×

On Bendixson’s criterion. (English) Zbl 0786.34033

Summary: For autonomous differential equations in \(\mathbb{R}^ n\) criteria are developed which preclude the existence of invariant closed curves such as periodic or homoclinic trajectories. The technique is based on the study of functionals on 2-surfaces. Results generalize to higher dimensions a criterion of Bendixson for the non-existence of nonconstant periodic solutions in the case \(n=2\). As an example, an application to the Lorenz system in \(\mathbb{R}^ 3\) is given.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI