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

AUTHORS:
Ainoa Murillo^(1), Arturo Vieiro^(1),

(1) Department de Matematiques i Informatica,
    Universitat de Barcelona (UB), Barcelona, Spain.

ABSTRACT:

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 behaviour of the exponentially small splitting of
invariant manifolds in analytic near-integrable maps $F_\varepsilon$. For a
given $\varepsilon$, 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.