i have this pitchfork bifurcation f=x^4-mx^3+x^2, is there a way to colour the equilibria according to their stability? [closed]

Question

V = x^4 - m*x^3 + x^2
sol = Solve[D[V, x] == 0, x] // Simplify
Plot[{x/.sol[[1]],x/.sol[[2]],x/.sol[[3]]},{m,0.1,4}, PlotStyle->{Blue,Blue,Blue}]

as far as I'm concerned sol[[1]]=0 at first is stable, when m>mbif it becomes unstable

Answer 1

I do not remember by heart the bifurcation value, MB, of the parameter m in this case, and too lazy to calculate it now. Let us assume that mb=2. Later you can substitute the correct value. With mb=2 try this:

ClearAll["Global`*"] 
V = x^4 - m*x^3 + x^2; 
sol = Solve[D[V, x] == 0, x] // Simplify;
mb = 2;
Show[{
  Plot[x /. sol[[1]], {m, 0.1, mb}, PlotStyle -> Blue, 
   PlotRange -> {{0.1, 4}, {-0.1, 2.5}}],
  Plot[x /. sol[[1]], {m, mb, 4}, PlotStyle -> Red],
  Plot[x /. sol[[3]], {m, 0.1, mb}, PlotStyle -> Red],
  Plot[x /. sol[[3]], {m, mb, 4}, PlotStyle -> Blue],
  Plot[x /. sol[[2]], {m, 0.1, 4}, PlotStyle -> Orange]
  }]

yielding the following

Have fun!