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Reconstructing the global dynamics of attractors via the Conley index

Christopher McCord (1999)

Banach Center Publications

Given an unknown attractor 𝓐 in a continuous dynamical system, how can we discover the topology and dynamics of 𝓐? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from 𝓐 onto a model system 𝓜 whose topology and dynamics are known. The complexity of 𝓜 then provides a lower bound for the complexity of 𝓐. The Conley index can be used to construct a simplicial model and a surjective semi-conjugacy for a large...

Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

Marian Mrozek (1994)

Fundamenta Mathematicae

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

Shape index in metric spaces

Francisco R. Ruiz del Portal, José M. Salazar (2003)

Fundamenta Mathematicae

We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally...

Simple connection matrices

Piotr Bartłomiejczyk (2007)

Annales Polonici Mathematici

We introduce simple connection matrices. We prove the existence of simple connection matrices for filtered differential vector spaces and Morse decompositions of compact metric spaces.

The Conley index and countable decompositions of invariant sets

Marian Gidea (1999)

Banach Center Publications

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.

The Conley index for flows preserving generalized symmetries

Artur Pruszko (1999)

Banach Center Publications

Topological spaces with generalized symmetries are defined and extensions of the Conley index of a compact isolated invariant set of the flow preserving the structures introduced are proposed. One of the two new indexes is constructed with no additional assumption on the examined set in terms of symmetry invariance.

The Conley index theory: A brief introduction

Konstantin Mischaikow (1999)

Banach Center Publications

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.

The omega limit sets of subsets in a metric space

Changming Ding (2005)

Czechoslovak Mathematical Journal

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the ω -limit set ω ( Y ) of Y is the limit point of the sequence { ( C l Y ) · [ i , ∞ ) } i = 1 ∞ in 2 X and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.

Topology and dynamics of unstable attractors

M. A. Morón, J. J. Sánchez-Gabites, J. M. R. Sanjurjo (2007)

Fundamenta Mathematicae

This article aims to explore the theory of unstable attractors with topological tools. A short topological analysis of the isolating blocks for unstable attractors with no external explosions leads quickly to sharp results about their shapes and some hints on how Conley's index is related to stability. Then the setting is specialized to the case of flows in ℝⁿ, where unstable attractors are seen to be dynamically complex since they must have external explosions.

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