I need to determine the stability of the origin of the following system of differential equations $$\pmatrix{\dot x \\ \dot y} = \pmatrix{-y^2 \\ -y + x^2 + xy},$$ using the Local Center Manifold Theorem as defined below.
Local Center Manifold Theorem: Let ${\bf f} \in C^r(E)$, where $E$ is an open subset of $\mathbb R^n$ containing the origin and $r \ge 1$. Suppose that ${\bf f}({\bf 0}) = {\bf 0}$ and that $D{\bf f}({\bf 0})$ has $c$ eigenvalues with zero real parts and $s$ eigenvalues with negative real parts, where $c+s = n$. The system $\dot {\bf x} = {\bf f}({\bf x})$ then can be written in the diagonal form $$\begin{cases}\dot{\bf x} &= C{\bf x} + {\bf F}({\bf x}, {\bf y}) \\ \dot{\bf y} & = P{\bf y} + {\bf G}({\bf x}, {\bf y}),\end{cases}$$ where $({\bf x}, {\bf y}) \in \mathbb R^c \times \mathbb R^s$, $C$ is a square matrix with $c$ eigenvalues having zero real parts, $P$ is a square matrix with $s$ eigenvalues with negative real parts, and ${\bf F}({\bf 0}) = {\bf G}({\bf 0}) = {\bf 0}, D{\bf F}({\bf 0}) = D{\bf G}({\bf 0}) = O$; furthermore, there exists $\delta > 0$ and a function ${\bf h} \in C^r(N_\delta({\bf 0}))$ that defines the local center manifold $$W^c_{\text{loc}}({\bf 0}) = \{({\bf x}, {\bf y}) \in \mathbb R^c \times \mathbb R^s : {\bf y} = {\bf h}({\bf x}) \text{ for } |{\bf x}| < \delta\}$$ and satisfies $$D{\bf h}({\bf x})[C{\bf x} + {\bf F}({\bf x}, {\bf h}({\bf x}))] - P{\bf h}({\bf x}) - {\bf G}({\bf x}, {\bf h}({\bf x})) = 0$$ for $|{\bf x}| < \delta$; and the flow on the center manifold $W^c({\bf 0})$ is defined by the system of differential equations $$\dot {\bf x} = C{\bf x} + {\bf F}({\bf x}, {\bf h}({\bf x}))$$ for all ${\bf x} \in \mathbb R^c$ with $|{\bf x}| < \delta$.
I have begun to attempt to solve the problem, namely by letting $f({\bf x}) = \pmatrix{-y^2 \\ -y + x^2 + xy}.$ Then, $$D{\bf f}({\bf x}) = \pmatrix{0 & -2y\\2x + y & x - 1} \implies D{\bf f}({\bf 0}) = \pmatrix{0 & -2\\0 & -1},$$ which has eigenvalues of $0$ and $-1$. Thus, following the above theorem, $C = O$ and $P = [-1]$, as well as $F(x,y) = -y^2$ and $G(x,y) = x^2 + xy$ (just take the nonlinear part of the original systems).
Then the system is now in diagonal form.
Here's where I am unfortunately stuck. I'm trying to follow examples in my textbook like the one found here on the first page. Almost magically, they come up with the next part which is the function $h$ with no explanation of how they come up with the terms that they do.
I understand that we omit the linear terms $1$ and $x$ because this would mean $(x,h(x))$ is tangent to the origin, as I have discovered here. But how do I know what terms to include or even when to stop? I've been looking at some examples and sometimes they have an $xy$ term in there in addition to the powers of $x$, especially when the dimension of the system gets higher.
The reason being is that the next step is to plug $h(x) = ax^2 + bx^3 + \cdots$ into the equation we have above, i.e., $D{\bf h}({\bf x})[C{\bf x} + {\bf F}({\bf x}, {\bf h}({\bf x}))] - P{\bf h}({\bf x}) - {\bf G}({\bf x}, {\bf h}({\bf x})) = 0$ to find the unknown coefficients. Short of just guessing can anyone provide some insight on how we come up with the form of $h$? (short of just guessing).
You want a Taylor series $y = a_2 x^2 + a_3 x^3 + \ldots$ with, as you said, no $x^0$ or $x^1$ term. Plug that in to the equation for $\dot{x}$, and you get $$ \dot{x} = -( a_2 x^2 + a_3 x^3 + \ldots)^2 = -a_2^2 x^4 - 2 a_2 a_3 x^5 + \ldots $$ Thus $$ \dot{y} = (2 a_2 x + 3 a_3 x^2 + \ldots) \dot{x} = - 2 a_2^3 x^5 - 7 a_2^2 a_3 x^6 + \ldots $$ But also $$ \dot{y} = - y + x^2 + x y = \left( -a_{{2}}+1 \right) {x}^{2}+ \left( -a_{{3}}+a_{{2}} \right) { x}^{3}+ \left( -a_{{4}}+a_{{3}} \right) {x}^{4}+ \left( -a_{{5}}+a_{{4 }} \right) {x}^{5}+ ( - a_6 + a_5) x^6 + \ldots$$ Equating coefficients of powers of $x$ and solving, we must have $$ \eqalign{a_2 &= 1\cr a_3 &= a_2 = 1\cr a_4 &= a_3 = 1\cr a_5 &= a_4 + 2 a_2^3 = 3\cr a_6 &= a_5 + 7 a_2^2 a_3 = 10\cr} $$ etc. You can take this to as many terms as you wish.