On Bendixson’s criterion. (English) Zbl 0786.34033

Li, Yi; Muldowney, James S.

J. Differ. Equations 106, No. 1, 27-39 (1993).

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

Keywords:

autonomous differential equations; existence of invariant closed curves; periodic or homoclinic trajectories; criterion of Bendixson; periodic solutions; Lorenz system