Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [δ + e (cos ω1 t + cos ω2 t)] x + αx3 = 0.We assume that e ≪ 1 and that the coefficient of the nonlinearterm, α, is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the usual trigonometric functions – to study subharmonic resonances associated with orbits in 2m:1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, e, ω1, ω2) at which subharmonic resonance bands in a Poincare section of action space begin to overlap. These results are used in combination with Chirikov's overlap criterion to obtain an overview of the O(e) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and e fixed.