I'm wondering about the following statement:
Consider $f:E\subset \mathbb{R}^{2}\to \mathbb{R}^{2}$, $f \in C^{1}(E)$, and the system $$\dot{x}= f(x)\hspace{10mm}(1)$$ ans supose that $f(0)=0$, could you classify the qualitative behavior of statiotanary point $x=0$?
My attempt: Consider the linearized system associated of (1), i.e.,
$$Df(0), \hspace{10mm} (2)$$
So we have the following cases
- The eigen values have real part non zero, Thus from Hartman-Grobman we have that the behavior of (1) is ''equivalent'' to (2).
2.If only one eigenvalue of (2) have a real part zero, we can use the Central Manifold theory for analyze the behavior of (1).
- if both eigen values have real part zero, we can have the center, focus or center-focus.
3a. If the system (1) is symmetric respcect to axis $x$ or $y$ the system is a center.
3b. If the system (1) is analytic, then the center is either the center or a focus.
3c. If (1) is an Hamiltonian system, we have a center. (The hamiltonian the critical points are saddles or centers.)
Could have more results in this case? Thanks in advance!