[minor] modifying demo_VanDerPol

[ares7823]

Hi VBA developers,
I recently had a chance to go through 'demo_VanDerPol' and 'f_vanDerPol' specifying the evolution function.
There are two things I want to discuss which may help improve the demo:

1. Practical assumption for the precision rates of hidden states

• Van Der Pol has two distinct hidden states (say x1 and x2), and the innovation for x1 is simply the replication of the second time series one time step earlier. The current demo assumes the precision rate (alpha) to be equal to both time series. I think that having the first time series being deterministic isn't a bad idea, by adding modified iQx{t}.

2. dF_dX in 'f_vanDerPol'

• This is more like a question, but I found that the Jacobian matrix for f(x) doesn't have a part for x in line 25-28. Is there reason that the partial derivatives of the evolution function only go on the innovation part? I was confused because in other scripts (e.g. line 18-19 in 'f_lin2D.m') I saw the entire f(x) partially derived with regard to x and had an identity matrix for the previous level. Can you clarify this?

I tried using the modified iQx{t} and dF_dX and got the better fit from the VBA inversion, hope this helps the consideration.

Thanks,
Jungmin