So let us say we have a function $\dot{x} = f(x,r)$ that has some critical point at $(x_0,r_0)$ such that $f(x_0,r_0)=0$. The question now is: when is this a bifurcation point? I understand that $\frac{\partial{f(x_0,r_0)}}{\partial{x}} = 0$ works in practice, but I have two questions.
1.) Intuitively, why is this the case?
2.) The proof for this appealed to the implicit function theorem and said that if the derivative was non-zero, then there would be a (local) solution $x=X(r)$ such that $X(r_0)=x_0$, which cannot happen if the point is a bifurcation point. This doesn't click with me, why would this be not work if the point was a bifurcation? Also, how is it the case that the Jacobian is zero if only one entry is zero (namely the x-derivative entry)? Shouldn't it be the case that any 2 entries on opposite columns need to be zero?
Thanks