On the stability of limit cycles for planar differential systems

We consider a planar differential system $\dot{x}=P(x,y)$, $\dot{y} = Q(x,y)$,where$P$and$Q$are$$\mathcal{C}$^1$functionsinsomeopenset$$\mathcal{U}$ \subseteq $\mathbb{R}$^2$,and$\dot{}=\frac{d}{dt}$\mathrm{.}Let$\gamma$beaperiodicorbitofthesystemin$$\mathcal{U}$$\mathrm{.}Let$f(x,y): $\mathcal{U}$ \subseteq $\mathbb{R}$^2 \to $\mathbb{R}$$bea$$\mathcal{C}$^1$ function such that $$P(x,y)\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(x,y)+Q(x,y)\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(x,y)=k(x,y)f(x,y),$$ where $k(x,y)$ is a ${\mathcal{C}}^1$ function in $\mathcal{U}$ and $\gamma \subseteq {(x,y) | f(x,y) = 0}$\mathrm{.}Weassumethatif$p \in $\mathcal{U}$$issuchthat$f(p)=0$and$\nabla f(p)=0$,then$p$ is a singular point. We prove that $\int_{0}^{T} (\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y})(\gamma(t)) dt= \int_0^{T} k(\gamma(t)) dt$,where$T>0$ is the period of $\gamma$. As an application, we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems.