Here is the RHS of the SOI DiffEq model based on two different techniques. The representation in blue is from maximizing the correlation coefficient of a DiffEq solution via Mathematica (see [Comment #17](https://forum.azimuthproject.org/discussion/comment/14461/#Comment_14461) ). The red dotted profile is based on letting loose the Eureqa symbolic regression solver on the LHS constraint and unknown RHS:

![eureqa](http://imageshack.com/a/img909/4924/k298G8.png)

The solver then combined algebraic operators and sinusoids to arrive at an analytical expression which matched the residual RHS data. This is an extremely powerful technique.

![fit comparison](http://imageshack.com/a/img911/6858/BaRrOP.gif)

Amazing that so little guidance is required for Eureqa to find a solution that essentially maps to the hypothesized aggregate input of QBO, TSI, and CW that was fed into the Mathematica DiffEq solution.

The only RHS constraint that was provided was a *suggestion* that a conditional building block centered at 1974 (i.e Time > 1974 or Time < 1974) could be used as a "no penalty" factor in arriving at a solution. The transition between 1974 and 1980 is obvious and shows a clear shift from an approximate 22 year beat envelope to a stronger 11 year beat envelope.

This is also clearly seen in the wavelet spectrum that I described in the white paper. In the figure below, month 1200 corresponds to the year 1980. Above ~1980, a strong lower frequency beat component is seen in the scalogram.

![wavelet](http://imageshack.com/a/img904/714/jkgkWf.gif)

Things are still holding together, and if anything the ENSO solution is as far from being a chaotic mess as I can imagine. Certainly, the answer is more complex than a single sinusoid, but not that unusual considering what is routinely solved in other scientific and engineering disciplines -- e.g. band structures, etc.

![eureqa](http://imageshack.com/a/img909/4924/k298G8.png)

The solver then combined algebraic operators and sinusoids to arrive at an analytical expression which matched the residual RHS data. This is an extremely powerful technique.

![fit comparison](http://imageshack.com/a/img911/6858/BaRrOP.gif)

Amazing that so little guidance is required for Eureqa to find a solution that essentially maps to the hypothesized aggregate input of QBO, TSI, and CW that was fed into the Mathematica DiffEq solution.

The only RHS constraint that was provided was a *suggestion* that a conditional building block centered at 1974 (i.e Time > 1974 or Time < 1974) could be used as a "no penalty" factor in arriving at a solution. The transition between 1974 and 1980 is obvious and shows a clear shift from an approximate 22 year beat envelope to a stronger 11 year beat envelope.

This is also clearly seen in the wavelet spectrum that I described in the white paper. In the figure below, month 1200 corresponds to the year 1980. Above ~1980, a strong lower frequency beat component is seen in the scalogram.

![wavelet](http://imageshack.com/a/img904/714/jkgkWf.gif)

Things are still holding together, and if anything the ENSO solution is as far from being a chaotic mess as I can imagine. Certainly, the answer is more complex than a single sinusoid, but not that unusual considering what is routinely solved in other scientific and engineering disciplines -- e.g. band structures, etc.