Modified Lindsted-Poincare methods for some strongly nonlinear oscillations Part III: Double series expansion

In this paper, we propose a new perturbation technique for strongly nonlinear oscillations with two parameters, which need not to be small in the present study. In this new method, the solution is expanded into double series of the two parameters. In order to avoid the secular terms, a constant in the equation is also expressed in a double series expansion. The present technique can be widely applied to nonlinear systems with multitude nonlinear terms. The preliminary study shows that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly nonlinear systems, but also for strongly nonlinear systems.