I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the Jacobian of the fixed point $(1,0)$, which is


$$A\_{(1,0)} =
 \begin{pmatrix}
 -1 & -a\\
 0 & 0 
\end{pmatrix}$$


It is clear that we have a zero real part eigenvalue, so we need to find the center manifold. Here is the problem, every time i assume $y=h(x)=c\_1 x^2 + c\_2 x^3 +...$ I get all the coefficients zero! What is the problem



Why all the coefficients of the center manifold of this system are zeros?
The center manifold for the first problem is not $y = O(x^2)$. It is
tangent to the center subspace of the linearized operator, which is
not $x$-axis in this case.
Please find an eigenvector associated with zero eigenvalue and change
the coordinates so that the center subspace corresponds one of the
coordinate axes. In the new coordinates, you will get the answer soon.


