# Proof of Dulac’s Criteria

Consider the the planar system $\dot{x}=f(x)$, where $f=(X,Y)^{t}$ and $x=(x,y)^{t}$. Consider the vector field $(-\rho Y,\rho X)$. Suppose that there is a periodic orbit contained in $E$ associated to the planar system. Let $\gamma$ be that periodic orbit. We have:

$\displaystyle\int_{\gamma}(-\rho Y,\rho X)ds=\int_{0}^{\tau}(-\rho Y(x,y),\rho X% (x,y))\cdot(\dot{x},\dot{y})dt=\int_{0}^{\tau}-\rho Y(x,y)X(x,y)+\rho X(x,y)Y(% x,y)=0$

On the other hand, the region within $E$ that is limited by $\gamma$ is simply connected because $E$ is simply connected. Let $\tilde{E}$ be the region limited by $\gamma$. Then, by Green’s theorem, we have:

 $\int_{\gamma^{+}}(-\rho Y,\rho X)ds=\int\int_{\tilde{E}}\frac{\partial}{% \partial x}(\rho X)-\frac{\partial}{\partial y}(-\rho Y)dxdy=\int\int_{\tilde{% E}}\frac{\partial}{\partial x}(\rho X)+\frac{\partial}{\partial y}(\rho Y)dxdy$

Because $\tilde{E}$ has positive area and the integrand function has constant signal, then this integral is different from zero. This is a contradiction. So there are no periodic orbits. \qed

Title Proof of Dulac’s Criteria ProofOfDulacsCriteria 2013-03-11 19:17:10 2013-03-11 19:17:10 Filipe (28191) (0) 9 Filipe (0) Proof msc 34C25