A remark on the flow near a compact invariant set

On the attraction and stability of sets with respect to solutions of ordinary differential equations

“…The following theorem is a corollary to Theorem 4. It is an analogue of a theorem of Ura and Kimura proved for dynamical systems (see [21], cited in [6], [13]).…”

Topological characterizations of recurrence, Poisson stability, and isometric property of flows on surfaces

“…By results in [15,16,17] (cf. [3,Theorem 1.6] or [7,Theorem]), since {x} is neither positively asymptotically stable nor negative asymptotically stable with respect to w, the singular point x is an ω-limit or α-limit set of a non-singular point with respect to w. Poisson stability implies that there is a non-singular point y with respect to w with ω w (y) = α w (y) = x. Then y is not recurrent, which contradicts that w is recurrent.…”

Fixed points, bounded orbits and attractors of planar flows

“…The dynamical structure near isolating invariant sets shall play an important role in this paper and it is described by the Theorem 1.2 (Ura-Kimura-Egawa [20,8]) Let M be a locally compact separable metric space and ϕ a flow on M . Suppose K = M is a non-empty isolated invariant compactum.…”