I am given a nonlinear dynamical system:
$\dot x = x^2y-x$
$\dot y = 4xy^2+4y^2-x^2y-xy$
I was able to find all the equilibrium points and classify them, expect $e_0=(0,0)$. I found that the Jacobian evaluated at $e_0$ is $$ \begin{pmatrix} -1 & 0 \\ 0 & 0 \\ \end{pmatrix}$$ Which gives eigenvalues $\lambda_{1,2}=0,-1$. I have seen that if $\lambda_2>0$ then the classification is trivial, however this is not the case. The only way to do it (apparently) is to use the Central Manifold Theorem and the Hartman-Grobman Theorem, however I have difficult time applying it (I have never studied it). So my question is:
$1.$ Is the application of the Central Manifold and Hartman-Grobman theorems necessary ? Are there any other way to do classify $e_0$ ?