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

TL;DR

This paper investigates the exponentially small splitting of separatrices in a family of area-preserving maps near a 1:3 resonance, providing an asymptotic formula linked to the Stokes constant.

Contribution

It proves that in a generic family, the separatrices split when the Stokes constant is non-zero, and derives an asymptotic expression for the splitting size.

Findings

Separatrices split when the Stokes constant does not vanish.

An asymptotic formula for the splitting size is derived.

The leading term of the splitting is proportional to the Stokes constant.

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.