"Spaghetti"-solutions for ODE nonautonomous system and reduced vector field

Question

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

Multidimensional obstacle avoidance in ODE (Visualization)

Given simple system of ODE:

$\begin{cases} \dot{x}=g \\ \dot{o}=2 \cdot(-o+x) \\ \dot{g}=(1+\sin(3 t)) \cdot (-g+\frac{df}{do}) \\ \dot{h}=-h+\frac{d^2f}{d^2o} \end{cases}$

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

It is not difficult to construct a 3D trajectory using the command ParametricPlot3D.

Clear["Derivative"]

ClearAll["Global`*"]

pars = {xs = -1, k = (1 + 1 Sin[3 t])};

f = Exp[-(o[t])^2];

s = NDSolve[{x'[t] == g[t], o'[t] == 2 (-o[t] + x[t]), 
    g'[t] == k (-g[t] + D[f, {o[t], 1}]), 
    h'[t] == -h[t] + D[f, {o[t], 2}], x[0] == xs, o[0] == xs, 
    g[0] == 0.01, h[0] == 0}, {x, o, g, h}, {t, 0, 200}, 
   MaxSteps -> \[Infinity]];
   
ParametricPlot3D[Evaluate[{o[t], g[t], h[t]} /. s], {t, 0, 200}, 
 PlotPoints -> 100, ColorFunction -> (Hue[#4] &), 
 BoxRatios -> {1, 1, 1}, PlotRange -> Full]

Questions:

1. How to combine multiple solutions for different initial conditions on one ParametricPlot3D?
2. How to plot the final point on the ParametricPlot3D?
3. How to build a vector field around the trajectory?

Answer 1

Several plots and the endpoints can be drawn by:

Clear["Derivative"]
ClearAll["Global`*"]
pars = {xs = -1, k = (1 + 1 Sin[3 t])};
f = Exp[-(o[t])^2];
s = Table[
   NDSolve[{x'[t] == g[t], o'[t] == 2 (-o[t] + x[t]), 
     g'[t] == k (-g[t] + D[f, {o[t], 1}]), 
     h'[t] == -h[t] + D[f, {o[t], 2}], x[0] == xs, o[0] == xs, 
     g[0] == g0, h[0] == 0}, {x, o, g, h}, {t, 0, 200}, 
    MaxSteps -> \[Infinity]], {g0, 0, 4, 0.8}];
funs[t_] = {o[t], g[t], h[t]} /. (Transpose[s][[1]]);
end = funs[200];

Show[{
  ParametricPlot3D[Evaluate[funs[t]], {t, 0, 200}, PlotPoints -> 100, 
   ColorFunction -> (Hue[#4] &), BoxRatios -> {1, 1, 1}, 
   PlotRange -> Full],
  Graphics3D[{PointSize[0.03], Point[end]}]
  }]

If you want different colors for different trajectories you simply eliminate the ColorFunction command:

Show[{ParametricPlot3D[Evaluate[funs[t]], {t, 0, 200}, 
   PlotPoints -> 100, 
   BoxRatios -> {1, 1, 1}, PlotRange -> Full], 
  Graphics3D[{PointSize[0.03], Point[end]}]}]

The velocity field is actually 4 dimensional and can not simply be drawn. What we can do is to draw velocity vectors along the trajectories:

velocities = 
  Table[Arrow[{{o[t], g[t], 
        h[t]}, {o[t], g[t], h[t]} + 0.5 {2 (-o[t] + x[t]), 
         k (-g[t] + D[f, {o[t], 1}]), -h[t] + D[f, {o[t], 2}]}}], {t, 
      0, 200}] /. (Transpose[s][[1]]) // Flatten;
Show[{
  ParametricPlot3D[Evaluate[funs[t]], {t, 0, 200}, PlotPoints -> 100, 
   ColorFunction -> (Hue[#4] &), BoxRatios -> {1, 1, 1}, 
   PlotRange -> Full],
  Graphics3D[{Thickness[Tiny], Arrowheads[0.02], velocities, 
    PointSize[0.03], Point[end]}]
  }]

Answer 2

The "final point" follows from

ode = {x'[t] == g[t], o'[t] == 2 (-o[t] + x[t]),g'[t] == k (-g[t] + D[f, {o[t], 1}]),h'[t] == -h[t] + D[f, {o[t], 2}]}
Reduce[ode/.   s_'[t] -> 0]
(*(g[t] == 0 && Sin[3 t] == -1 && h[t] == 2 E^-x[t]^2 (-1 + 2x[t]^2) &&o[t] == x[t]) || (x[t] == 0 && o[t] == 0 && h[t] == -2 && g[t] == 0)*)

FixedPoint: x[t]==0,o[t]==0,g[t]==0,h[t]==-2

Answer 3

With respect to 1., I suggest using ParametricNDSolveValue

Clear["Derivative"]
Clear[f, k, s]
f = Exp[-(o[t])^2];
k = (1 + 1 Sin[3 t]);
s = ParametricNDSolveValue[{x'[t] == g[t], o'[t] == 2 (-o[t] + x[t]), 
   g'[t] == k (-g[t] + D[f, {o[t], 1}]), 
   h'[t] == -h[t] + D[f, {o[t], 2}], x[0] == xs, o[0] == xs, 
   g[0] == 0.01, h[0] == 0}, {o[t], g[t], h[t]}, {t, 0, 200}, {xs}, 
   MaxSteps -> \[Infinity]
  ]
ParametricPlot3D[s /@ {-1, -1.5, -2} // Evaluate, {t, 0, 200}, PlotPoints -> 100, BoxRatios -> {1, 1, 1}, PlotRange -> Full]