Local center manifold theorem.

Question

Local center manifold theorem, under certain assumptions, state that for the \begin{cases} \dot x = Cx+F(x,y) \\ \dot y = Py+G(x,y)\\ \end{cases} there exist a function $h(x)$ such that $$Dh(x)\left[Cx+F(x,h(x))\right]-\left[P\ h(x)+G(x,h(x))\right]=0$$ I noticed, doing exercises, that what one does is to develop the function $h(x)$ as $$h(x)=a x^2+b x^3+...$$ My question is: why we start from the second order and not by first?

Answer 1

The absence of linear term in $$h(x) = ax^2 + bx^3 + \dots$$ means that curve $(x, h(x))$ is tangent to $Ox$ axis at origin. And that definitely makes sense when $E_{c}$ (a linear subspace which is spanned by eigenvectors of Jacobi matrix with purely imaginary or zero eigenvalues) coincides with $Ox$ axis too. Roughly speaking, local center (or stable/unstable) manifold is an invariant manifold which is tangent to $E_c$ (or $E_s$/$E_u)$ at origin. Moreover, this local manifold is a graph over $E_c$, you can take a look at this question for better understanding. If we suppose that $$\frac{\partial F}{\partial x}(0,0) = \frac{\partial F}{\partial y}(0,0) = \frac{\partial G}{\partial x}(0,0) = \frac{\partial G}{\partial y}(0,0) = 0,$$ then eigenvectors of Jacobi matrix just coincide with $Ox$ and $Oy$ axes. I think this explains pretty well why excercises suggest searching function of this form