Consider the system:
$\left\{ \begin{array}{lcc} \dot{x_1}=x_1y-x_1x_2^2 \\ \\ \dot{x_2}=x_2y-x_2x_1^2\\ \\ \dot{y}=-y+x_1^2+x_2^2 \end{array} \right.$
Obtain an approximation for the stable and center varieties and make the phase portrait. (Get the orbits on the center variety here)
I know how to calculate the center variety for a system of two equations but I do not know how to do it for three, could someone help me please? When there are two equations one makes $x_2=h(x_1)$ and so $\dot{x_2}=\frac{\partial h(x_1)}{\partial x_1}\dot{x_1}$, then helping each other by Taylor series, one gets explicitly to $h(x_1)$. How do I find the stable variety? I know that the stable variety of the center variety are tangent to the stable space and the center space respectively, but how do I find them in this case? Thank you very much.