Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications

A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincaré sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (“one-half” Melnikov functions) and the distance between perturbed stable and unstable manifolds (“full” Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.