I am looking for a place to read the proofs of the Bendixson and Dulac-Bendixson theorems. Namely, let $D$ be a simply connected set and the following system be defined in $D$, and \begin{align} \dot x &=P(x,y), \\ \dot y &=Q(x,y), \end{align} then the following theorems hold.
Theorem: (Bendixson) Given that $P_x+Q_y$ doesn’t change sign in $D$, the system does not have a non-constant periodical solution.
Theorem: (Dulac-Bendixson) If a function $B(x,y)\in C^1(D)$ exists, such that $(BP)_x+(BQ)_y$ doesn't change sign in $D$. Then the system does not have a non-constant periodical solution.
You can read the proof right here, in short form:
Assuming a periodic orbit $C$ with interior $U$, $$\oint_C(BQ\,dx-BP\,dy)=\oint_C B(\dot y\,dx-\dot x\,dy)=0$$ because $\dot y\,dx-\dot x\,dy=0$ along the orbit. Now use Green's theorem to conclude $$\iint_U\bigl((BP)_x+(BQ)_y\bigr)\,dx\,dy=0.$$You now have your contradiction. This proves Dulac–Bendixson; Bendixson follows by setting $B=1$.