Global behavior of a nonlinear quasiperiodic Mathieu equation
Here, the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation \[ \ddot x + [\delta + \varepsilon (\cos \omega_1t+\cos \omega_2t)]x + \alpha x^3=0 \] is studied, where \(\varepsilon \ll 1 \) and the coefficient of the nonlinear term \(\alpha\) is positive but not necessarily small. By using Lie transform perturbation theory with elliptic functions, the authors study subharmonic resonances associated with orbits in \(2m:1\) resonance with a respective driver. In particular, the authors derive analytic expressions that put conditions on the parameters \((\delta, \varepsilon, \omega_1, \omega_2)\) at which subharmonic resonance bands in a Poincaré section of action space begin. The authors obtain an overview of the \(\vartheta(\varepsilon)\) global behavior of above equation as a function of \(\delta\) and \(\omega_2\) with \(\omega_1, \alpha\) and \(\varepsilon\) fixed.
scientific article; zbMATH DE number 1747905