Subharmonic resonance in the non-linear Mathieu equation

Abstract

In this paper, we present an O(ε) perturbation method that utilizes Lie transform perturbation theory and elliptic functions to investigate subharmonic resonances in the non-linear Mathieu equation x ̈ +(δ+ε cos ωt) x+αx 3=0. It is assumed that the parametric perturbation, ε cos ωt , is small and that the coefficient of the non-linear term, α, is positive but not necessarily small. We derive analytic expressions for features (width and location of equilibria) of resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. In contrast to previous perturbation treatments of this problem, the unperturbed system is non-linear and the transformation to action-angle variables involves elliptic functions. We are, therefore, not restricted to a neighborhood of the origin in our investigation. The Hamiltonian structure of the unperturbed vector field, an integrable vector field, provides us with a framework for developing an analysis of the perturbed orbit structure. The methodolgy revolves around employing Lie transform perturbation theory for constructing the so-called “resonance Kamiltonian”, K r, whose level curves correspond to invariant curves of a Poincaré map for the non-linear Mathieu equation. Explicit knowledge of K r enables us to derive analytic expressions for the resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. Predictions of the perturbation method are compared to results obtained by direct numerical integration of the non-linear Mathieu equation. The integrable nature of the unperturbed ( ε=0) non-linear Mathieu equation is preserved under the perturbation method. Consequently, the method is unable to predict the appearance of chaos.