Existence of quasi-periodic solutions of fast excited van der Pol-Mathieu-Duffing equation

Abstract

The van der Pol-Mathieu-Duffing equation ẍ+(Ω02+h1cosΩ1t+h2cosΩ2t)x−(α−βx2)ẋ−h3x3=h4Ω32cosxcosΩ3t is considered in this paper, where α, β, h1, h2, h3, h4, Ω1, Ω2 are small parameters, α, β > 0, the frequency Ω3 is large compared to Ω1 and Ω2, the above parameters are real. For ∀α, β > 0, we use KAM (Kolmogorov-Arnold-Moser) theory to prove that the van der Pol-Mathieu-Duffing equation possesses quasi-periodic solutions for most of the parameters Ω0, Ω1, Ω2, Ω3, it verifies some phenomenon of Fahsi and Belhaq [Commun. Nonlinear Sci. 14, 244-253 (2009)] and can be regarded as a extension of Abouhazim et al. [Nonlinear Dyn. 39, 395-409 (2005)].