Capture observed dependence on unknowns in simplified nonlinear systems

[hersle]

This fixes the incorrect Jacobian in #2858.

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

[hersle]

Would it be better if, for example,

using ModelingToolkit
@variables x y z
@parameters σ ρ β
@mtkbuild ns = NonlinearSystem([
    0 ~ σ * (y - x)
    0 ~ x * (ρ - z) - y
    0 ~ x * y - β * z
])
J = calculate_jacobian(ns)

did not expand observed derivatives like

2×2 Matrix{Num}:
 (-1 - z + ρ)*σ        -x*σ
    2x*(-z + ρ)  -β - (x^2)

but rather kept them in a form like

2×2 Matrix{Num}:
 (-1 + Differential(x)(y))*σ     Differential(z)(y)*σ
    y + x*Differential(x)(y)  x*Differential(z)(y) -β

?

[ChrisRackauckas]

This won't catch a chain of observed, it would need a fixed point sub I think

[hersle]

Like in the most recent commit? Can you think of a simple example on to test?

[ChrisRackauckas]

z ~ y, y ~ x, x^2 - 2z` should hit it, since if you just plug in z you get y.

[hersle]

Not quite after structural simplification, it seems.
But I added 0 ~ x^2 + 2*z + y, z ~ y, y ~ x, which seems to do the trick.