Can any general statements be made about the fixed points of coupled systems of holomorphic functions?

Consider the system:

$\displaystyle z'(t)=f(z);\quad z\in \mathbb{C}$

where $\displaystyle f(z)$ is holomorphic.

For example:

$\displaystyle z'(t)=z\Rightarrow \begin{cases} x'(t)=x \\ y'(t)=y\end{cases}$

or:

$\displaystyle z'(t)=\sin(z)\Rightarrow \begin{cases} x'(t)=\textbf{Re}(\sin z) \\ y'(t)=\textbf{Im}(\sin z)\end{cases}$

Can I predict which types of fixed points I would encounter with these systems strictly on the analytic properties of $\displaystyle f(z)$?