This is an interesting fit to the Unified ENSO Proxy time-series using Eureqa, which uses a search algorithm somehow related to Differential Evolution


The higher-complexity sinusoidal formulation it finds on the Pareto front (red dot) is bizarre

$ UEP = sin(7.824 \cdot Year + sin(1.812 \cdot Year)) \cdot cos(0.01399 \cdot Year + sin(25.36 \cdot Year)) $

The correlation coefficient is still just 0.42.

Finding solutions of sine waves modulated by sine waves leads to the idea (via the chain rule) that a differential equation parameter may have a sinusoidal dependence with time. Of course the solution is iterative, as the chain rule doesn't converge. But iterating further is what gives rise to the Mathieu basis functions.

So there are tantalizing hints that a pattern exists.