For an ODE $\dot{x} = f(x)$, it is known that if an equilibrium $x^{*}$ is hyperbolic (none of eigenvalues of $Df(x^{*})$ have zero real part), then it persists for small $O(\epsilon)$ perturbations as a function of $\epsilon$ as can be shown by the implicit function theorem. So it still exists, although it can move slightly. By the Hartman-Grobman, the perturbed equilibrium also has (un)stable manifolds. For a diffeomorphism $\bar{x} = F(x)$, the situation with hyperbolic fixed points for perturbations $F_{\epsilon}(x)$ is similar.
What about the case of parabolic / elliptic points? KAM theory states that for a diffeomorphism, elliptic fixed points persist under sufficient non-resonant assumptions on their eigenvalues. However, in the general case, what happens under an arbitrary (in some sense) perturbation to the elliptic or parabolic point itself? Does it move slightly, vanish, or undergo a bifurcation?