How to apply centre manifold theorem in the following system?

Question

I am learning the centre manifold. One theorem is as follows

Suppose the zero solution of our centre manifold system is stable, then the zero solution of our originating system is also stable.

I am looking the following example: $$ x'=x^2y-x^5, y'=-y+x^2 $$

Clearly, $(0,0)$ is a fixed point. I am confused how to check its stability.

We have the matrix of linearization $$\begin{bmatrix} 0 & 0\\ 0 & -1 \end{bmatrix}$$ with eigenvalues $\lambda_1=0$ and $\lambda_2=-1$.

But why it said that "we find $y=h(x)=x^2+o(x^4)$, it follows that the dynamics on the centre manifold is dictated by $x'=x^4-x^5+o(x^8)$."

So I am confused if $(0,0)$ stable or not?

Answer 1

Assuming $y = y(x) = \sum_{k=1}^n a_k x^k$ we have

$$ \dot y = y_x\dot x = \left(\sum_{k=1}^n k a_k x^{k-1}\right)\left(x^2\left(\sum_{k=1}^n a_k x^k\right)-x^5\right)=-\sum_{k=1}^n a_k x^k+x^2 $$

having

$$ a_1+(a_2-1)x+(a_1^2+a_3)x^2+\cdots + = 0\ \ \forall x $$

thus obtaining

$$ y = x^2 + O(x^5) $$

then the origin dynamics is given by

$$ \dot x = x^2(x^2+O(x^5))-x^5 $$

NOTE

$\dot x = x^4-x^5$ is clearly unstable.