Complex normal form for strongly nonlinear vibration systems exemplified by Duffing–van der Pol Equation

Abstract

We extend the normal form method to study the asymptotic solutions of strongly non-linear oscillatorsu+ω²u=f](u,u), wheref(u,u) contains only linear and cubic non-linear terms. The novel contribution is the ansatzu=ξ+ξ,u=iω1(ξ−ξ) where ω1is to be determined, allowing for the change of the fundamental frequency during the course of vibration, rather than usingu=ξ+ξ,u=iω(ξ−ξ) as suggested by Nayfeh. With the present method, not only the stability of the periodic solutions but also the asymptotic expressions for the periodic solutions can be obtained easily. The results obtained by the method presented coincide very well with the results obtained by numerical integration for the Duffing–van der Pol oscillator withf(u,u)=μ(1−u²)u−βu³. When ω=μ=β=1, Nayfeh's method givers qualitatively different results from the numerical integration while our method works well even when ω=1, μ=β=3, since Nayfeh's method is based on weak non-linearities and ω=1, μ=β=3 is beyond the valid range of assumption.