[Invariants analytiqués pour le 1:-1 résonance]
Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1367-1426.

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.

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.

DOI : 10.5802/aif.2806
Classification : 37J20, 34M40, 34M30
Keywords: analytic classification, Stokes phenomenon, splitting of separatrices
Mot clés : classification analytique, phénomène de Stokes, l’écart des séparatrices
Gaivão, José Pedro 1

1 Cemapre Rua do Quelhas 6 1200-781 Lisboa Portugal
@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},
     pages = {1367--1426},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     doi = {10.5802/aif.2806},
     zbl = {06359592},
     mrnumber = {3137358},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2806/}
}
TY  - JOUR
AU  - Gaivão, José Pedro
TI  - Analytic invariants for the $1:-1$ resonance
JO  - Annales de l'Institut Fourier
PY  - 2013
SP  - 1367
EP  - 1426
VL  - 63
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2806/
DO  - 10.5802/aif.2806
LA  - en
ID  - AIF_2013__63_4_1367_0
ER  - 
%0 Journal Article
%A Gaivão, José Pedro
%T Analytic invariants for the $1:-1$ resonance
%J Annales de l'Institut Fourier
%D 2013
%P 1367-1426
%V 63
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2806/
%R 10.5802/aif.2806
%G en
%F AIF_2013__63_4_1367_0
Gaivão, José Pedro. Analytic invariants for the $1:-1$ resonance. Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1367-1426. doi : 10.5802/aif.2806. http://www.numdam.org/articles/10.5802/aif.2806/

[1] Baldomá, I.; Seara, T. M. The inner equation for generic analytic unfoldings of the Hopf-zero singularity, Discrete Contin. Dyn. Syst. Ser. B, Volume 10 (2008) no. 2-3, pp. 323-347 | MR | Zbl

[2] Burgoyne, N.; Cushman, R. Normal forms for real linear hamiltonian systems with purely imaginary eigenvalues, Celestial Mechanics, Volume 8 (1974), pp. 435-443 | DOI | MR | Zbl

[3] Chapman, S. J.; King, J. R.; Adams, K. L. Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 454 (1998) no. 1978, pp. 2733-2755 | DOI | MR | Zbl

[4] Devaney, Robert L. Homoclinic orbits in Hamiltonian systems, Differential Equations, Volume 21 (1976) no. 2, pp. 431-438 | MR | Zbl

[5] Dieudonne, J. Foundations of modern analysis, Academic Press, New York and London, 1969 | MR | Zbl

[6] Écalle, J. Singularités non abordables par la géométrie, Ann. Inst. Fourier (Grenoble), Volume 42 (1992) no. 1-2, pp. 73-164 | DOI | EuDML | Numdam | MR | Zbl

[7] Folland, Gerald B. Introduction to partial differential equations, Princeton University Press, Princeton, NJ,, 1995 | MR | Zbl

[8] Gaivao, J. P. Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation, University of Warwick (2010) (Ph. D. Thesis)

[9] Gaivao, J. P.; Gelfreich, V. Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example, Nonlinearity, Volume 24 (2011) no. 3, pp. 677-698 | DOI | MR | Zbl

[10] Gelfreich, V. Reference systems for splittings of separatrices, Nonlinearity, Volume 10 (1997) no. 1, pp. 175-193 | DOI | MR | Zbl

[11] Gelfreich, V. A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., Volume 101 (1999), pp. 155-216 | DOI | MR | Zbl

[12] Gelfreich, V.; Naudot, V. Analytic invariants associated with a parabolic fixed point in 2 , Ergodic Theory and Dynamical Systems, Volume 28 (2008) no. 6, pp. 1815-1848 | DOI | MR | Zbl

[13] Gelfreich, V.; Sauzin, D. Borel summation and splitting of separatrices for the hénon map, Annales de l’institut Fourier, Volume 51 (2001) no. 2, p. 513-267 | DOI | Numdam | MR | Zbl

[14] Gelfreich, V. G. Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps, Phys. D, Volume 136 (2000) no. 3-4, pp. 266-279 | DOI | MR | Zbl

[15] Glebsky, L. Yu.; M., Lerman On small stationary localized solutions for the generalized 1d swift-hohenberg equation, Chaos: Internat. J. Nonlin. Sci., Volume 5 (1995) no. 2 | DOI | Zbl

[16] Lazutkin, V. F. An analytic integral along the separatrix of the semistandard map: existence and an exponential estimate for the distance between the stable and unstable separatrices, Algebra i Analiz, Volume 4 (1992) no. 4, pp. 110-142 | MR | Zbl

[17] Lazutkin, V. F. Splitting of separatrices for the chirikov standard map, VINITI, (Moscow 1984) no. 6372/84

[18] Lerman, L. M. Dynamical phenomena near a saddle-focus homoclinic connection in a Hamiltonian system, J. Statist. Phys., Volume 101 (2000) no. 1-2, pp. 357-372 | DOI | MR | Zbl

[19] Lerman, L. M.; Belyakov, L. A. Stationary localized solutions, fronts and traveling fronts to the generalized 1d swift-hohenberg equation, EQUADIFF 2003: Proceedings of the International Conference on Differential Equations (2003), pp. 801-806 | MR | Zbl

[20] Lerman, L. M.; Markova, A. P. On stability at the Hamiltonian Hopf bifurcation, Regul. Chaotic Dyn., Volume 14 (2009) no. 1, pp. 148-162 | DOI | MR | Zbl

[21] Lochak, J.-P. P. Marco; Sauzin, D. On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163, 2003, no. 775 | MR | Zbl

[22] Martín, P.; Sauzin, D.; Seara, T. M. Resurgence of inner solutions for perturbations of the McMillan map, Discrete Contin. Dyn. Syst., Volume 31 (2011) no. 1, pp. 165-207 | DOI | MR | Zbl

[23] Martinet, J.; Ramis, J.-P. Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math., Volume 55 (1982), pp. 63-164 | DOI | Numdam | MR | Zbl

[24] McSwiggen, P. D.; Meyer, K. R. The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations, Journal of Differential Equations, Volume 189 (2003), pp. 538-555 | DOI | MR | Zbl

[25] Olivé, C.; Sauzin, D.; Seara, T. M. Resurgence in a Hamilton-Jacobi equation, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 4, pp. 1185-1235 | DOI | Numdam | MR | Zbl

[26] Sauzin, D. A new method for measuring the splitting of invariant manifolds, Ann. Sci. École Norm. Sup. (4), Volume 34 (2001) no. 2, pp. 159-221 | Numdam | MR | Zbl

[27] Sauzin, D. Resurgent functions and splitting problems, New Trends and Applications of Complex Asymptotic Analysis: Around Dynamical Systems, Summability, Continued Fractions (RIMS Kôkyûroku), Volume 1493, Res. Inst. Math. Sci., Kyoto, 2006, pp. 48-117

[28] Sokol’skiĭ, A. G. On the stability of an autonomous Hamiltonian system with two degrees of freedom in the case of equal frequencies, Prikl. Mat. Meh., Volume 38 (1974), pp. 791-799 | MR | Zbl

[29] Stolovitch, L. Classification analytique de champs de vecteurs 1-résonnants de (C n ,0), Asymptotic Anal., Volume 12 (1996), pp. 91-143 | MR | Zbl

[30] van der Meer, Jan-Cees The Hamiltonian Hopf bifurcation, Lecture Notes in Mathematics, 1160, Springer-Verlag, Berlin, 1985 | MR | Zbl

[31] Zung, N. T. Convergence versus integrability in Birkhoff normal form, Ann. of Math. (2), Volume 161 (2005) no. 1, pp. 141-156 | DOI | MR | Zbl

Cité par Sources :