Yes, the MathieuC and MathieuS functions are derived as Fourier coefficients, but they are recursively established by the +/- perturbation of the $\omega$ frequency on the wave equation. It does take lots of computations to get the Mathieu functions to converge properly ( [see here](http://dlmf.nist.gov/28.4) ). It definitely is not something as simple as the Taylor series expansion of a Sine function, which is what I meant by not "easily understood".

The line between quasi-periodicity versus aperiodicity is a fine one. In the limit of small q (which is the amplitude of the Mathieu modulation), the MathieuC and MathieuS functions converge to Sine and Cosine functions, which are known to be periodic. The way to think about this is that as the modulation gets stronger, the period of repeat starts to extend, and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence.

Here is an example of a MathieuC function where one can pick out some of the repeat period

![MathieuC](http://imageshack.com/a/img742/1842/wYewOa.gif)

That repeat sequence appears to be about 57 time units, but even that evolves somewhat. It is all a matter of practical application. Look up the Floquet theorem and I think there are more references to the solutions of these equations being quasiperiodic than aperiodic, but I might be wrong.

Actually that might not be a bad idea for machine learning from paleo records -- look for patterns of a repeat sequence in the historical data.

The line between quasi-periodicity versus aperiodicity is a fine one. In the limit of small q (which is the amplitude of the Mathieu modulation), the MathieuC and MathieuS functions converge to Sine and Cosine functions, which are known to be periodic. The way to think about this is that as the modulation gets stronger, the period of repeat starts to extend, and unless one has a long-enough time series to deal with, one might not pick up the repeat sequence.

Here is an example of a MathieuC function where one can pick out some of the repeat period

![MathieuC](http://imageshack.com/a/img742/1842/wYewOa.gif)

That repeat sequence appears to be about 57 time units, but even that evolves somewhat. It is all a matter of practical application. Look up the Floquet theorem and I think there are more references to the solutions of these equations being quasiperiodic than aperiodic, but I might be wrong.

Actually that might not be a bad idea for machine learning from paleo records -- look for patterns of a repeat sequence in the historical data.