Homoclinic orbits to invariant tori near a homoclinic orbit to center-center-saddle equilibrium
The paper deals with the behaviour of the perturbed Hamiltonian \(H= H_0+\varepsilon H_1\) on a neighborhood of one loop in the phase space. First, the authors consider the local map for the perturbed Hamiltonian. With the help of the center manifold theory and KAM theory, the authors introduce new symplectic coordinates such that the persistent hyperbolic invariant tori, as well as their stable and unstable manifolds, become flat. Further, the authors consider the perturbed global map and study the existence of homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic tori on the perturbed center manifold. Depending on the preserving (or not preserving) of a homoclinic loop, the authors analyse a number of transverse homoclinic loops. Using the Poincaré-Melnikov method, they prove that a first approximation to the splitting distance between the invariant manifolds can be expressed in terms of a Melnikov potential defined on a torus \(\mathbb{T}^2\). Hence, the transverse homoclinic orbits can be detected as nondegenerate critical points of the Melnikov potential.
scientific article; zbMATH DE number 2167901