2:1:1 Resonance in the Quasi-Periodic Mathieu Equation

TLDR: In this paper, a small e perturbation analysis of the quasi-periodic Mathieu equation is presented, where the scaling factors are conjectured to determine the size of instability regions as we go from one resonance to another in the δ−ω parameter plane.

Abstract: We present a small e perturbation analysis of the quasi-periodic Mathieu equation $$ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 $$ in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales including terms of O(e2) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows good agreement for e = 0.1. Then we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine the size of instability regions as we go from one resonance to another in the δ−ω parameter plane.