Modification of Variable In Parametric Plot 3D

Question

I am plotting a 3D plot of my set of ODEs. The ODEs are as follows:

H = Sqrt[(1/
3)*((0.5/(a^6*V^2 *Exp[6*t]))*(P[t]^2 + 0.5*sig^2) + (0.25*m^2*
M^2)*(1 - Exp[-1/(sig^2*M^2)]*Cos[2 *\[Psi][t]/M]))]; 
eqs = {\[Psi]'[t] == P[t] Exp[-3 t]/(H a^3  V), 
P'[t] == (-0.5/H) ( 
a^3 V Exp[3 t]*M* m^2 Exp[-1/(M^2*sig^2)] Sin[2 \[Psi][t]/M])};

I want to plot a phase portrait with following variables on my axes, (t, $\frac{P}{e^{3t}}$, $\phi$). The thing that confuses me here is I have tried to plot it by using the 3d parametric plot command and my code is here as follows,

The solution to equations is:

sol = ParametricNDSolve[{eqs /. {M -> 1, m -> 100, V -> 10, a -> 1, 
sig -> 15}, \[Psi][-10] == c, P[-10] == b}, {\[Psi], 
P}, {t, -10, 12}, {c, b}, MaxSteps -> 100000];

The plot is done by using:

  plot1 = ParametricPlot3D[
  Flatten[Table[{t, \[Psi][c, b][t], P[c, b][t]/Exp[3*t]}, {c, -3, 
  3}, {b, -3, 3}], 1] /. sol // Evaluate, {t, -10, 12}, 
  BoxRatios -> {1, 1, 1}, AxesLabel -> {"N", "\[Phi]", "P"}, 
  Boxed -> False, PlotLabel -> "m=100,M=5", 
  ColorFunction -> Function[t, Hue[t]]]

This plot vanishes the visualization of initial conditions, i.e. the plots don't start from t=-10 instead from somewhere at about t=0. The initial conditions are at t=-10 so the trend should be visible at t=-10. Have I done something wrong by writing: P[c, b][t]/Exp[3*t], in the parametric3d plot command?` Secondly, I tried to plot H vs t, by using the parametric plot command but an empty plot occurs. Any leads would be appreciated.
The sample plot(researchgate.net/figure/3-dimensional-phase-portrait-of-the-system-51-assuming-a-quadratic-potential-with-g_fig2_265967212) has all its trajectories starting from t=-10 (there, t is represented by N). I tried to plot H vs t by using the following code:

ParametricPlot[
Flatten[Table[H, {c, -3, 3}, {b, -3, 3}], 1] /. sol // 
Evaluate, {t, -10, 9}

]

Answer 1

Mathematica is "helpfully" adjusting the ranges of the plot (particularly the range of P) to allow for more detail in the region $N > 0$. You can force Mathematica to show the entire plot by using the PlotRange -> All option.

Note the difference in "vertical" range between the two plots. It appears that most of your phase space trajectories grow exponentially if you project them back to negative time. Whether or not this is reasonable given the physical situation you're looking at, I cannot say; but that seems more likely be a problem translating the physics into the software rather than a problem with the software itself.