Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique

For the strongly nonlinear equations without small parameter, a transformation u = βv can be introduced for the nonlinear equation L(u) + N(u) = 0, where L and N are general linear and nonlinear differential operators respectively, β is sufficiently small. Therefore, a strongly non-linear system is transformed into a small parameter system with respect to the new introduced parameter β, thus various traditional perturbation techniques can be applied. By Lindstedt-Poincare method, a perturbation solution for Dufing equation with 5th order nonlinearity is obtained, which is valid not only for the small parameter ε in the equation, but also for very large values of ε.