Sauzin, David Résurgence paramétrique et exponentielle petitesse de l’écart des séparatrices du pendule rapidement forcé. (Parametric resurgence and exponential smallness of the splitting of the separatrices of the rapidly forced pendulum.). (French) Zbl 0826.30004 Ann. Inst. Fourier 45, No. 2, 453-511 (1995). Summary: Henri Poincaré had already noticed that the stable and unstable manifolds of the perturbed pendulum defined by the Hamiltonian \[ H(q,p,t)=p^2/ 2+(-1+\text{cos} q)(1- \mu \text{sin}(t/\varepsilon)), \] do not coincide when parameter \(\mu\) is not equal to zero, and that the same formal divergent series in powers of \(\varepsilon\) may be associated with both of them. Here this divergence is analyzed by means of the recent theory of resurgence and alien calculus which allows to estimate asymptotically the size of the splitting of the manifolds as \(\varepsilon\) tends to zero - at least this is proven for the simplified problem where \(\text{sin}(t/\varepsilon)\) is replaced with \(e^{it/\varepsilon}\). Cited in 8 Documents MSC: 30B99 Series expansions of functions of one complex variable 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 40C99 General summability methods 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:resurgence; divergent series; splitting of separatrices; asymptotic analysis; alien derivations; Hamilton-Jacobi equation × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [P] , Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris, 1893. · JFM 25.1847.03 [2] [H] , A Multiplier Theorem for Schrödinger operators, Colloquium Math., LX/LXI (1990), 659-664. · Zbl 0779.35025 [3] [KES] , and , Transcendentally Small Transversality on the Rapidly Forced Pendulum, Preprint (1992), 40 p. · Zbl 0782.34052 [4] [DS] and , An Asymptotic Expression for the Splitting of Separatrices of the Rapidly Forced Pendulum, Commun. Math. Phys., 150 (1992), 433-463. · Zbl 0765.70016 [5] [E1] , Les fonctions résurgentes, vol. 3, L’équation du pont et la classification analytique des objets locaux, Publ. Math. Université Paris-Sud, Orsay, 1985. · Zbl 0602.30029 [6] [E1’] , Singularités non abordables par la géométrie, Ann. Inst. Fourier, Grenoble, 42, 1-2 (1992), 73-164. · Zbl 0940.32013 [7] [E2] , Weighted products and parametric resurgence, Prépub. Math., 54, Université Paris-Sud, Orsay (1992), 43 p., à paraître dans Proc. Franco-Japanese Colloq. on Stokes Phenomena (Luminy, Déc. 1990), Boutet de Monvel ed., Lectures Notes in Math., Springer-Verlag. · Zbl 0834.34067 [8] [L1] , Exponential splitting of separatrices and an analytical integral for the semistandard map, Prépub. Math., 7, Université Paris 7 (1991), 53 p. [9] [L2] , Resurgence of the separatrices of the semistandard map, Preprint Forschungsinstitut für Mathematik, ETH, Zürich (1991), 14 p. [10] [T] , An averaging method for Hamiltonian systems, exponentially close to integrable ones, Preprint Moscow State University (1993), 23 p. [11] [S] , Résurgence paramétrique et exponentielle petitesse de l’écart des séparatrices du pendule rapidement forcé, Thèse de doctorat à l’Université Paris 7 Denis Diderot. [12] [CNP] , et , Approche de la résurgence, Actualités Math. Hermann, Paris, 1993. · Zbl 0791.32001 [13] [V] , The return of the quartic oscillator - The complex WKB method, Ann. Inst. Henri Poincaré, 39, n° 3 (1983), 211-338. · Zbl 0526.34046 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.