I have found the following research articles vital to formulating a basic model for ENSO.
The first citation finds the disturbance after 1980 leading to the identification of a phase reversal in the ENSO behavior. They apply Takens embedding theorem (which works for linear and non-linear systems such as Mathieu and Hill) to the time series, reconstructing current and future behavior from past behavior.
H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.
The following identifies the characteristic frequency of 1/5.25 cycles per year to ENSO. This is critical to applying the wave equation to standing wave ENSO time series such as the SOI. In general, the research from FSU from Clarke, Kim, etc is the most mathematically comprehensive in trying to describe the dynamics of ENSO.
A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007. PDF
Another paper from a former FSU researcher establishes the role of the biennial mode to the equatorial Pacific. More refs in the citations at the end of this post where Kim discusses the concept of cyclostationarity..
Kim, Jinju, and Kwang-Yul Kim. “The tropospheric biennial oscillation defined by a biennial mode of sea surface temperature and its impact on the atmospheric circulation and precipitation in the tropical eastern Indo-western Pacific region.” Climate Dynamics, 2016: 1-15. PDF : kim2016.compressed
The following paper finds an additional wobble term to the Chandler wobble, corresponding to the non-spherical nature to the earth, essentially creating a triaxial system. They estimate this at 14.6 years.
Wang, Wen-Jun, W.-B. Shen, and H.-W. Zhang, “Verifications for Multiple Solutions of Triaxial Earth Rotation,” IERS Workshop on Conventions Bureau International des Poids et Mesures (BIPM), Sep. 2007. http://www1.bipm.org/utils/en/events/iers/Wang.pdf
The possibility of the moon’s longitudinal free libration contributing to a 2.9 year oscillation
Rambaux, N., and J. G. Williams. “The Moon’s physical librations and determination of their free modes.” Celestial Mechanics and Dynamical Astronomy 109.1 (2011): 85-100. PDF
Research that is asserting that ENSO is not chaotic based on the distribution of the magnitude of peaks.
Serykh, Ilya, and Dmitry Sonechkin. “Evidence of a strange nonchaotic attractor in the El Niño dynamics.” EGU General Assembly Conference Abstracts. Vol. 17. 2015. ResearchGate
Li, G., Zong, H., & Zhang, Q. (2011). 27.3-day and average 13.6-day periodic oscillations in the Earth’s rotation rate and atmospheric pressure fields due to celestial gravitation forcing. Advances in Atmospheric Sciences, 28, 45-58.
Gross’s research on the Chandler wobble. Nastula and Gross recently came up with a value of 430.9 days from newly available satellite measurements.
R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000.
Ray’s research identifying long-period lunar tidal factors
R. D. Ray and S. Y. Erofeeva, “Long‐period tidal variations in the length of day,” Journal of Geophysical Research: Solid Earth, vol. 119, no. 2, pp. 1498–1509, 2014.
R. D. Ray, “Decadal climate variability: Is there a tidal connection?,” Journal of climate, vol. 20, no. 14, pp. 3542–3560, 2007.
Memos and proposals from JPL pointing to the importance of lunar dynamics
C. Perigaud, “Importance of the Earth-Moon system for reducing uncertainties in climate modelling and monitoring.” NASA JPL proposal submitted 10/31/2009.
To: Lunar Investigators
From: James G. Williams, Dale H. Boggs and William M. Folkner
Subject: DE430 Lunar Orbit, Physical Librations, and Surface Coordinates PDF
More on lunar libration
Bois, E., F. Boudin, and A. Journet. “Secular variation of the Moon’s rotation rate.” Astronomy and Astrophysics 314 (1996): 989-994 [ PDF ]
And the general physics of sloshing research, which describes how the Mathieu formulation figures in to the wave equation
J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004. ResearchGate
O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.
Each one of these citations has been involved in formulating a basic model of ENSO — this uses a concise set of 4 parameters (2 for wobble and 2 for lunar tides) and biennial modulation to capture the stationary behavior.
The yellow highlights show regions of disagreement. The fidelity in capturing the fine detail is approaching and may have surpassed what I found with the forced response to the QBO model. The fitting approach based on a wave equation transform will be discussed in more detail next.
R. Yeo and K.-Y. Kim, “Global warming, low-frequency variability, and biennial oscillation: an attempt to understand the physical mechanisms driving major ENSO events,” Climate Dynamics, vol. 43, no. 3–4, pp. 771–786, 2014. http://link.springer.com/article/10.1007/s00382-013-1862-1/fulltext.html
Kim, Kwang-Yul, James J O’Brien, and Albert I Barcilon. “The Principal Physical Modes of Variability over the Tropical Pacific.” Earth Interactions 7, no. 3 (2003): 1–32
Sonechkin, D. M., and R. Brojewski. “ENSO: a quasiperiodic forced dynamical system.” International worhsop on the low-frequency modulation of ENSO, Touluse (2003): 23-25.