Plotting a time delay in mathematica?

Question

I'm having some difficulty plotting a phase space of a function f that exhibits time delay.

i.e.

s = NDSolve[{x'[t] == f[x[t - \[Tau]]], x[0] == 1 }, {x}, {t, 100}];
ParametricPlot[{x[t] /. s, x[t-\[Tau]] /. s}, PlotRange -> All]

i.e. how do I plot x(t-\[Tau]) vs x(t) ?

Answer 1

References:

• Tutorial: How To Solve Delay Differential Equations
• Syntax for ParametricPlot

The following examples where (1) s is defined as a function of a numeric argument $\tau$, and (2) ParametricPlot is supplied with the parameter argument ({t,0,10}) that it needs, work as you intended:

 ClearAll[s];
 f = Cos;
 s[\[Tau]_?NumericQ] :=   NDSolve[{x'[t] == f[x[t - \[Tau]]], x[0] == t}, {x},
  {t, 100}];
 Quiet@ParametricPlot[Evaluate[{ x[t - #] /. First@s[#], x[t] /. First@s[#]} & /@ 
    {1, 2, 3, 4}],  {t, 0, 100}, PlotRange -> All]

Another example with a non-constant initial function:

  ClearAll[s2]; 
  s2[\[Tau]_?NumericQ] := NDSolve[{x'[t] == f[x[t - \[Tau]]], 
        x[t /; t <= \[Tau]] == t}, {x}, {t, 100}]; 
  Quiet@ParametricPlot[ Evaluate[{x[t - #] /. First@s2[#], x[t] /. 
     First@s2[#]} & /@ {1, 2, 3, 4}], {t, 0, 100}, PlotRange -> All]

and, for the constant initial function in your example:

 ClearAll[s3];
 s3[\[Tau]_?NumericQ] := NDSolve[{x'[t] == f[x[t - \[Tau]]], x[0] == 0.},
   {x}, {t, 100}]; 
 Quiet@ ParametricPlot[Evaluate[{x[t - #] /. First@s3[#], x[t] /. 
    First@s3[#]} & /@ {1, 2, 3, 4}], {t, 0, 100}, PlotRange -> All]