Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations, some superfluous hypotheses are abandoned and some conclusions are proved in a simpler way.
In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system in Ω, in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....
This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.
We introduce connection graphs for both continuous and discrete dynamical systems. We prove the existence of connection graphs for Morse decompositions of isolated invariant sets.
This paper is an introduction to connection and transition matrices in the Conley index theory for flows. Basic definitions and simple examples are discussed.
We discuss the ideas of Morse decompositions and index filtrations for isolated invariant sets for both single-valued and multi-valued maps. We introduce the definition of connection matrix pairs and present the theorem of their existence. Connection matrix pair theory for multi-valued maps is used to show that connection matrix pairs obey the continuation property. We conclude by addressing applications to numerical analysis. This paper is primarily an overview of the papers [R1] and [R2].
In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems. The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse sets.
We prove that the index defined by Szymczak in [9] has an additivity property. Moreover we give an abstract theorem for extending coproducts from an initial category to the Szymczak category, which provides a setting for the proof of additivity.