Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the kernel of the Hessian at $x$ is non-trivial). An example is $x = 0$ for $f : \mathbb{R} \to \mathbb{R} : x \mapsto x^3$.
Then, if I perturb my function $f$ generically, I should observe such a phenomenon:
Called a "birth-death bifurcation": my degenerate critical point will either bifurcate into multiple non-degenerate critical points (birth), or die. (In the picture, I drew the bifurcations $f(x) \pm \varepsilon x$ for $\varepsilon > 0$. We can easily show that, for $f(x) = x^3$, $f - \varepsilon x$ has two critical points near $0$, while $f + \varepsilon x$ has none.)
My question is then the following: is it standard knowledge that my degenerate critical point will either die, or bifurcate into strictly more than one critical point? (For a generic bifurcation). If so, where can I find a proof of this statement?
It seems to be the picture that everyone always draws when speaking of birth-death bifurcations, but I haven't been able to find a proof, or an intuition behind this.
PS: provided that the degenerate critical point does indeed bifurcate into multiple critical points, I already know that these will be (generically) non-degenerate. So I only need the statement that:
"If the degeneracy doesn't die along the perturbation, then it will bifurcate into $N > 1$ critical points."
Any ideas, or references would be greatly appreciated :). Have a nice day!