export data points from differential equation system?

Question

I am solving a dynamical three equation system. Besides plotting the individual effects for each of the state variables in an array and in a tridimensional graph, I would like to export the data points to a .csv file. However, when I try to export the table to a .txt file via the "Export" command, the output is not suitable (as it exports the output in a very disorganized way). I was wondering if there could be a way to do this. The code I am using is below:

f = {L, \[Psi], d} /. NDSolve[{
  d'[t] == (1/z) (\[Alpha] - 
       X \[Beta] (d[t]/Y[t]) - \[Gamma] L[t] - \[Delta] \[Psi][
         t] - \[Tau] Y[t]) + \[Beta] (d[t]/Y[t]) d[t],
  L'[t] == j (Y[t]/K) - e,
  \[Psi]'[t] == p L[t] + 0.035 \[Psi][t],
  Y[t] == (I + E + \[Alpha] - 
      X \[Beta] ( d[t]/Y[t]) - \[Gamma] L[t] - \[Delta] \[Psi][
        t])/(1 - (1 - \[Tau]) (\[Psi][
          t] + (1 - s) (1 - \[Psi][t]) - m z)),
  L[0] == 0, \[Psi][0] == 0, 
  d[0] == 0} /. { \[Alpha] -> 1, \[Gamma] -> 10, \[Delta] -> 10, 
  X -> 1, \[Beta] -> 0.5, j -> 0.8, \[Tau] -> 0.01, I -> 1, 
  s -> 0.85, p -> 0.8, E -> 1, m -> 1.3, e -> 0.035, w -> 0.5, 
  z -> 1.2, K -> 1}, {L, \[Psi], d, Y}, {t, 0, 200}, MaxSteps -> 1000000 ][[1]]
l = {L[t], \[Psi][t], d[t]};
Show[GraphicsArray[Table[Plot[f[[i]][t], {t, 0, 200}, PlotRange -> All,
PlotStyle -> Blue,
AxesLabel -> TraditionalForm /@ {t, l[[i]]}, 
DisplayFunction -> Identity], {i, 3}]]]
Export["table.txt", f[t] /. sol, "Table"]
ParametricPlot3D[Evaluate[Append[#[t] & /@ f, Red]], {t, 0, 200}, BoxRatios -> {1, 1,0.5}, PlotRange -> All, PlotPoints -> 1500, AxesLabel -> TraditionalForm /@ l]

Answer 1

Result of NDSolve is an InterpolatingFunction, which is a Mathematica object, and is NOT a list of points. To export numerical list of points for any chosen step of t:

data1 = Table[Evaluate[f[[#]][t] & /@ {1, 2, 3}], {t, 0, 200, .1}];
Export["data1.csv", data1];

This creates data file "data1.csv" in default directory found by evaluating function Directory[]. The data file is now fine, it's easy to check:

data2 = Import["data1.csv"];

data1 === data2

True

Choosing step t small enough will make your data smooth:

ListLinePlot[Transpose@data2, Filling -> 0]

Graphics3D[Point[data2], BoxRatios -> 1]

Answer 2

If you want the actual points NDSolve is using, you can extract that from the Interpolating functions.

    Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
    coords = Map[ First[InterpolatingFunctionCoordinates[#]] &, f];
    output = MapThread[Transpose[{#1, #2@#1}] &, {coords, f}];
    ListPlot[output]