Attractors of Local Semiflows on Topological Spaces

Abstract

Abstract. In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.

On Relative Category and Morse Decompositions for Infinite-Dimensional Dynamical Systems

“…The first conclusion is similar to the results of Lemmas 4.6 and 4.7 in [20]. The proof is just a slight modification of the proofs of the two lemmas, by using the framework of topological dynamical systems ( [12]). We thus omit it.…”

“…Ω(A) is called the region of attraction (or attraction basin) of A. One can easily verify that Ω(A) is open; moreover, A attracts each compact subset of Ω(A), see [12]. In the case when Ω(A) = X, we simply call A the global attractor of Φ.…”

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

“…In this section we collect some necessary notions and results in the theory of topology and dynamical systems on Hausdorff spaces (see [13]). The reader is supposed to be familiar with basic knowledge of algebraic topology.…”

“…Here we use the attractor theory of topological spaces stated in [13], which is a generalisation of the attractor theory in metric spaces [16,26].…”

“…Thirdly, due to the lack of metric on the quotient space, the quotient flow used in this paper is merely defined on normal Hausdorff spaces. Therefore, the relevant results for quotient flows are of new versions and all based on the dynamical systems on topological spaces (see [13]). Moreover, in order to give the Morse equations with respect to H-shape index, we develop somehow cohomological theory and the exactness property for compactly generated shape in this paper.…”