Stability boundaries analysis of non-autonomous systems with resonant solutions based on subharmonic Melnikov functions

Abstract

This paper addresses stability boundaries in non-autonomous systems. An analytical criterion for stability boundaries in one degree of freedom (time-periodic) perturbed Hamiltonian systems was recently proposed. The criterion evaluates basin boundaries of non-resonant solutions. This paper discusses the stability boundaries with respect to the resonant solutions based on the above result and subharmonic Melnikov functions. At first one degree of freedom perturbed (time-independent) Hamiltonian systems for the resonant solutions is derived using coordinates transformations and second order averaging. Then an approximate expression for the basin boundaries of the resonant solutions is obtained based on the above analytical criterion. This paper also exhibits the effectiveness of the approximate expression through a simple example.