# Projections of the 3-dimensional phase-space of a non-autonomous ODE system

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.

• 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. Jun 3 at 19:41
• @Michael E2 Sorry. Mistake. I have corrected $k$ to $t$.
– dtn
Jun 3 at 19:44
• 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.) Jun 3 at 20:03
• "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? Jun 3 at 20:10
• Jun 3 at 20:15