Consider the following system \begin{align*} \dot{x} &= -2x - y - xy\\ \dot{y} &= 2x + y + 2xy \end{align*} I am trying to determine its center manifold around point $(0,0)$.
I tried to use the approximation $h(x) = ax^2 + bx^3 + \cdots$, however, there will be an $\mathcal{O}(x)$ term in the center manifold formula. I also tried to directly solve $y$ for $x$ by considering $dy/dx$, but the integrand is way too complicated and I didn't see a way to solve it.
So could any one give me some hints?
Trial 1 The linearized system at $(0,0)$ has matrix \begin{bmatrix} -2 & -1 \\ 2 & 1 \end{bmatrix} and the eigenvectors are $$ \begin{bmatrix} -1 \\ 1 \end{bmatrix}\;, \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$ Then the transformation $$ \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -1 & 1\\1 & -2 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} \Longleftrightarrow \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} -2 & -1\\-1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$ which gives the transformed system $$ \begin{bmatrix} \dot{u} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} -2\dot{x} - \dot{y} \\ -\dot{x} - \dot{y} \end{bmatrix} = \begin{bmatrix} 2x+y \\ -xy \end{bmatrix} $$ Plugging in $(x,y) = (-u-v,u-2v)$ gives \begin{align*} \dot{u} &= -u \\ \dot{v} &= 2v^2 - 3uv + u^2 \end{align*}