Suppose we have a dynamical system 

$\dot{x} = f(x,r)$  

in which x is a state variable and r is a bifurcation parameter. How to figure out which kind of bifurcation(s) (e.g. saddle-node, transcritical, pitchfork, hopf and etc) the system undergoes?


Edit 1: consider the space as 1D or 2D.