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:
even and 
odd characteristic value curves--- that is, the stability region--- becomes vanishingly small for large q.
-periodic odd and even solutions
for q=0 require 
. This result
follows immediately from inspection of Eq. .
to within a specified number of significant figures for q less than some critical value. For example, for n>15, 
to within 5 significant figures for 
.[28] Thus, for large values of n and small values of q, the stability region dominates, since the 
even and 
odd characteristic value curves are ``slow'' (with regard to q) to converge.