nad said

> "... that's been one of the things which I found hadn't been really adressed in your type of reasoning."

I think the confusion may be in where the periodicities lie. For the forcing functions, there are certainly strict periodicities at play, and these involve factors such as seasonal cycles, QBO, Chandler wobble, tidal cycles, etc. However, in the *response* functions, these periodicities can be quickly obscured.

One way for this to happen is if there are many forcing cycles at work and they combine to form a tangled mat of beat frequencies. Ordinarily, a Fourier series decomposition can straightforwardly isolate these frequencies. This is obviously not happening with ENSO, otherwise somebody would have discovered the specific series for ENSO long ago !

The second way for the time series to diverge from a periodic characteristic is if the transfer function shows non-linear effects. In the case of a Mathieu or Hill DiffEq, this is a special case of non-linearity in only the temporal modification of the DiffEq coefficients (not in the differential order terms). It is important to note that a Fourier series decomposition via a Fourier transform will not easily isolate the characteristics of the response as the spectral lines will show multiple bifurcations.

IMO, the second way is operational here and I have been pushing this since I first wrote a blog post on ENSO [last year]( Read that post and you will understand why I too thought that using conventional analyses to find the underlying periodicities was going to be a challenging task. It took me a while to understand the sloshing literature, where they too discover the same kind of challenges in trying to deconvolute sloshing dynamics.

*Thuvanismail, Nasar, Sannasiraj Sannasi, and Sundar Vallam. "A numerical study: liquid sloshing dynamics in a tank due to uncoupled sway, heave and roll ship motions." Journal of Naval Architecture and Marine Engineering 10.2 (2013): 119-138.*

So over the last year, I have developed quite an arsenal of techniques to infer the behavior. What is most interesting is that in that first post I inferred these characteristics of a Mathieu equation

a = 2.83
q = 2.72
T = 6.30 years

The $a$ term is $\omega^2$ which I have as ~2.2 right now. The $q$ shows a similar relative amplitude now as then, and the $T$ term is now close to the Chandler Wobble of ~6.5 years. The progress has been slow but it is gradually converging.

Does that address your concern?