I am trying to verify that a ODE system, $\dot x=f(x,y), \dot y=g(x,y)$, has no periodic orbits in the plane (presumed to be in $\mathbb R^2$).
Thus I need to show for a function $h(x,y)$, that $\dfrac{\partial hf}{\partial x}+\dfrac{\partial hg}{\partial y}$ is either positive or negative throughout the plane.
However, I have found if I choose $h=1$, then I obtain $-2(x^2+y^2) \le 0 $ in the plane. Thus my question is whether this would satisfy Dulac’s negative criteria due to the presence of zero.
Thank you for your time.