## 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
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