# How to figure out the type of the bifurcation in a dynamical system?

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.

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It depends: if you know $f$ explicitely then working out critical points and normal forms will tell you so, otherwise you'd have to use a specialized program. Have a look at this scholarpedia article and the other articles, books and programs mentionned there (start with saddle-node) .