An example equation for a Transcritical Bifurcations is given by:
$$\dfrac{dx}{dt} = f(x, r) = r x - x^2$$
In Mathematica, we can define the function as:
f[x_, r_] := r x - x^2
We can create a grid of plots to show the Transcritical bifurcation as:
p1 = Plot[f[x, 0], {x, -3, 3}, PlotRange -> {{-3, 3}, {-4, 3}}, Frame -> True,
FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r=0 case"}}, BaseStyle -> 12,
RotateLabel -> False, PlotTheme -> "Classic",
PlotStyle -> Thick, ImageSize -> 250];
p2 = Plot[f[x, 3], {x, -5, 5}, PlotRange -> {{-5, 5}, {-4, 3}}, Frame -> True,
FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r>0 case"}}, BaseStyle -> 12,
RotateLabel -> False, PlotTheme -> "Classic",
PlotStyle -> Thick, ImageSize -> 250];
p3 = Plot[f[x, -3], {x, -5, 5}, PlotRange -> {{-5, 5}, {-4, 3}}, Frame -> True,
FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r<0 case"}}, BaseStyle -> 12,
RotateLabel -> False, PlotTheme -> "Classic",
PlotStyle -> Thick, ImageSize -> 250];
Grid[{{p1, p2, p3}}, Frame -> True, FrameStyle -> LightGray]
However, what is the best approach to having it look like the grid below by adding the arrows and circles for stability and type of stability?
Is there a way to generalize this for different type of bifurcations (Hopf, Supercritical ...)?