I'm currently brushing up on my ODE theory by reading through some texts.
I was told that the system
has a rather interesting property:
Apparently, all periodic solutions of this system are of the form $\varphi=(\varphi_1,\varphi_2)$, where $\varphi_1(t)=\sin(t+c)$, $\varphi_2(t)=\cos(t+c)$, and $c$ is an arbitrary constant. (This is excluding the trivial periodic solution $\varphi=0$ of course.)
Is there an easy proof to see why this is true? It looks like a really interesting result that comes out of nowhere (or in my opinion at least), and I'm hoping that the reasoning behind it could help me understand periodic solutions better.