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Given classical system of ODE:

$\begin{cases} \dot{x}=g \\ \dot{g}=t \cdot (-g+\frac{df}{dx}) \\ \dot{h}=-h+\frac{d^2f}{d^2x} \end{cases}$

where $f = e^{-x^2}$

I am constructing a three-dimensional phase space as follows:

    f = Exp[-x^2]; dfdx = D[f, {x, 1}]; dfdx2 = D[f, {x, 2}];

    Manipulate[
 VectorPlot3D[{g, t (-g + dfdx), -h + dfdx2}, {x, -2, 2}, {g, -2, 
   2}, {h, -2, 2}, PlotLabel -> Row[{"t = ", t}]], {t, 1, 15}]

My mathematica version is 12.0, so the StreamPlot3D command is not supported for me, but I hope my results are correct.

Questions:

  1. How to visualize the steady state point on this graph?
  2. How to get projections of $x-g$,$x-h$ and $g-h$ planes?
  3. Is it possible in an older version to build an analogue of the command StreamPlot3D.

Best regards to all specialists.

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    $\begingroup$ Isn't the system autonomous? (Does the dot indicate time-derivative? There's no time variable apparent in the equations.) — Won't the projections look like a pile of spaghetti? The basic idea would be to compute a bunch of trajectories and drop one of the coordinates when you ParametricPlot. $\endgroup$
    – Michael E2
    Jun 3 at 19:41
  • $\begingroup$ @Michael E2 Sorry. Mistake. I have corrected $k$ to $t$. $\endgroup$
    – dtn
    Jun 3 at 19:44
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    $\begingroup$ The x-g projection, for instance, of a single trajectory could be done ParametricPlot[{x[t],g[t]} /. sol, {t, 1, 15}], no? (I'm talking about Q2.) $\endgroup$
    – Michael E2
    Jun 3 at 20:03
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    $\begingroup$ "Phase space" of this nonautonomous system only makes sense to me as having the underlying coordinates {t, x, g, h}, with the corresponding vector field {1, dx/dt, dg/dt, dh/dt}. Projection onto a coordinate plane involves plotting the corresponding coordinates. Does that correspond to your idea? $\endgroup$
    – Michael E2
    Jun 3 at 20:10
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    $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – Michael E2
    Jun 3 at 20:15

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