# Dulac’s criteria

Let

$\dot{\textbf{x}}=\textbf{f}(\textbf{x})$

be a planar system where $\textbf{f}=(\textbf{X},\textbf{Y})^{t}$ and $\textbf{x}=(x,y)^{t}$. Furthermore $\textbf{f}\in C^{1}(E)$ where $E$ is a simply connected region of the plane. If there exists a function $p(x,y)\in C^{1}(E)$ such that $\frac{\partial(p(x,y)\textbf{X})}{\partial x}+\frac{\partial(p(x,y)\textbf{Y})% }{\partial y}$ (the divergence of the vector field $p(x,y)\textbf{f}$, $\nabla\cdot p(x,y)\textbf{f}$) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system. In addition, if $A$ is an annular region contained in $E$ on which the above condition is satisfied then there exists at most one periodic solution in $A$.

Title Dulac’s criteria DulacsCriteria 2013-03-22 13:37:09 2013-03-22 13:37:09 Daume (40) Daume (40) 5 Daume (40) Theorem msc 34C25