On Bendixson′s Criterion

For autonomous differential equations in 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 nonexistence of nonconstant periodic solutions in the case n = 2. As an example, an application to the Lorenz system in 3 is given