Consider the DE $\begin{cases} \dot{x}=-2x(x-1)(2x-1)\\ \dot{y}=-2y \end{cases}$. Determine all equilibrium points and classify these. Choose between a saddle point, (in)stable nod, center or a (in)stable spiral.
I have found $(0,0), (1,0), (\frac{1}{2},0)$ to be the equilibrium points of the system, but I am stuck at classifying them.
To classify them, I determined the following Jacobian matrix: $$J=\begin{bmatrix}-12x^{2}+12x-2 & 0\\ 0 & -2 \end{bmatrix}$$.
For $(0,0)$, I found:
$$J_{(0,0)}=\begin{bmatrix}-2 & 0\\ 0 & -2 \end{bmatrix}\Rightarrow\lambda=\begin{vmatrix}-2-\lambda & 0\\ 0 & -2-\lambda \end{vmatrix} =(-2-\lambda)(-2-\lambda)=0 \iff\lambda_{1,2}=-2$$
The corresponding eigenvectors are: $v_{1}=\begin{bmatrix}1\\ 0 \end{bmatrix},\,v_{2}=\begin{bmatrix}0\\ 1 \end{bmatrix}$
Which are linearly independent, so the corresponding phase portrait is a stable star. However, this option wasn't listed in the question, so I must have made a mistake. Could anyone give me a hint here?