Putting NDSolve into ParametricPlot

Question

I am having issues using Manipulate to plot the (numeric) solution to an ODE for different parameter values.

I have a code that has several stages, which seem to all work when I do them one after another. This code solves a system of ODEs (for particular parameter values), then does a parametric plot of the solution. The problem is I need to put the lines together to wrap them around with Manipulate (for I can do this code again easily for different parameter values), and this is causing me a lot of pain.

My initial code is:

(*this is my ODE*)

unforced[x0_, p0_, α_:α, δ_:δ] := 
{x'[t] == p[t], 
 p'[t] == -α x[t] - δ p[t] + α (x[t])^3, 
 x[0] == x0, 
 p[0] == p0
}

(*Choose some parameter values*)

α = -1, δ = 1

(*Solve my ODE*)

s = NDSolve[unforced[x0 = 1, p0 = 1], {x, p}, {t, 20}]

(*Plot It*)

ParametricPlot[Evaluate[{x[t], p[t]} /. s], {t, 0, 20}]

But now I want to be able to Manipulate the parameter values α and δ. So I start putting the lines of code together... and problems happen.

ParametricPlot[Evaluate[{x[t], p[t]}/.
               NDSolve[{x'[t] == p[t], 
                        p'[t] == -α x[t] - δ p[t] + α (x[t])^3, 
                        x[0] == 1, p[0] == 0}, {x, p}, {t, 20}]], {t, 0, 20}]

This plots an empty graph. This confuses me because it seems like all I did was substitute into my previous code. Because this doesn't work, I can't put a Manipulate around this. If it worked then I would have tried:

Manipulate[ParametricPlot[Evaluate[{x[t], p[t]} /.
                NDSolve[{x'[t] == p[t],   
                         p'[t] == -α x[t] - δ p[t] + α (x[t])^3, 
                         x[0] == 1, p[0] == 0}, {x, p}, {t, 20}]], {t, 0, 20}], 
           {{α, -1, "α"}, -2, 0}, {{δ, 0, "δ"}, 0, 2}]

How do I get around this problem?

Answer 1

You can use ParametricNDSolve for this:

pf = ParametricNDSolveValue[{x'[t] == p[t], 
    p'[t] == -a x[t] - d p[t] + a (x[t])^3, x[0] == 1, 
    p[0] == p0}, {x[t], p[t]}, {t, 20}, {a, d, p0}];

Manipulate[
 ParametricPlot[pf[a, d, p0], {t, 0, 20}], {{a, -1, "Alpha"}, -2, 
  0}, {{d, 0, "Delta"}, 0, 2}, {{p0, .01, "p[0]"}, 0, 2}]

This allows you to pre-compute as much as is possible and then evaluate with specific parameters. Also, you could compute the sensitivity of the parameters. This is discussed in the documentation for ParametricNDSolve.

Answer 2

As mentioned by @spawn1701d, use a different initial condition.

This worked:

Manipulate[
 ParametricPlot[
  Evaluate[{x[t], p[t]} /. 
    NDSolve[{x'[t] == p[t], 
      p'[t] == -α x[t] - δ p[t] + α (x[t])^3, 
      x[0] == 1, p[0] == p0}, {x, p}, {t, 20}]], {t, 0, 20}],
 {{α, -1, "Alpha"}, -2, 0}, {{δ, 0, "Delta"}, 0, 
  2}, {{p0, .01, "p[0]"}, 0, 2}]