A Comparison of Perturbation Methods

Perturbation techniques are frequently used in the study of nonlinear dynamical systems, in particular, for bifurcation and stability analyses. In this paper, four most popular perturbation methods – Lindstedt-Poincaré (LP) approach, timeaveraging (TA), multiple time scales (MTS) and the intrinsic harmonic balancing (IHB) – are particularly considered to compare their accuracy of the bifurcation solutions and the efficiency of the computation of normal forms. The results show that the MTS is the best method, which can be used to find not only the accurate bifurcation solution but also the normal form. The LP yields the same accurate approximate solution, but is not applicable for stability analysis. The TA method, on the other hand, has the simplest procedure to generate the normal form, but its approximate solution is less accurate. The IHB technique does not need to solve differential equations, but is not easy to obtain the normal form, especially for higher codimensional systems.