Borel summation and splitting of separatrices for the Hénon map
[Sommation de Borel et écart des séparatrices de l'application de Hénon]
Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 513-567.

L’application de Hénon est une transformation symplectique de 2 possédant un point fixe parabolique auquel sont associées deux variétés invariantes complexes (les séparatrices). Une unique série formelle correspond à ces deux variétés, et nous étudions sa transformée de Borel formelle. Nous prouvons qu’elle définit un germe analytique et étudions sa surface de Riemann, ainsi que les singularités de son prolongement analytique. Nous donnons en particulier une description complète de la “première singularité”, et démontrons qu’une certaine constante qui détermine l’écart des séparatrices n’est pas nulle. Ces résultats sont aussi présentés dans le langage de la théorie de la résurgence.

We study two complex invariant manifolds associated with the para\-bolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the "first" singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation in terms of Resurgence theory is also given.

DOI : 10.5802/aif.1831
Classification : 37J10, 37D30, 40G10, 37E30, 37F45
Keywords: Hénon map, difference equations, splitting of separatrices, Borel summation, Laplace transform, resurgence
Mot clés : application de Hénon, équations aux différences, écart des séparatrices, sommation de Borel, transformation de Laplace, résurgence
Gelfreich, Vassili 1 ; Sauzin, David 2

1 The Steklov Mathematical Institute at St. Petersburg, St. Petersburg (Russie)
2 Institut de Mécanique Céleste, Astronomie et systèmes Dynamiques, 77 avenue Denfert-Rochereau, 75014 Paris (France)
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Gelfreich, Vassili; Sauzin, David. Borel summation and splitting of separatrices for the Hénon map. Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 513-567. doi : 10.5802/aif.1831. http://www.numdam.org/articles/10.5802/aif.1831/

[BSSV98] C. Bonet; D. Sauzin; T. M. Seara; M. Valéncia Adiabatic invariant of the harmonic oscillator, complex matching and resurgence, SIAM J. Math. Anal., Volume 29 (1998) no. 6, pp. 1335-1360 | DOI | MR | Zbl

[Che98] V. Chernov On separatrix splitting of some quadratic area-preserving maps of the plane, Regular \& Chaotic Dynamics, Volume 3 (1998) no. 1, pp. 49-65 | DOI | MR | Zbl

[CNP93] B. Candelpergher; J.C. Nosmas; F. Pham Approche de la résurgence, Actualités Math., Hermann, Paris, 1993 | MR | Zbl

[Eca81] J. Écalle Les fonctions résurgentes, vol. 2, Publ. Math. d'Orsay, Paris, 1981

[Eca93] J. Écalle; D. Schlomiuk (ed.) Six lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's conjecture, Bifurcations and Periodic Orbits of Vector Field (1993), pp. 75-184 | Zbl

[FS90] E. Fontich; C. Simó Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Th. and Dynam. Sys., Volume 10 (1990), pp. 319-346 | MR | Zbl

[Gel00] V.G. Gelfreich Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps, Physica D, Volume 136 (2000), pp. 266-279 | DOI | MR | Zbl

[Gel91] V.G. Gelfreich; M. Sh. Birman (ed.) Separatrices splitting for polynomial area-preserving maps (Topics in Math. Phys.), Volume vol. 13 (1991), pp. 108-116

[Gel99] V.G. Gelfreich A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., Volume 201 (1999), pp. 155-216 | DOI | MR | Zbl

[GLS94] V. G. Gelfreich; V. F. Lazutkin; N. V. Svanidze A refined formula for the separatrix splitting for the standard map, Physica D, Volume 71 (1994) no. 2, pp. 82-101 | DOI | MR | Zbl

[GLT91] V.G. Gelfreich; V.F. Lazutkin; M.B. Tabanov Exponentially small splitting in Hamiltonian systems, Chaos, Volume 1 (1991) no. 2, pp. 137-142 | DOI | MR | Zbl

[HM93] V. Hakim; K. Mallick Exponentially small splittings of separatrices, matching in the complex plane and Borel summation, Nonlinearity, Volume 6 (1993), pp. 57-70 | DOI | MR | Zbl

[Laz84] V.F. Lazutkin Splitting of separatrices for the standard map, VINITI (1984)

[Laz93] V.F. Lazutkin Resurgent approach to the separatrices splitting, Equadiff91, International conference on differential equations, Barcelona 1991, Volume vol. 1 (1993), pp. 163-176 | Zbl

[LST89] V.F. Lazutkin; I.G. Schachmanski; M.B. Tabanov Splitting of separatrices for standard and semistandard mappings, Physica D, Volume 40 (1989), pp. 235-348 | DOI | MR | Zbl

[Mal95] B. Malgrange Resommation des séries divergentes, Expo. Math., Volume 13 (1995), pp. 163-222 | MR | Zbl

[Sur94] Yu.B. Suris On the complex separatrices of some standard-like maps, Nonlinearity, Volume 7 (1994) no. 4, pp. 1225-1236 | DOI | MR | Zbl

[Tov94] A. Tovbis Asymptotics beyond all orders and analytic properties of inverse Laplace transforms of solutions, Comm. Math. Phys., Volume 163 (1994), pp. 245-255 | DOI | MR | Zbl

[TTJ98] A. Tovbis; M. Tsuchiya ; C. Jaffe Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, Volume 8 (1998), pp. 665-681 | DOI | MR | Zbl

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