We consider a planar differential system $\dot{ x}= P(x,y)$, $\dot{ y} = Q(x,y)$, where $P$ and $Q$ are $\mathcal{ C}^1$ functions in some open set $\mathcal{ U} \subseteq \mathbb{ R}^2$, and $\dot{ }=\frac{ d}{ dt}$. Let $\gamma$ be a periodic orbit of the system in $\mathcal{ U}$. Let $f(x,y): \mathcal{ U} \subseteq \mathbb{ R}^2 \to \mathbb{ R}$ be a $\mathcal{ C}^1$ function such that \[ P(x,y) \frac{ \partial f}{ \partial x}(x,y) + Q(x,y) \frac{ \partial f}{ \partial y} (x,y) = k(x,y) f(x,y), \] where $k(x,y)$ is a $\mathcal{ C}^1$ function in $\mathcal{ U}$ and $\gamma \subseteq \{ (x,y) | f(x,y) = 0\}$. We assume that if $p \in \mathcal{ U}$ is such that $f(p)=0$ 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.
Categories: math.DS