Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation

Abstract

Quasi-Periodic (QP) solutions are investigated for a weakly dampednonlinear QP Mathieu equation. A double parametric primary resonance(1:2, 1:2) is considered. To approximate QP solutions, a double multiple-scales method is applied to transform the original QP oscillator to anautonomous system performing two successive reductions. In a first step,the multiple-scales method is applied to the original equation to derive afirst reduced differential amplitude-phase system having periodiccomponents. The stability of stationary solutions of this reduced systemis analyzed. In a second step, the multiple-scales method is applied again tothe first reduced system (RS) to obtain a second autonomous differentialamplitude-phase RS. The problem for approximating QP solutions of theoriginal system is then transformed to the study of stationary regimesof the induced autonomous second RS. Explicit analytical approximationsto QP solutions are obtained and comparisons to numerical integrationare provided.