Reduced-order modelling of parameter-dependent systems with invariant manifolds: Application to Hopf bifurcations in follower force problems

The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest is investigated, and the ability to predict the amplitudes of the limit cycle oscillations after the bifurcation is demonstrated. The cases of the Ziegler pendulum and Beck's column, both of which have a follower force, are considered for applications. By comparison with the eigenvalue trajectories in the conservative case, it is advocated that using two master modes to derive the ROM, instead of only considering the unstable one, should give more accurate results. Also, in the specific case where an exceptional bifurcation point is met, a numerical strategy enforcing the presence of Jordan blocks in the Jacobian matrix during the procedure is devised. The ROMs are constructed for the Ziegler pendulum having two and three degrees of freedom, and then Beck's column is investigated, where a finite element procedure is used to spatially discretise the problem. The numerical results show the ability of the ROMs to correctly predict the amplitude of the limit cycles up to a certain range, and it is shown that computing the ROM after the Hopf bifurcation gives the most satisfactory results. This feature is analysed in terms of phase space representations, and the two proposed adjustments are shown to improve the validity range of the ROMs.