I have a dynamical system with one of its equilibrium points at $(0,0)$.
The Jacobian matrix, I have calculated as \begin{pmatrix} 2yx-1 & x^2\\ -y-2yx+4y^2 & -x-x^2+8y+8xy \end{pmatrix}
So, evaluated at $(0,0)$, this becomes \begin{pmatrix} -1 & 0\\ 0 & 0 \end{pmatrix}
And the eigenvalues of this matrix are $\lambda_1=0$ and $\lambda_2=-1$.
How do I classify this equilibrium point? I know if the eigenvalues are real and different and of the same sign they are stable or unstable nodes, and if they are real, different and of opposite signs they are saddle points. Am I missing some information here?
Edit: The system in question is
$$\dot{x}=yx^2-x$$ $$\dot{y}=-xy-x^2y+4y^2+4xy^2$$