Since is determined by a and q, the Mathieu equation has stable--- corresponding to bounded trajectories--- and unstable--- corresponding to unbounded trajectories--- solutions as a function of a and q. Stability regions for the one-dimensional Mathieu equation (Eq. ) correspond to regions of a,q parameter space for which there exist bounded solutions for . In order to understand the a, q values which comprise the Mathieu stability regions, we must consider the integral-order solutions to the Mathieu equation.
Figure: Stability regions for the one-dimensional Mathieu equation. Note the failure of curves , and , to converge. The published series approximations are inaccurate for ``large'' values of q.
For each value of q, there corresponds a countably infinite set of characteristic values of a for which is an odd or even function that is -periodic in , n being an integer. Series approximations for the characteristic values are obtained by expressing the integral-order Mathieu function as a series of harmonic oscillations, plugging the resultant expression into Eq. , and equating coefficients of each (orthogonal) frequency component to zero. These laborious calculations yield infinite series in q where each coefficient of q can be expressed as a continued fraction.[39,25] Several inconsistencies in published values of numerical coefficients of the series, as well as a lack of uniformity in the notation of the Mathieu equation and its parameters, were encountered in Mathieu equation and Paul-trapping literature.
As stated in Sec. , stable solutions of the Mathieu equation include fractional-, as well as integral-, order Mathieu functions. Fractional-order stable solutions are attained when, for a given value of q, the parameter a lies in the region between the characteristic values for the even and odd integral-order Mathieu functions. That is, is bounded iff:
where and are the even and odd characteristic values for a as a function of q. Several features of the characteristic values curves (Fig. ) are noteworthy: