2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224783A 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.17 pages, 4 figuresDynamical SystemsMathematical Physicstext