Asymptotic series for the splitting of separatrices near a Hamiltonian bifurcation

2008/06/14 by Vassili Gelfreich, Gelfreich, Vassili, Niklas Brännström +2
Computer Science · Mathematics · Physics and Astronomy · #37J20 #70K44 #70K50 #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #math.DS #msc:37J20 #msc:70K44 #msc:70K50

paper · pdf · doi:10.48550/arxiv.0806.2403

arxiv created 2008/06/14 · openalex publication_date 2008/06/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a proof of an asymptotic formula which describes exponentially small splitting of separatrices in a generic analytic family of area-preserving maps near a Hamiltonian saddle-centre bifurcation. As a particular case and in combination with an earlier work on a Stokes constant for the Hénon map (Gelfreich, Sauzin (2001)), it implies exponentially small transversality of separatrices in the area-preserving Hénon family when the multiplicator of the fixed point is close to one.

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