The splitting of separatrices for analytic diffeomorphisms

Abstract

Abstract. We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given. IntroductionIn a previous paper [5] we considered families of differentiable diffeomorphisms with hyperbolic points which reduce to the identity for a certain value of the parameter. We studied the separation between the stable and unstable manifolds near homo-heteroclinic points associated with the hyperbolic ones. Now we shall study the analytic and conservative case. We refer the reader to [5] for the motivations of this problem. The results were inspired by the previous work by Lazutkin [10] for the standard map.We study the complex invariant manifolds through the Birkhoff normal form of a diffeomorphism in a neighbourhood of a hyperbolic point. For that we need uniform behaviour of the normal form with respect to the parameter of the family. As in the differentiable case we use an auxiliary family of diffeomorphisms defined through the flow of an autonomous vector field with a homoclinic orbit. We compare the invariant manifolds of the two families, first locally and then globally. From the Birkhoff normal form we get a local first integral for the diffeomorphisms, which can be extended in a neighbourhood of the invariant manifolds as a multivalued function. With a suitable parametrization of the invariant manifolds, the evaluation of the first integral extended along one manifold on the other manifold gives a periodic function. The Fourier coefficients of that function can be bounded by using integration over complex paths. From that we get the bound of the separation between the invariant manifolds. Now we state the main result in a precise way.

The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems

“…The complete proof of it can be found in [14]. In this context in [12] exponentially small upper bounds for the splitting of separatrices are proved for analytic families of diffeomorphisms close to the identity. In [6], is proved an asymptotic expression for the splitting of separatrices for some perturbations of the McMillan map, which is also exponentially small and, in fact, coincides with the prediction given by the Poincaré-Melnikov function.…”

Towards global models near homoclinic tangencies of dissipative diffeomorphisms

“…2 ) terms maintaining the conservative character. This generically gives an exponentially small splitting of the separatrices (see [21]) such that, in a suitable parametrization, the unstable manifold with respect to the stable manifold, is locally given by an expression of the form n 1 a n sin(nz + ϕ n ), where a n = O(exp(−nc To model a return map we split the diffeomorphism as the composition of two maps, one near the saddle, the other being the reinjection (figure 71).…”

“…Then area-preserving arguments show the existence of homoclinic points. Generically (see again [21]) the oscillation of W u with respect to W s is modelled by a sinusoidal function (higher-order harmonics being much less important). figure 85 shows W u , W s near the origin.…”

A normally elliptic Hamiltonian bifurcation

“…In real analytic cases sometimes also more explicit, viz. exponential, estimates can be given, on the splitting of the separatrices, for instance compare Holmes, Scheurle and Marsden [26] or Fontich and Sim6 [20]. A question is how to apply these methods in the present setting.…”