Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds
The existence of transverse intersection of stable and unstable manifolds of hyperbolic invariant manifolds for a nonlinear dynamical system is, as well known, associated with chaotic behaviour. In the case of nearly integrable Hamiltonian system one deals with whiskered tori associated with resonances, the whiskers, being stable and unstable manifolds. A perturbative expansion shows that stable and unstable manifolds coincide up to any finite order in the perturbation parameter \(\varepsilon\), thus giving rise to separatrices. The whiskers in general, do not coincide, but their distance is exponentially small in \(\varepsilon\); one says that there is an exponentially small splitting of separatrices. This phenomenon was first remarked by Poincaré (by what is now known as the Poincaré-Melnikov method). The computation of the splitting of separatrices is a very relevant problem. Indeed, the existence of splitting with transverse homoclinic orbits implies the existence of heteroclinic orbits connecting wiskered tori (whose reciprocal distance is of the same order as the splitting distance), and is thus essential in Arnold diffusion. In the exponentially small case, application of Poincaré-Melnikov method (to detect the splitting) becomes rather delicate, due to its singular character. The present paper deals precisely with this case. Flow-box coordinates are introduced in a small neighbourhood having nonzero intersection with both whiskers (but zero intersection with the whiskered torus); in this way one straightens all of the coordinates (and not only half of them as in other approaches), obtaining coordinates which are global in the angular variables of the whiskers. These allow to take advantage of the quasiperiodicity properties of the splitting. With these coordinates, the authors are able to provide sharp upper bounds for exponentially small splitting of separatrices. They also provide an accurate upper bound for the error term of Melnikov function. The general results provided here are applied in a companion paper [\textit{A. Delshams} and \textit{P. Gutiérrez}, Discrete Contin. Dyn. Syst. 11, 757--783 (2004; Zbl 1062.37059)] to some concrete three-degree-of-freedom systems.
scientific article; zbMATH DE number 2139315