This a verbatim copy of an example on center manifold reduction on nonlinear dynamical system I found on some lecture note:
Consider the system \begin{align*} \dot{x}&=x^2y-x^5\\ \dot{y}&=-y+x^2 \end{align*} Again $(0,0)$ is an equilibrium point and the system is in canonical form $$J_X(0,0)=\begin{bmatrix}0&0\\0&-1\end{bmatrix}.$$ Consider $y=h(x)=ax^2+bx^3+O(x^4)$. Then as in previous example $$\dot{y}=h'(x)\dot{x}=2a^2x^5+[2a(b-1)+3ab]x^6+O(x^7)$$ and $$\dot{y}=-h(x)+x^2=-(a-1)x^2-bx^3+O(x^4).$$ We deduce that $a=1$ and $b=0$. The center manifold is given by $$y=h(x)=x^2+O(x^4)$$ and the dynamics are governed by $$\dot{x}=x^4+O(x^5).$$
My question is how do we obtain the last equation? I was guessing that we substitute $y=x^2+O(x^4)$ into $\dot{x}=x^2y-x^5$. But if we do so we should have $$\dot{x}=x^2(x^2+O(x^4))-x^5=x^4+O(x^6)-x^5.$$ I am completely lost at the moment.