Splitting of separatrices in a family of area-preserving maps that unfolds a fixed point at the resonance of order three

Abstract

We study the exponentially small splitting of separatrices in an analytic one-parameter family of area-preserving maps that generically unfolds a 1:3 resonance. Near the resonance the normal form theory predicts existence of a small triangle formed by separatrices of a period three hyperbolic point. We prove that in a generic family the separatrices split, provided that the Stokes constant of the map does not vanish. This constant describes the distance between the analytical continuations of invariant manifolds associated with the degenerate saddle of the map at the exact resonance. We provide an asymptotic formula which describes the size of the splitting. The leading term of this asymptotic formula is proportional to the Stokes constant.