  A periodic perturbation generates a complicated dynamics close to
separatrices and saddle points. We construct an asymptotic solution which is
close to the separatrix for the unperturbed Duffing's oscillator over a long
time. This solution is defined by a separatrix map. This map is obtained for
any order of the perturbation parameter. Properties of this map show an
instability of a motion for the perturbed system.
