Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the corresponding bifurcation parameter value, can we classify the bifurcation point using the same criteria as is used to classify the bifurcation points of iterated maps (see below)?
Context: To classify bifurcations for iterated maps $x_{n + 1} = f(x_n;\; \mu)$ we have the following criteria. In the following the partial derivatives of $f$ are evaluated at $(x^\star, \mu_c)$,
Saddle node bifurcations: $f_x = 1$, $f_{xx} \neq 0$, $f_\mu \neq 0$
Transcritical bifurcations: $f_x = 1$, $f_{xx} \neq 0$, $f_\mu = 0$, $f_{\mu x}^2 - f_{xx} f_{\mu \mu} \neq 0$
Pitchfork bifurcations: $f_x = 1$, $f_{xx} = 0$, $f_\mu = 0$, $f_{\mu x} \neq 0$, $f_{xxx} \neq 0$
Period doubling bifurcations: $f_x = -1$, $2f_{\mu x} + f_{\mu} f_{xx} \neq 0$, $f_{xx}^2/2 + f_{xxx}/3 \neq 0$.
Remarks: Of course, the $f_x = \pm 1$ criteria will have to be replaced with $f_x = 0$.
Related question: Do period doubling bifurcations occur in continuous systems of the form $\dot{x} = f(x;\; \mu)$, or do they only occur in iterated maps?