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 are 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 space discretize 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 analyzed in terms of phase space representations, and the two proposed adjustments are shown to improve the validity range of the ROMs.

翻译：调整不变流形的直接参数化方法以考虑参数变化。具体而言，研究了在关注参数范围内经历Hopf分岔的系统，并证明了该方法预测分岔后极限环振荡幅值的能力。以具有随从力的Ziegler摆和Beck柱作为应用案例进行研究。通过与保守情况下特征值轨迹的对比，论证了使用两个主模态（而非仅考虑不稳定模态）推导降阶模型可获得更精确的结果。此外，针对遇到异常分岔点的特殊情况，设计了一种在计算过程中强制雅可比矩阵存在Jordan块的数值策略。分别为具有两个和三个自由度的Ziegler摆构建了降阶模型，随后研究了Beck柱案例——该问题采用有限元方法进行空间离散化。数值结果表明，降阶模型能在一定范围内准确预测极限环幅值，且证明在Hopf分岔后计算降阶模型可获得最理想的结果。此特性通过相空间表征进行分析，并证实所提出的两项调整扩展了降阶模型的有效适用范围。