Bifurcation diagram with first-order differential equation

Question

I have code from question 96004, and I made same changes:

s1 = ParametricNDSolveValue[{D[r[t], t, 
       t] - μ r[t] (r[t] + μ)^2 == 0, D[θ[t], t] == 1, 
    r[0] == 1, θ[0] == 1}, {r[t] Cos[θ[t]], r[t] Sin[θ[t]]}, {t, 0, 
    100}, {μ}, MaxSteps -> ∞];
coll = {};
Table[pp = 
   ParametricPlot[Through[s1[a][t]], {t, 0, 100}, 
    PlotRange -> {{-4, 4}, {-4, 4}}, PerformanceGoal -> "Quality", 
    MeshFunctions -> (#2 &), Mesh -> {{0.}}, 
    MeshStyle -> {Red, PointSize[0.02]}];
  pts = pp[[1, 1]];
  AppendTo[
   coll, {a, #[[1]]} & /@ 
    pts[[First@Cases[pp[[1]], Point[x__] :> x, -1]]]];, {a, -1, -0.5, 
   0.005}];
lp = ListPlot[Join @@ coll, Frame -> True, PlotStyle -> Red];
Manipulate[
 Column[{ParametricPlot[Through[s1[τ][t]], {t, 0, 100}, 
    PlotRange -> {{-4, 4}, {-4, 4}}, PerformanceGoal -> "Quality", 
    MeshFunctions -> (#2 &), Mesh -> {{0.}}, 
    MeshStyle -> {Red, PointSize[0.02]}, Frame -> True], 
   Show[lp, 
    Graphics[{Gray, 
      Line[{{τ, -2}, {τ, 2}}]}]]}], {τ, -1, -0.5, 
  0.01}]

In the original question the differential equation is second order. I found the code doesn't work for first order differential equation. Could someone tell me how to change the above code?