Some properties of invariant sets of a flow

In this paper, some new properties of the regions of weak attraction, the regions of attraction, and the Ω-limit sets are proved. A necessary and sufficient condition for which a component of an attractor is also an attractor is given. Some results of Conley are generalized (cf. Conley C. Isolated invariant sets and the morse index. CBMS regional conference series in mathematics, vol. 38. Providence, RI: AMS, 1978; Conley C. Ergodic Theory Dyn Syst 1988;8:11–26).