Free vibration of a strong non-linear system described with complex functions

In this paper two approximate analytical methods for solving strong non-linear differential equations with complex functions are developed. Besides the adopted elliptic Krylov–Bogolubov procedure, an alternative analytical method based on the variation of the parameters as well as the elliptic Krylov–Bogolubov is introduced. The difference is that the alternative method considers two approximate first order differential equations. The suggested procedures are compared and tested for the system with strong cubic non-linearity and a small non-linearity of Van der Pol type. Comparing the approximate analytical solutions with exact numerical solution it is concluded that the difference is negligible.