Problem :
Find the Stationary Points and discuss their behavior using Linearisation for the following system of differential equations :
$$x' = \sin y, y' = -\sin x$$
Discussion :
Solving the system :
$$\begin{cases} \sin y = 0 \\ -\sin x =0 \end{cases}$$
yields :
$$\begin{cases} x= k\pi, k\in \mathbb Z \\ y=h\pi, h\in \mathbb Z \end{cases}$$
So, every point $A_k = (k\pi,h\pi)$ is a stationary point of the initial system of differential equations.
Now, the Jacobian of the given system is :
$$J(x,y) = \begin{bmatrix} 0 & \cos y \\ -\cos x & 0\end{bmatrix}$$
The Jacobian of any given stationary point that we found, calculates as :
$$J(k\pi, k\pi) = \begin{bmatrix} 0 & \cos (k\pi) \\ -\cos (h\pi) & 0\end{bmatrix} $$
It's clear that $\det(J(k\pi, k\pi)) = \cos^2(k\pi) \neq 0 $
and that the eigenvalues of the stationary point Jacobian is :
$$\det(J(k\pi, k\pi) - λI) = 0 \Leftrightarrow λ^2 + \cos(k\pi)\cos(h\pi) = \begin{cases} λ^2 +1 = 0 \Leftrightarrow λ = \pm i \\ λ^2 - 1 = 0 \Leftrightarrow λ = \pm 1 \end{cases}$$
So, since the eigenvalues are purely imaginary, according to notes, this tells us that the stationary points $A_k = (k\pi,h\pi)$ are verified as centers of our differential equation system if the first case holds, or as a different kind of critical point ( I do not know it's name in English), which is unstable.
My question is, am I correct ? I found a theorem in our book stating that :
Let $(ξ,n)$ be a hyperbolic stationary point (the Jacobian at the point $(ξ,n)$ is not equal to $0$) of the almost linear system : $$x'=f(x,y), y' = g(x,y).$$ Then, in a neighborhood of $(ξ,n)$ the almost linear system and the linearised system of it, have topologically equal vector fields, which means that thay have the same kind of stability, except of the case when $(ξ,n)$ is a center of the linearised system.
The last part of this theorem (except of the case when $(ξ,n)$ is a center of the linearised system) got me wondering if my solution and hence my conclusion about the stationary points and their behavior, of the given system of differential equations is correct.
I would really appreciate if anyone could clarify me the above and help me get a better grasp of understanding how the theorem correlates with such cases of systems.