Global bifurcations in periodically perturbed gyroscopic systems with application to rotating shafts

In this paper, we examine global bifurcations in two degree of freedom conservative nonlinear gyroscopic systems which are periodically perturbed. We study the effect of these periodic perturbations near a double zero eigenvalue of the linear system in the presence of symmetry-breaking. After determining the normal form for the Hamiltonian, we study the unperturbed system and find that parameter regions exist in which homoclinic and heteroclinic cycles are present. Using the Melnikov method for perturbations of Hamiltonian systems, we determine that, under perturbation, the homoclinic cycles break, and the stable and unstable manifolds of the normally hyperbolic invariant manifold intersect transversally. These transverse intersections generate Smale horseshoes, which result in chaotic phenomena.