The x-limit sets of a flow and periodic orbits

In this paper we discuss the x-limit sets of a flow using the Conley theory, chain recurrence and Morse decompositions. Our results generalize and improve the related result in [Schropp J. A reduction principle for x-limit sets. Z Angew Math Meth 1996;76(6):349–56], and we also show how they can be used as a basis for some new criteria for the existence of periodic orbits.