Consider a planar dynamical system described in polar coordinates as $$ \left\{ \begin{array}{ll} \dot{\theta}=\Delta - r \sin \theta,\\ \dot{r} = - r + 1 + \cos \theta, \end{array} \right. $$ where $\Delta$ is a constant such that $|\Delta|< \max_{0\le x\le 2\pi}(1+\cos x)\sin x$. I need to show that the system does not have any limit cycle.

Remark: The system has a stable node and a saddle node.

Maybe a better question is that what are the ideas, methods and tricks to prove of nonexistence of limit cycle, especially for planar systems. For example, Dulac- Bendixson is one of them. Is there any other?