Transition Curves for the Quasi-Periodic Mathieu Equation

Abstract

In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu equation given by \[ \ddot{\psi} + [\delta + \eps \,( \cos t + \cos \omega t)]\, \psi = 0 \] for small $\eps$ and irrational $\omega$. Of interest is the generation of stability diagrams that identify the points or regions in the $\delta$-$\omega$ parameter plane (for fixed $\eps$) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots ofLyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves $\delta = \delta(\omega; \eps)$ are each expanded in powers of $\eps$, and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that predominant regions of instability are clearly coincident. And, both analytic techniques enable us to gain insight into the structure of the corresponding numerical plots. However, the perturbation method fails in the neighborhood of resonant values of $\omega$ due to the problem of small divisors; the results obtained by harmonic balance do not display this undesirable feature.