# Bendixson’s negative criterion

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 $\frac{\partial\textbf{X}}{\partial x}+\frac{\partial\textbf{Y}}{\partial y}$ (the divergence of the vector field f, $\nabla\cdot\textbf{f}$) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system.

Title Bendixson’s negative criterion BendixsonsNegativeCriterion 2013-03-22 13:31:01 2013-03-22 13:31:01 Daume (40) Daume (40) 5 Daume (40) Theorem msc 34C25