Harmonic balance, Melnikov method and nonlinear oscillators under resonant perturbation

The Subharmonic Melnikov's method is a classical tool for the analysis of subharmonic orbits in weakly perturbed nonlinear oscillators, but its application requires the availability of an analytical expression for the periodic trajectories of the unperturbed system. On the other hand, spectral techniques, like the harmonic balance, have been widely applied for the analysis and design of nonlinear oscillators. In this manuscript we show that bifurcations of subharmonic orbits in perturbed systems can be easily detected computing the Melnikov's integral over the harmonic balance approximation of the unperturbed orbits. The proposed method significantly extend the applicability of the Melnikov's method since the orbits of any nonlinear oscillator can be approximated by the harmonic balance technique, and the integrability of the unperturbed system is no more required.