Detecting quasi-periodic properties of the splitting of separatrices via simultaneous approximation

The relation between the linear and the simultaneous approximation of a frequency vector leads to a methodology for detecting changes in the dominant harmonics of the asymptotic behavior of the exponentially small splitting of invariant manifolds in analytic near-integrable maps Fε. For a given ε, this reduces to computing the iterate of the map that is closest to the identity near the invariant manifolds. Using this idea, we describe the quasi-periodic properties of the splitting of two-dimensional invariant manifolds of fixed points in concrete families of near-integrable 3D volume-preserving and 4D symplectic maps.