The splitting of separatrices for analytic diffeomorphisms. (English) Zbl 0706.58061

Continuing the study of the families of differentiable diffeomorphisms with hyperbolic points which reduce to the identity for a certain value of the parameter, from their previous paper ibid., 319-346 (1990; see the review above), the authors consider here the analytic and conservative case. The complex invariant manifolds are studied through the Birkhoff normal form of a diffeomorphism in a neighbourhood of a hyperbolic point. The main result states that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter, the exponent being related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given: the conservative (orientation preserving) Hénon map, the Duffing equation perturbed with a high frequency periodic function, a generalized standard map and the Hénon-Heiles problem.
Reviewer: L.Maxim-Răileanu


37C80 Symmetries, equivariant dynamical systems (MSC2010)
37C10 Dynamics induced by flows and semiflows
37D99 Dynamical systems with hyperbolic behavior


Zbl 0706.58060