Phase Asymptotic Semiflows, Poincaréʹs Condition, and the Existence of Stable Limit Cycles

A concept of phase asymptotic semiflow is defined. It is shown that any Lagrange stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. A Lagrange stable solution of aC1differential equation is considered. When the second compound of the variational equation with respect to this solution is uniformly asymptotically stable and the omega limit set contains no equilibrium, then the semiflow is phase asymptotic at the orbit of the solution and the omega limit set is a stable periodic orbit. Analogous results are obtained for discrete semiflows and periodic differential equations.