Here's a very convincing data analysis that shows that ENSO (and El Nino) is far from being chaotic. I ran an auto-correlation of the the Fourier series of an ENSO time-series and found a strongly correlated one-year shift from all spectral components
This can only happen if the Green's function (impulse) response derives from a frequency modulation or mixing of an annual impulse with another external forcing.
It's crucial to understand that this is not an auto-correlation in the time-domain but an auto-correlation in the frequency-domain, which is rarely used under most circumstances.
The analysis is related to deciding whether a detected radio signal is noisy/ chaotic or AM/FM -- it's really a matter of demodulating the signal with the carrier (mixing) signal to find out what the information content is. What the auto-correlation in the frequency-domain does is to demodulate the signal; i.e. convolution in the frequency domain leads to multiplication in the time domain. In other words, this analysis is directly showing what the time-domain mixing or multiplication function, which is an annual impulse train.
I also haven't found anything close to this kind of analysis in the research literature.