Modified Lindstedt–Poincare methods for some strongly non-linear oscillations

In this paper, a modified Lindstedt–Poincare method is proposed. In this technique, we introduce a new transformation of the independent variable. This transformation will also allow us to avoid the occurrence of secular terms in the perturbation series solution. Some examples are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly non-linear systems, but also for strongly non-linear systems.