So I have just started taking a course in bifurcations and finding some parts of the course rather tricky when I don't think they should be. I was wondering if anyone could explain this question.
Find a transformation putting each of the f(r, x) in normal form for a saddle-node bifurcation in a vicinity of the bifurcation point.
$$x'=1+r/2+x^2$$
From what I've researched, the normal form is the simplest form of expressing a saddle node bifurcation which is $x'=r+x^2$ and so the question is asking me to put the equation into normal form.
This is what I have done,
I have completed the square and got the equation into:
$$(x+r/4)^2+1-r^2/16$$
Thus, I let $X=(x+r/4)$ and $R=(1-r^2/16)$ and therefore can express it as:
$$X'=X^2+R$$
From here I am very confused because I believe that I need to incorporate the use of critical points but I am not sure. If someone could explain the procedure(?) in finding the normal form for a saddle node bifurcation, it would help clarify things so much for me