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# Stability Regions for the 1-Dimensional Mathieu Equation

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:

• This statement implies that the region between the even and odd characteristic value curves--- that is, the stability region--- becomes vanishingly small for large q.

• This is simply an assertion that the -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.

Next: Stability Regions for Up: Theory and Simulation Previous: Classes of Solutions

Wolfgang Christian
Fri May 12 10:36:01 EDT 1995