TRANSITION CURVES FOR THE QUASI-PERIODIC MATHIEU EQUATION

In this work we investigate an extension of Mathieu's equation, the qu asi-periodic (QP) Mathieu equation given by psi + [delta + epsilon(cos t + cost omega t)] psi = 0 for small epsilon and irrational omega. Of interest is the generation of stability diagrams that identify the poi nts or regions in the delta-omega parameter plane (for fixed epsilon) for which all solutions of the QP Mathieu equation are bounded. Numeri cal integration is employed to produce approximations to the true stab ility diagrams both directly and through contour plots of Lyapunov exp onents. In addition, we derive approximate analytic expressions for tr ansition curves using two distinct techniques: (1) a regular perturbat ion method under which transition curves delta = delta(omega;epsilon) are each expanded in powers of epsilon, 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 sens e 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 metho d fails in the neighborhood of resonant values of omega due to the pro blem of small divisors; the results obtained by harmonic balance do no t display this undesirable feature.