Here is another clincher for long-term deterministic stationary properties of ENSO. Some time ago I was running machine-learning on the Universal ENSO Proxy (UEP) records. This goes back to the year 1650 and is an annual record. Interestingly, the primary component that the symbolic reasoning finds is the aliased anomalistic tide!
If I overlay the part of the UEP fit that overlaps the modern day SOI records, we get this.
Note that this is trivial to do as the symbolic reasoner provides a sinusoidal function.
The aliased frequency of 7.821 rads/year that the UEP fit maps to is precisely the aliased anomalistic frequency factor of 4.085 rads/year shifted by 2$\pi$, which is equivalent under yearly sampling.
$ 2 \pi (1+1/4.085) = 7.8213 $