Consider $\dot x =f(x)$ where $x\in\mathbb{R}^n$ and $f:\mathbb{R}^n\to\mathbb{R}^n$ is smooth enough. I am studying Fold and Hopf bifurcations in such ordinary differential equations. We know that a bifurcation happens when we break the hyperbolicity of the Jacobian matrix $\frac{\partial f}{\partial x}$. Now, if we break the hyperbolicity via one (or several) zero eigenvalue(s), Fold bifurcation takes place. Moreover, if we we break the hyperbolicity via one (or several) pair(s) of pure imaginary eigenvalue, Hopf takes place. I was wondering if there is any relationship between the algebraic/geometric multiplicty of these eigenvalues and the bifurcations (e.g., their stability, type, etc.)?
Any comment or response is greatly appreciated!