# 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]

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


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

• From your question it is unclear what you are asking for. Jan 23 at 12:32

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[], {m, 0.1, mb}, PlotStyle -> Blue,
PlotRange -> {{0.1, 4}, {-0.1, 2.5}}],
Plot[x /. sol[], {m, mb, 4}, PlotStyle -> Red],
Plot[x /. sol[], {m, 0.1, mb}, PlotStyle -> Red],
Plot[x /. sol[], {m, mb, 4}, PlotStyle -> Blue],
Plot[x /. sol[], {m, 0.1, 4}, PlotStyle -> Orange]
}]
`

yielding the following Have fun!