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$$f(x)= rx-\frac{x}{1+x}$$

Find the values of $r$ at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation.

I found the fixed points as $x^*=0,\frac{1}{r}-1$.

I am not sure how to identify what TYPE of bifurcation it is out of the three types. Is there a way to test which one?

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    If you have access to the book by Elaydi, Discrete Chaos, the author gives conditions in terms of various partial derivatives of $f$ with respect to $r$ and $x$ for distinguishing among the various bifurcations.2012-10-30

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