On the stability of limit cycles for planar differential systems

Abstract

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.