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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

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 \(e^{it/\varepsilon}\).

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
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References:

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