Pendulum: Separatrix Splitting

An exact expression for the determinant of the splitting matrix is derived: it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in Ann. Inst. H. Poincaré, B-60, 1, 1994. The asymtotic validity of Mel-nokov's formulae is proved for the class of models considered, which include polynomial perturbations. §1. Introduction. Recently V. Gelfreich noted that a " theorem " in [CG] contains an error. The theorem gave a lower bound on the splitting angles in a three degrees of freedom system and it was needed to show the existence of heteroclinic chains in a class of hamiltonian systems with the aim of an application to a celestial mechanics problem. We correct it here by providing the correct lower bound and, at the same time, exposing again and in a more meditated form some ideas of [CG]. In the present paper we do not discuss the existence of Arnold's diffusion, [A2], in our systems. We do not discuss the celestial mechanics application of [CG] either, as two parts of it (see below) relied on the erroneous statement, and more work is needed. The present paper is, therefore, not a correction of the implications of the error in [CG] but only of the error itself. In order to derive the same implications further work is necessary as the erroneous result was used several times in the last three sections of [CG]. Each use has therefore to be treated separately. Our analysis deals mainly with systems with three different time scales with the ratio between the smallest to the largest being ≪ 1: e.g. the largest is η −1/2 , the intermediate is 1 and the smallest is η a , a ≥ 0 (the a = 0 case being a limiting two scales problem). The result will be called the large angles theorem, (theorem 2 in §5, proved in §6, §7 for the systems defined in (2.1),(7.1) below), and describes a property of a pendulum subject to both a slow periodic force and to a rapid periodic force (incommensurate to the former) that we imagine generated by a pair of rotators, whose positions are given by two angles α, λ while the pendulum position is given by an angle ϕ. The angles α, λ will be respectively called slow and fast. When the system is perturbed some of the quasi periodic motions performed by rotators (or …