Let $$\dot x = f(x), \; x\in \mathbb{R}^n$$ be an ODE and let the Jacobian $\frac{\partial f}{\partial x}$ be singular at the point $x_0$. Is this sigularity sufficient for saying that a bifurcation occurs at $x_0$?
Note: In this question, I am thinking about the role of the "parameter" in bifurcation (i.e., we can define $\dot x = f(x,p), \; x\in \mathbb{R}^n, p\in\mathbb{R}$, where $p$ is a parameter). Particularly, I was thinking what happens if we do not consider any parameter for the vector field, and just look at the singularity of the vector field.
No, namely in general a singular Jacobian at an equilibrium point yields an inconclusive stability analysis. For example consider
$$ f(x,p) = -p^2\,x-x^3 $$
which for $p=0$ has a singular Jacobian at $x=0$, but that equilibrium remains stable.