How to extract data from the solution of NDsolve with specific parameter range

Question

So, I have solved a coupled differential equation using NDsolve here:

I want the functions x[t] and y[t] for some specific parameter range, say for t>0 or t<0, and store it in some array perhaps for some computation. Can anyone tell me how this can be done?

Furthermore, How can I use different plot styles in Parametric plot for parameter ranges t>0 and t<0, How to go about doing this?

Thank you.

Answer 1

I'll use the Lotka-Volterra model as an example since I can't copy your code.

First way: Use Show to get the forward t>0 and backward t<0 solutions to be different colors.

{xsol, ysol} = NDSolveValue[{
   x'[t] == x[t] - 2 x[t] y[t],
   y'[t] == x[t] y[t] - y[t],
   x[0] == y[0] == 1},
  {x, y}, {t, -20, 20}]


Show[{
  ParametricPlot[{xsol[t], ysol[t]}, {t, -4, 0}, PlotStyle -> Red, 
   PlotRange -> {{0, 2.5}, {0, 1.5}},
   Prolog -> {PointSize[Large], Blue, Point[{1, 1}]}],
  ParametricPlot[{xsol[t], ysol[t]}, {t, 0, 4}, PlotStyle -> Green]}]

Second way: Have a forward and a backward NDSolve.

Forward solution

{xsolf, ysolf} = NDSolveValue[{
   x'[t] == x[t] - 2 x[t] y[t],
   y'[t] == x[t] y[t] - y[t],
   x[0] == y[0] == 1},
  {x, y}, {t, 0, 20}]

Backward solution

{xsolb, ysolb} = NDSolveValue[{
   x'[t] == -(x[t] - 2 x[t] y[t]),
   y'[t] == -(x[t] y[t] - y[t]),
   x[0] == y[0] == 1},
  {x, y}, {t, 0, 20}]

Notice that the backward solve has the same initial conditions and time range as the forward solve, but that the equations are negated.

This lets you use only a single ParametricPlot for both forward and backward solutions.

ParametricPlot[{{xsolf[t], ysolf[t]}, {xsolb[t], ysolb[t]}}, {t, 0, 4}, 
  PlotStyle -> {Red, Green}, 
  Prolog -> {PointSize[Large], Blue, Point[{1, 1}]}]

I've also gone with NDSolveValue instead of the regular NDSolve since you asked to store the data in a table which you can do pretty easily with

xfData = Table[xsolf[t], {t, 0, 5, .1}]

If you use the second way, remember that for the backward solve time is "negative time" so you might need to do something like

xbData = Table[xsolb[t], {t, 10, 0, -.1}]