Finding the local center manifold Question:
Consider the system
\begin{align}
\frac{dx}{dt} & = x^2+xy+y^2 \\
\frac{dy}{dt} & = x^2+xy-y
\end{align}


*

*Find a quadratic approximation for the center manifold about $(0,0)$.

*Find a quadratic vector field that approximates the dynamics on the center manifold.

Attempt: Q1
The linear approximation to the system about $(0,0)$ is
$$\frac{d}{dt}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$
where the eigenvalues are $0$ and $1$, and the eigenvector for the zero-eigenvalue is $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
After some plug and chug, I found that the center manifold is
$$y(x) = x^2 + \mathcal O(x^3)$$
Is this what is meant by "a quadratic approximation" for the center manifold? I ask this because I have seen an example somewhere (forgot where though) where the quadratic center manifold is expressed in the form $h(x,y) = ax+by+Ax^2+Bxy+Cy^2+\mathcal O(x^3)$.

Attempt: Q2
Supposing that my answer to the first question is correct, then to do the second question, do I just plug in $y(x) = x^2 + \mathcal O(x^3)$ into the equation for $\frac{dx}{dt}$?
Namely, is the solution for the second question simply
$$\frac{dx}{dt} = x^2+x\big(x^2 + \mathcal O(x^3)\big) + \big(x^2 + \mathcal O(x^3)\big)^2 = x^2 + \mathcal O(x^3)$$
But this is not a "vector field" though? (At least, not in the sense that I think the question is asking for)
Any hints would be much appreciated. Thanks!

Given the system
$$
\dot x = C x + F(x,y)\\
\dot y = D y + G(x,y)
$$
we are looking for a function $h(x) = a x^2+b x^3+\cdots$ such that
$$
h'(x)\left(C x+F(x,h(x))\right) = D h(x) + G(x,h(x))
$$
or
$$
\left(2ax+3bx^2+\cdots\right)\left(x^2+x\left(a x^2+b x^3+\cdots\right)+\left(a x^2+b x^3+\cdots\right)^2\right) = -\left(a x^2+b x^3+\cdots\right)+x^2+x\left(a x^2+b x^3+\cdots\right)
$$
so equating we obtain from $a-1 = 0$
$$
h(x) = x^2 + O(x^3)
$$
now the flux approximation is given by
$$
\dot x = x^2+O(x^3)\\
\dot y = -y + x^2 + O(x^3)
$$
Follows the stream flow for the complete system with the $h(x)$ in red defining the unstable center manifold

