Displaying similar documents to “Dynamical systems and shapes.”

The Conley index theory: A brief introduction

Konstantin Mischaikow (1999)

Banach Center Publications

Similarity:

A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.

On foundations of the Conley index theory

Roman Srzednicki (1999)

Banach Center Publications

Similarity:

The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the...

On a generalization of the Conley index

Marian Mrozek, James Reineck, Roman Srzednicki (1999)

Banach Center Publications

Similarity:

In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain...

The Conley index in Hilbert spaces and its applications

K. Gęba, M. Izydorek, A. Pruszko (1999)

Studia Mathematica

Similarity:

We present a generalization of the classical Conley index defined for flows on locally compact spaces to flows on an infinite-dimensional real Hilbert space H generated by vector fields of the form f: H → H, f(x) = Lx + K(x), where L: H → H is a bounded linear operator satisfying some technical assumptions and K is a completely continuous perturbation. Simple examples are presented to show how this new invariant can be applied in searching critical points of strongly indefinite functionals...