Jim, I did coax Per to come over to this forum and share his neural network model. So something good may come out of my excursion to foreign territory :)
I've found that straightforward periodicity analysis does not work for ENSO and certainly does not give a huge amount insight, apart from potentially isolating the strongest signals.
I am thinking that instead of periodicity analysis, we need something more like a bifurcation analysis. What Graham Jones is suggesting may give hints at a potential direction. With these nonlinear equations such as Mathieu and Hill, the periods can bifurcate and then provide feedback to the forcing. I have found once the fit starts locking in, it gets better when the modulating factors are duplicated as forcings.