This is the most impressive set of graphs that I have ever created based on the result of an automated solver:


The significance is that the red and blue time-series profiles individually decompose from **non-overlapping intervals** of the ENSO record from 1880 to 2013. Each set of curves took over an hour to calculate by running the Excel solver. This is an agonizingly slow process as it iterates over likely billions of calculations, yet, in the end the sets of curves asymptotically converge to essentially the same result, apart from a ~1.5 month shift between the red and the blue sets.

The coloration pattern is an artifact of the blue lines being drawn in the foreground and the red lines in the background. As the fit is highly coincident, the red lines hide behind the blue lines and only pop out where there is a discrepancy between the values. So the more blue you see, the higher the correlation coefficient. And based on my experiences, any coefficient above 0.8 is excellent agreement for highly oscillatory waveforms such as these.

This is one of those results that is well beyond requiring a statistical significance check since a coincidentally random convergence would obviously be extremely remote.

At the very least, this result should put to rest that ENSO is either a (1) chaotic or (2) random behavior. I am convinced the reason that the deterministic behavior has not been revealed in previous research is because applying a Mathieu modulation of a two-year period was never considered. This never occurred to anyone despite the fact that every text on the hydrodynamics of liquid sloshing asserts that a Mathieu-based wave equation represents the physics of the process, although not considered on an oceanic scale. Each of the following books and papers touches on that aspect:

* Benjamin, T. Brooke, and F. Ursell. "The stability of the plane free surface of a liquid in vertical periodic motion." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 225. No. 1163. The Royal Society, 1954.
* O. M. Faltinsen and A. N. Timokha, *Sloshing*. Cambridge University Press, 2009.
* O. M. Faltinsen, “A numerical nonlinear method of sloshing in tanks with two-dimensional flow,” Journal of Ship Research, vol. 22, no. 3, 1978.
* G. Wu, Q. Ma, and R. Eatock Taylor, “Numerical simulation of sloshing waves in a 3D tank based on a finite element method,” Applied Ocean Research, vol. 20, no. 6, pp. 337–355, 1998.
* R. A. Ibrahim, *Liquid sloshing dynamics: theory and applications*. Cambridge University Press, 2005.
* J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.
* F. Dubois and D. Stoliaroff, “Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics,” arXiv preprint arXiv:1407.1829, 2014.

The one recent paper that may be the supporting groundbreaker, along with a cited paper, is

* J. Rajchenbach and D. Clamond, “Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited,” Journal of Fluid Mechanics, vol. 777, p. R2, 2015. [PDF here](
* Skeldon, A. C., and A. M. Rucklidge. "Can weakly nonlinear theory explain Faraday wave patterns near onset?." Journal of Fluid Mechanics 777 (2015): 604-632.

They make some strong claims concerning our understanding of the dispersion of waves, in particular when placed in the context of a Mathieu modulation.