We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions.

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

In this article, we study the forward dynamical behaviour of nonautonomous lattice systems. We first construct a family of sets {Aϵ(σ)}σ∈Σ in the arbitrary small neighbourhood of a global attractor...

On the forward dynamical behaviour of nonautonomous lattice dynamical systems

On relative category and Morse decompositions for infinite-dimensional dynamical systems

In this paper, we establish a theory of compactly generated shape index for local semiflows on complete metric spaces via very ordinary shape index pairs. The main advantage is that the quotient space $N/E$ is not necessary to be metrizable for the shape index pair $(N,E)$. In the new shape theory, we can calculate the shape index of a compact isolated invariant set $K$ in any closed subset that contains a local unstable manifold of $K$, and define the shape cohomology index of $K$ to develop the Morse equations. This provides more effective ways to calculate shape indices and Morse equations theoretically and specifically for infinite dimensional systems, without particular requirements on the index pairs or the unstable manifolds.

Compactly Generated Shape Index for Infinite-dimensional Local Dynamical Systems on Complete Metric Spaces

Compactly generated shape index theory and its application to a retarded nonautonomous parabolic equation

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.

Jintao Wang

Papers

Attractors of Local Semiflows on Topological Spaces