Displaying similar documents to “Connection matrix theory for discrete dynamical systems”

Connection matrix pairs

David Richeson (1999)

Banach Center Publications

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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...

The Conley index and countable decompositions of invariant sets

Marian Gidea (1999)

Banach Center Publications

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We define a new cohomological index of Conley type associated to any bi-infinite sequence of neighborhoods that satisfies a certain isolation condition. We use this index to study the chaotic dynamics on invariant sets which decompose as countable unions of pairwise disjoint (mod 0) compact pieces.

Foreword, Contents

Konstantin Mischaikow, Marian Mrozek, Piotr Zgliczyński (1999)

Banach Center Publications

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Index filtrations and Morse decompositions for discrete dynamical systems

P. Bartłomiejczyk, Z. Dzedzej (1999)

Annales Polonici Mathematici

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On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.

Connection matrices and transition matrices

Christopher McCord, James Reineck (1999)

Banach Center Publications

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This paper is an introduction to connection and transition matrices in the Conley index theory for flows. Basic definitions and simple examples are discussed.

The Conley index theory: A brief introduction

Konstantin Mischaikow (1999)

Banach Center Publications

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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.

Directional transition matrix

Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka (1999)

Banach Center Publications

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We present a generalization of topological transition matrices introduced in [6].

On foundations of the Conley index theory

Roman Srzednicki (1999)

Banach Center Publications

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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...