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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) .

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  • $\begingroup$ thanks but I need more elaborate answer here. I've read the resources you've mentioned. but some parts of them are blurred. by the way, I made the question more specific. $\endgroup$
    – Kamran Bigdely
    Feb 28, 2010 at 23:16
  • $\begingroup$ Well, at that point you really need to sit down and do some reading and computations for yourself on simple examples, over several days. Another excellent reference, a complete course textbook this time, is Elements of applied bifurcation theory (second edition) by Kuznetsov, there are even some excerpts on google books. I'm sure there are also good sets of lecture notes on the web for free. $\endgroup$
    – Thomas Sauvaget
    Mar 1, 2010 at 6:08

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