Bifurcations are my favorite part of dynamical systems study. It took me a long time to get an intuitive feeling of their caracterisation and classification, and to connect it to what I saw in the wild. In part because I had trouble finding systematic presentations of simple bifurcations.

Here are interactive visualisations of some codimention 1 bifurcations.

We consider the one dimentional dynamical system given by the ordinary differential equation:

\frac{dy}{dt}=f_a(y)

For each bifurcation, three graphs are plotted, the bifurcation diagram (a \mapsto y^*), the flow (y \mapsto f_a(y)) and some trajectories (t\mapsto y(t)). In the bifurcation diagram, stable equlibria are represented by solid lines, and unstable equilibria are represented by dashed lines. You can mouse-over the bifurcation diagram to change the value of the parameter a.

Saddle-node bifurcation

Also called limit point bifurcation.

Normal form: f(y)=a+y^2

Transcritical Bifurcation

Normal form: f(y)=ay-y^2

Pitchfork bifurcation

Normal form: f(y)=ay \pm y^3

Supercritical

Normal form: f(y)=ay - y^3

Subcritical

Normal form: f(y)=ay + y^3

Source: main.js Software used: d3js.