Saddle-node bifurcation

Manipulate[Plot[{E^(x + r), x}, {x, -10, 10}, PlotRange -> {-5, 15}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Red}, {Black}}], {r, -5,5}]

Pitchfork bifurcation

Manipulate[Plot[{r ArcTan[x], x}, {x, -5, 5}, PlotRange -> {-8, 8}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Red}, {Black}}], {r,0,5}]

Transcritical bifurcation

Manipulate[Plot[{r x(1-x), x}, {x, -3,3}, PlotRange -> {-5, 3}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Red}, {Black}}], {r,0,2}]

Period-doubling bifurcation

Manipulate[Plot[{r x(1-x), x, (2 - r)(x - ((-1+r)/r)) + ((-1+r)/r) , -(x - ((-1+r)/r)) + ((-1+r)/r)}, {x, -.5, 1.5}, PlotRange -> {-.5, 1.5}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Red}, {Black}, {Blue}, {Black}}], {r,2.5,3.5}]

Manipulate[Plot[{r x(1-x), x, r r x(1-x)( 1 - r x(1-x))}, {x, -.5, 1.5}, PlotRange -> {-.5, 1.5}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Red}, {Black}, {Purple, Thick}}], {r,2.5,3.5}]