Stability of a complex valued system using Jacobian

Question

I want to find the conditions for stability of a system by constructing the Jacobian matrix and finding what values of my parameters will give me negative eigenvalues. My system is of the form $ \dot{x} = f(x) $ , and $f(x)$ is complex. I have seen something similar to this done, and apparently when $f(x)$ is complex we must split it up into its real and complex components and making new equations. I am a bit confused as to how to do this, and why it is necessary to break up the complex components. (Also, if anyone can point me to a text discussing the Jacobian in the context I am using it - finding stability conditions, I would much appreciate it.)