Let's say I have a 4D system of autonomous ODEs
\begin{equation} \begin{split} \dot{u} = f(u,v,w,z)\\ \dot{v} = g(u,v,w,z)\\ \dot{w} = h(u,v,w,z)\\ \dot{z} = i(u,v,w,z)\\ \end{split} \end{equation}
Let $\dot{x} = (\dot{u}, \dot{v}, \dot{w}, \dot{z})$ and $x^{*} = (u^{*}, v^{*}, w^{*}, z^{*})$ be a fixed point. Therefore, Taylor's expansion of the RHS yields
\begin{equation} \dot{x} = f(x^*) + \frac{\partial f}{\partial x} |_{x^*}(x - x^*)+ ... = \frac{\partial f}{\partial x} |_{x^*}(x - x^*) ... \end{equation}
By interpreting $\frac{\partial f}{\partial x} |_{x^*}$ as a Jacobian and setting $\delta x = x - x^*$, we get a linearized system
\begin{equation} \dot{\delta x} = J^{*}\delta x \end{equation}
Let's say the characteristic polynomial of the characteristic equation $det(\lambda I - J^{*})$ is 4th degree polynomial with 4 roots such that $\lambda _{1,2}$ are real and $\lambda _{3,4}$ are complex. How do I classify this fixed point?