Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation

Quasi-periodic (QP) solutions of a weakly damped non-linear QP Mathieu equation are investigated near a double primary parametric resonance. A double multiple scales method is applied to reduce the original QP oscillator to an autonomous system performing two successive reduction. The problem for approximating QP solutions of the original system is then transformed to the study of stationary regimes of the induced autonomous system. Explicit analytical approximations to QP oscillations are obtained and comparisons to numerical integration of the original QP oscillator are provided.