Center manifold of dynamical system

Question

Studying this dynamical system $\begin{cases}\dot x =(a-1)xy\\\dot y=-y^2\end{cases}$ ; I found that points along $y=0$ are equilibrium points, and linearization is $J=\begin{pmatrix}0& (a-1)x\\0& 0\end{pmatrix}$. So i have to look at the center manifold to see if they are stable or unstable.\ But how can I do? I tried this: center manifold shoul be \begin{equation}y=h(x)\end{equation} the derivative is $\dot y=h'\dot x=h'(a-1)xh=-h^2$ so i get $h'=-\frac{h}{(a-1)x}$. But I read that center manifold should be tangent to $x-$axis, and the solution of this equation is not. Where is the error? How can i study the stability of this critical points when $a$ is varying?

Answer 1

You can solve this one explicitly to see what the stability is. The centre manifold is the whole space.