I have an ODE which can be written as
$x' = g(x) = x^3 + px + q$
where $p=-\frac{3c}{A}$ and $q=-\frac{3d}{A}$ (we can assume $c>0,d\neq. 0$) and I am trying to find and classify the bifurcation points relating to the bifurcation parameter A. I am not entirely sure on how this is done - I have tried to use the discriminant to find the bifurcation point (which I found to be $A=\frac{4c^3}{9d^2}$). However there is also a bifurcation point at $A=0$ (I am not sure how this is derived). Can someone please explain this and how to classify the bifurcation points once we find them?