On the stability of limit cycles for planar differential systems
| dc.creator | Giacomini, Hector | |
| dc.creator | Grau, Maite | |
| dc.date | 2005-05-02 | |
| dc.date.accessioned | 2026-07-07T05:19:34Z | |
| dc.date.available | 2026-07-07T05:19:34Z | |
| dc.description | 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 $γ$ 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 $γ\subseteq \{(x,y) | f(x,y) = 0\}$. 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 $\int_{0}^{T} (\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y})(γ(t)) dt= \int_0^{T} k(γ(t)) dt$, where $T>0$ is the period of $γ$. As an application, we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems. | |
| dc.description | 22 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0505027 | |
| dc.identifier | http://arxiv.org/abs/math/0505027 | |
| dc.identifier | J. Differential Equations 213 (2005) 368 -- 388 | |
| dc.identifier | doi:10.1016/j.jde.2005.02.010 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75065 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 34C05; 34C07; 34D20 | |
| dc.title | On the stability of limit cycles for planar differential systems | |
| dc.type | text |