Consider the small solutions of the following system
\begin{align} \dot{x}&=\epsilon x-x^3+xy \\ \dot{y}&=-y+y^2-x^2 \\ \dot{\epsilon}&=0 \end{align}
with $0<\epsilon\ll 1$. Show there exist a $\delta>0$ so that
$y=h(x,\epsilon), \quad \text{for} \quad |(x,\epsilon)|\leq\delta$
with $h$ smooth, is an attracting center manifold.
Initial calculations: We let $y=h(x,\epsilon)$ and substitute into the second equation
$$ h'(x)(\epsilon x - x^3 + xh) = -h+h^2-x^2 $$
Now let
\begin{align} h(x,\epsilon)&=ax^2+bx\epsilon+c\epsilon+O((x,\epsilon)^3) \\ D_xh &= 2ax+b\epsilon+O((x,\epsilon)^2) \end{align}
so that
\begin{align} O(x,\epsilon)^3 = x^2(a+1)+b\epsilon+c\epsilon^2+O(x,\epsilon)^3 \end{align}
This yields that $a=-1$ and $b=c=0$ or
\begin{align} h(x,\epsilon)=-x^2+O((x,\epsilon)^3) \end{align}
I am not sure how this translates into the claim that $h(x,\epsilon)$ is an attracting center manifold for $|(x,\epsilon)|\leq \delta$ (or if this is the correct procedure). Can someone give an explanation ? Thanks.