nad earlier asked:

>I assume that “First” plots out y(x) of NDSolve in particular I don’t have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +….) is that what you call a filter?

That term is borne out of experience by how additional small-perturbation Mathieu (or Hill) terms impact the solution. I indicated that this is a kind of "time dilation", which is what an additional periodic modulation term will do to the characteristic frequency. So a slight change in characteristic frequency will add a long-term modulation in time --i.e. the definition of frequency modulation.

If you don't believe this partly hand-wavy explanation, I moved the time dilation from the y solution term to a 3rd frequency modulation term in the LHS of the DiffEq and recalculated below. The correlation coefficient is even better -- compare to the chart in #2

![qbo3terms](http://imageshack.com/a/img540/2719/uYyO6E.gif)

Thus we have a single strong Mathieu modulation term along with a pair of weaker Hill terms operating on different scales (1) a strong biannual modulation (2) a weak 12 year modulation and (3) and even weaker 50 year modulation.

>I assume that “First” plots out y(x) of NDSolve in particular I don’t have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +….) is that what you call a filter?

That term is borne out of experience by how additional small-perturbation Mathieu (or Hill) terms impact the solution. I indicated that this is a kind of "time dilation", which is what an additional periodic modulation term will do to the characteristic frequency. So a slight change in characteristic frequency will add a long-term modulation in time --i.e. the definition of frequency modulation.

If you don't believe this partly hand-wavy explanation, I moved the time dilation from the y solution term to a 3rd frequency modulation term in the LHS of the DiffEq and recalculated below. The correlation coefficient is even better -- compare to the chart in #2

![qbo3terms](http://imageshack.com/a/img540/2719/uYyO6E.gif)

Thus we have a single strong Mathieu modulation term along with a pair of weaker Hill terms operating on different scales (1) a strong biannual modulation (2) a weak 12 year modulation and (3) and even weaker 50 year modulation.