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# 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:

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

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## Mathematics Subject Classification

34C25*no label found*

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