Floquet's theorem asserts that any solution to the Mathieu equation (Eq. ) is of the form:[4,29,28]
where is either real or imaginary and is a -periodic function of . This assertion implies that solutions of Eq. can be separated into the product of a term of the form and a series of harmonics, , each of which is -periodic in . For values of with a non-zero imaginary component and , Eq. indicates that:
Thus, , yields an unbounded solution to the Mathieu equation. On the other hand, values of which are purely real yield bounded, oscillatory solutions for .
There are two types of bounded solutions to Mathieu's equation: functions of fractional order and functions of integral order. Functions of fractional order are obtained for . The oscillatory periods of the frequencies associated with fractional-order Mathieu functions are non-integer multiples of . The special case of corresponds to an -periodic (in terms of ) solution. These -periodic solutions are Mathieu functions of integral order.
All bounded solutions to the Mathieu equation--- those of fractional as well as integral order--- are described (in accordance with Floquet theory) by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency:
The argument of in Eq. represents harmonics of frequency , while the term represents a (secular) frequency which is a function of the trap specifications (or, equivalently, of
the Mathieu parameters a, q). The Mathieu parameters and also determine the coefficients , which approach zero as n increases.