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Vassili Gelfreich; David Sauzin
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, 51 no. 2 (2001), p. 513-567, doi: 10.5802/aif.1831
Article PDF | Reviews MR 1824963 | Zbl 0988.37031 | 5 citations in Cedram
Class. Math.: 37J10, 37D30, 40G10, 37E30, 37F45
Keywords: Hénon map, difference equations, splitting of separatrices, Borel summation, Laplace transform, resurgence
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.
[BSSV98] C. Bonet, D. Sauzin, T. M. Seara & M. Valéncia, “Adiabatic invariant of the harmonic oscillator, complex matching and resurgence”, SIAM J. Math. Anal. 29 (1998) no. 6, p. 1335-1360 Article | MR 1638050 | Zbl 0913.58019
[CNP93] B. Candelpergher, J.C. Nosmas & F. Pham, Approche de la résurgence, Actualités Math., Hermann, Paris, 1993 MR 1250603 | Zbl 0791.32001
[Che98] V. Chernov, “On separatrix splitting of some quadratic area-preserving maps of the plane”, Regular \& Chaotic Dynamics 3 (1998) no. 1, p. 49-65 Article | MR 1652168 | Zbl 0924.58065
[Eca81] J. Écalle, Les fonctions résurgentes vol. 2, Publ. Math. d'Orsay, Paris, 1981 Article
[Eca93] J. Écalle, Six lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's conjecture, Kluwer Ac. Publishers, 1993, p. 75-184 Zbl 0814.32008
[FS90] E. Fontich & C. Simó, “Invariant manifolds for near identity differentiable maps and splitting of separatrices”, Ergod. Th. and Dynam. Sys. 10 (1990), p. 319-346 MR 1062761 | Zbl 0706.58060
[Gel91] V.G. Gelfreich, Separatrices splitting for polynomial area-preserving maps, Topics in Math. Phys., Leningrad State University (Russian), 1991, p. 108-116
[Gel99] V.G. Gelfreich, “A proof of the exponentially small transversality of the separatrices for the standard map”, Comm. Math. Phys. 201 (1999), p. 155-216 Article | MR 1669417 | Zbl 1042.37044
[Gel00] V.G. Gelfreich, “Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps”, Physica D 136 (2000), p. 266-279 Article | MR 1733057 | Zbl 0942.37016
[GLS94] V. G. Gelfreich, V. F. Lazutkin & N. V. Svanidze, “A refined formula for the separatrix splitting for the standard map”, Physica D 71 (1994) no. 2, p. 82-101 Article | MR 1264110 | Zbl 0812.70017
[GLT91] V.G. Gelfreich, V.F. Lazutkin & M.B. Tabanov, “Exponentially small splitting in Hamiltonian systems”, Chaos 1 (1991) no. 2, p. 137-142 Article | MR 1135901 | Zbl 0899.58016
[HM93] V. Hakim & K. Mallick, “Exponentially small splittings of separatrices, matching in the complex plane and Borel summation”, Nonlinearity 6 (1993), p. 57-70 Article | MR 1203052 | Zbl 0769.34036
[Laz84] V.F. Lazutkin, “Splitting of separatrices for the standard map”, VINITI (1984)
[Laz93] V.F. Lazutkin, Resurgent approach to the separatrices splitting, World Scientific, 1993, p. 163-176 Zbl 0938.37524
[LST89] V.F. Lazutkin, I.G. Schachmanski & M.B. Tabanov, “Splitting of separatrices for standard and semistandard mappings”, Physica D 40 (1989), p. 235-348 Article | MR 1029465 | Zbl 0825.58033
[Mal95] B. Malgrange, “Resommation des séries divergentes”, Expo. Math. 13 (1995), p. 163-222 MR 1346201 | Zbl 0836.40004
[Sur94] Yu.B. Suris, “On the complex separatrices of some standard-like maps”, Nonlinearity 7 (1994) no. 4, p. 1225-1236 Article | MR 1284689 | Zbl 0813.58024
[Tov94] A. Tovbis, “Asymptotics beyond all orders and analytic properties of inverse Laplace transforms of solutions”, Comm. Math. Phys. 163 (1994), p. 245-255 Article | MR 1284784 | Zbl 0804.44001
[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 8 (1998), p. 665-681 Article | MR 1645522 | Zbl 0987.37022
Vassili Gelfreich; David Sauzin
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, 51 no. 2 (2001), p. 513-567, doi: 10.5802/aif.1831
Article PDF | Reviews MR 1824963 | Zbl 0988.37031 | 5 citations in Cedram
Class. Math.: 37J10, 37D30, 40G10, 37E30, 37F45
Keywords: Hénon map, difference equations, splitting of separatrices, Borel summation, Laplace transform, resurgence
Résumé - Abstract
Bibliography
[CNP93] B. Candelpergher, J.C. Nosmas & F. Pham, Approche de la résurgence, Actualités Math., Hermann, Paris, 1993 MR 1250603 | Zbl 0791.32001
[Che98] V. Chernov, “On separatrix splitting of some quadratic area-preserving maps of the plane”, Regular \& Chaotic Dynamics 3 (1998) no. 1, p. 49-65 Article | MR 1652168 | Zbl 0924.58065
[Eca81] J. Écalle, Les fonctions résurgentes vol. 2, Publ. Math. d'Orsay, Paris, 1981 Article
[Eca93] J. Écalle, Six lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's conjecture, Kluwer Ac. Publishers, 1993, p. 75-184 Zbl 0814.32008
[FS90] E. Fontich & C. Simó, “Invariant manifolds for near identity differentiable maps and splitting of separatrices”, Ergod. Th. and Dynam. Sys. 10 (1990), p. 319-346 MR 1062761 | Zbl 0706.58060
[Gel91] V.G. Gelfreich, Separatrices splitting for polynomial area-preserving maps, Topics in Math. Phys., Leningrad State University (Russian), 1991, p. 108-116
[Gel99] V.G. Gelfreich, “A proof of the exponentially small transversality of the separatrices for the standard map”, Comm. Math. Phys. 201 (1999), p. 155-216 Article | MR 1669417 | Zbl 1042.37044
[Gel00] V.G. Gelfreich, “Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps”, Physica D 136 (2000), p. 266-279 Article | MR 1733057 | Zbl 0942.37016
[GLS94] V. G. Gelfreich, V. F. Lazutkin & N. V. Svanidze, “A refined formula for the separatrix splitting for the standard map”, Physica D 71 (1994) no. 2, p. 82-101 Article | MR 1264110 | Zbl 0812.70017
[GLT91] V.G. Gelfreich, V.F. Lazutkin & M.B. Tabanov, “Exponentially small splitting in Hamiltonian systems”, Chaos 1 (1991) no. 2, p. 137-142 Article | MR 1135901 | Zbl 0899.58016
[HM93] V. Hakim & K. Mallick, “Exponentially small splittings of separatrices, matching in the complex plane and Borel summation”, Nonlinearity 6 (1993), p. 57-70 Article | MR 1203052 | Zbl 0769.34036
[Laz84] V.F. Lazutkin, “Splitting of separatrices for the standard map”, VINITI (1984)
[Laz93] V.F. Lazutkin, Resurgent approach to the separatrices splitting, World Scientific, 1993, p. 163-176 Zbl 0938.37524
[LST89] V.F. Lazutkin, I.G. Schachmanski & M.B. Tabanov, “Splitting of separatrices for standard and semistandard mappings”, Physica D 40 (1989), p. 235-348 Article | MR 1029465 | Zbl 0825.58033
[Mal95] B. Malgrange, “Resommation des séries divergentes”, Expo. Math. 13 (1995), p. 163-222 MR 1346201 | Zbl 0836.40004
[Sur94] Yu.B. Suris, “On the complex separatrices of some standard-like maps”, Nonlinearity 7 (1994) no. 4, p. 1225-1236 Article | MR 1284689 | Zbl 0813.58024
[Tov94] A. Tovbis, “Asymptotics beyond all orders and analytic properties of inverse Laplace transforms of solutions”, Comm. Math. Phys. 163 (1994), p. 245-255 Article | MR 1284784 | Zbl 0804.44001
[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 8 (1998), p. 665-681 Article | MR 1645522 | Zbl 0987.37022
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310