A fifth-order multiple-scale solution for Hopf bifurcations

The subject of this paper is the multiple-time-scale analysis of Hopf bifurcations up to fifth-order nonlinearities. It is shown how an asymptotic fifth-order expansion captures the change in the nature of the limit cycle from stable to unstable and viceversa. The formulation is validated by applying it to a simple mechanical system for which there exists an analytical limit-cycle solution. Applications include the pre- and post-flutter behavior of a typical section with nonlinear spring having a stable limit cycle (supercritical Hopf bifurcation) that turns into an unstable one, because of fifth-order nonlinearities.