On R.A. smith’s autonomous convergence theorem

Abstract

R.A. Smith [19] showed that his higher dimensional generalization of Bendixson’s criterion for the nonexistence of periodic solutions to an autonomous differential equation implies that all bounded trajectories tend to an equilibrium. Here it is shown that a similar conclusion can be drawn from a generalized Dulac criterion. These conditions are also shown to have strong implications for the structure of invariant sets. © 1995 Rocky Mountain Mathematics Consortium.