Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits
The application of the Poincaré-Melnikov method to the effective detection of transverse homoclinic orbits, associated to an \(n\)-dimensional whiskered torus of a Hamiltonian with \(n+1\) degrees of freedom, needs to be carefully validated in the case of exponentially small splitting of separatrices, since this problem becomes singular with respect to the parameter \(\varepsilon \) of perturbation. Previous results are known only for \(n=1\). The present paper is devoted to an example of an integrable Hamiltonian with three degrees of freedom (i.e., \(n=2\)), possessing a two-dimensional hyperbolic invariant torus with fast frequencies \(\frac{\omega }{\sqrt{\varepsilon }}\) and coincident whiskers, plus a perturbation of order \(\varepsilon ^{p}\). \(\omega \) is chosen as the golden vector and then, the good arithmetic properties of \(\omega \) yield a splitting function with 4 simple zeros (corresponding to the nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits.
scientific article; zbMATH DE number 2139314