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The following chart is a Mathieu differential equation solution where a characteristic frequency of approximately $\sqrt{3}$ = 1.7 rads/year is modulated by a 1 year period

The modulation leads to a somewhat erratic solution as evidenced by the middle panes, but if we blow that up in the bottom pane, one can see where the peaks and shoulders line up with integral years at 1, 2, 3, 5 as indicated by the red down arrows. So even though the underlying period of approximately 3.5 years is clear, the Mathieu modulation pushes the peaks toward where $cos(2\pi t)$ hits extrema. In terms of the wave differential equation, the acceleration is strongest near these points and is enough to deflect the peak toward the integral years.

So the question is whether a machine-learning can detect this modulation on real data .

The following shows the results of a dispersion model applied to the NINO 3 time series as solved by Eureqa:

This actually finds a biannual modulation with peaks at January and June from the $cos(4\pi t)$ modulation.

If I place a 12-month smoothing averager (sma) on the time series, then the annual modulation appears instead of the biannual

So that the annual modulation is strong enough to not be completely smoothed out, but the biannual is.

The sporadic locking of ENSO to a seasonal cycle, and particularly to the beginning of the year, is well known [1], but this approach seems to be a sensitive way to detect the effect.

The complicating factor is that a longer Mathieu modulation between 6 and 10 years also appears if the waveform is smoothed further. Trying to pick up both of these modulations (annual and multi-year) simultaneously seems tricky using the Eureqa tool.

Yet the indications are that the annual modulation is what pushes an ENSO peak toward the beginning of the year, while the multi-year modulation causes the long-term erratic behavior that makes ENSO and El Ninos difficult to predict.

[1]E. Tziperman, M. A. Cane, and S. E. Zebiak, “Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos,” Journal of the Atmospheric Sciences, vol. 52, no. 3, pp. 293–306, 1995.

## Comments

This is a Mathieu sloshing differential equation solution used to model the SOI and Sydney tide gauge records as applied to the NINO 3 dataset

The time series extends back to 1856. The model has a $\pi$ phase shift in 1906 applied to the biennial forcing term, and so too the parameters were slightly adjusted to improve the correlation for NINO 3.

One feature that I have been playing with is applying a dilation to the output time. The period of the dilation is ~18 years and it has a significant enough magnitude that time can speed up or slow down by a year over the course of the 18 year span. This is one of those heuristics that machine learning discovered. Whether it means anything, perhaps related to a gravitational/tidal pull of the Saros cycle of 18 years, is purely conjecture. At this point, many of these fits are simply signposts along the path that could make this approach it practical for predictions.

One point at which the model fails is the strong heat pulse lasting from 1940 to 1944. This has been problematic in many of the models, with some fraction related to uncalibrated data collection during WWII.

The recent uptick in temperatures may be relevant to the increases in SST observed in the last 3 months, each of which was a global record.

`This is a Mathieu sloshing differential equation solution used to model the [SOI and Sydney tide gauge records](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12691#Comment_12691) as applied to the NINO 3 dataset ![NINO3](http://imageshack.com/a/img746/3192/nQN9nn.gif) The time series extends back to 1856. The model has a $\pi$ phase shift in 1906 applied to the biennial forcing term, and so too the parameters were slightly adjusted to improve the correlation for NINO 3. One feature that I have been playing with is applying a dilation to the output time. The period of the dilation is ~18 years and it has a significant enough magnitude that time can speed up or slow down by a year over the course of the 18 year span. This is one of those heuristics that machine learning discovered. Whether it means anything, perhaps related to a gravitational/tidal pull of the [Saros cycle](http://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html) of 18 years, is purely conjecture. At this point, many of these fits are simply signposts along the path that could make this approach it practical for predictions. One point at which the model fails is the strong heat pulse lasting from 1940 to 1944. This has been problematic in many of the models, with some fraction related to [uncalibrated data collection during WWII](http://contextearth.com/2013/11/16/csalt-and-sst-corrections/). The recent uptick in temperatures may be relevant to the increases in [SST observed in the last 3 months](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/?Focus=12795#Comment_12795), each of which was a global record.`

Paul

I got the attention of another genius like yourself who wants to code these diff EQ for John's climatology applications. He is a sharp programmer and he has same ideas as yourself.

So we might start showing some more support for your work.

Dara

`Paul I got the attention of another genius like yourself who wants to code these diff EQ for John's climatology applications. He is a sharp programmer and he has same ideas as yourself. So we might start showing some more support for your work. Dara`