Separatrix splitting for systems with three time scales

An exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales: it allows us t o analyze the asymptotic behaviour needed to amend the large angles theorem proposed in Ann. Inst, H. Poincare, B-60, 1 (1994). The asymptotic validit y of Mel'nikov's integrals is proved for the class of models considered, wh ich are polynomial perturbations. The technique for exhibiting cancellation s is inspired by renormalization theory in quantum electrodynamics and uses an analogue of Dyson's equations to prove an infinite family of identities , due to symmetries, that remind us of Ward's identities.