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.

22 pages, no figures