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, $\abla\\cdot\\textbf{f}$ ) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system.

单词	BendixsonsNegativeCriterion
释义	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, $\abla\\cdot\\textbf{f}$ ) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system.
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