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Resurgence of inner solutions for perturbations of the McMillan map. (English) Zbl 1230.37069

Summary: A sequence of “inner equations” attached to certain perturbations of the McMillan map was considered in [the authors, ibid. 31, No. 2, 301–372 (2011; Zbl 1230.37070)], where their solutions were used to measure an exponentially small separatrix splitting. We prove here all the results related to these equations which are necessary to complete the proof of the main result of [loc. cit.]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle’s alien derivations to measure the discrepancy between different Borel-Laplace sums.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations

Citations:

Zbl 1230.37070
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