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The discrete Conley index for non-invariant sets and detection of chaos.

机译：用于非不变集和混沌检测的离散Conley索引。

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The relationship between dynamical systems and topology dates from the work of Poincare. Since then, algebraic topology has provided significant techniques for the investigation of non-linear systems. An important contribution to this area has been made by Charles Conley, whose index is a topological generalization of the Morse index.;The first chapter of this dissertation briefly surveys the development of the Conley index theory from its origins to the more recently introduced discrete Conley index and its latest applications in detecting periodic orbits and chaos.;In the second chapter we present a cohomological Conley index for discrete-time dynamical systems in non-locally compact spaces, similar to the one constructed by Degiovanni and Mrozek. This is a discrete version of the continuous Conley index introduced by Benci and generalizes similar indices developed by Mrozek and Rybakowski. A few applications in studying hyperbolic and pseudo-hyperbolic operators are mentioned. However, the main purpose of this chapter is to produce the tools and techniques for the next chapter.;The third chapter constitutes the core of this dissertation. We define a new cohomological Conley index associated to some isolating neighborhood sequences, extending previous constructions of Easton, Srzednicki and of the author. Our construction uses a very general Leray functor on the category of graded directed systems of modules and homomorphisms, extending a similar construction of Mrozek. This index satisfies the fundamental Wazewski property, summation property and continuation property, common to all Conley index type of invariants. Under special conditions, it turns out to be an index of non-invariant sets, related to the Conley index for decompositions of isolated invariant sets considered by Szymczak. We apply our index to detect periodic orbits, extending results of Mrozek and Srzednicki and to detect chaos through a semi-conjugacy to a shift space, extending results of Mischaikow and Mrozek, Mischaikow, Kwapisz and Carbinatto, Srzednicki, Szymczak. We illustrate our methods on several types of horseshoes. In the end, we provide a unified Conley index theory approach to describe the chaotic behavior of some Julia sets.
机译：动力系统和拓扑之间的关系可以追溯到Poincare的工作。从那时起，代数拓扑已为研究非线性系统提供了重要的技术。查尔斯·康利（Charles Conley）在这一领域做出了重要贡献，其索引是莫尔斯（Morse）索引的拓扑概括。本论文的第一章简要概述了康利索引理论从其起源到最近引入的离散康利的发展。该指数及其在检测周期性轨道和混沌中的最新应用。在第二章中，我们介绍了非局部紧凑空间中离散动力系统的同调Conley指数，类似于Degiovanni和Mrozek构造的。这是Benci引入的连续Conley指数的离散版本，并概括了Mrozek和Rybakowski开发的类似指数。提到了一些在研究双曲和伪双曲算子中的应用。但是，本章的主要目的是为下一章提供工具和技术。第三章构成了本文的核心。我们定义了与某些孤立的邻域序列相关的新的同调Conley索引，扩展了Easton，Srzednicki和作者的先前构造。我们的构造使用了非常普通的Leray函子，用于模数和同态的有级定向系统类别，扩展了Mrozek的类似构造。该索引满足所有Conley索引类型不变式共有的基本Wazewski属性，求和属性和延续属性。在特殊条件下，它是非不变集的索引，与Szymczak所考虑的孤立不变集分解的Conley索引有关。我们应用我们的索引来检测周期性轨道，扩展Mrozek和Srzednicki的结果，并通过半共轭检测到移位空间的混沌，将Mischaikow和Mrozek，Mischaikow，Kwapisz和Carbinatto，Srzednicki，Szymczak的结果扩展。我们举例说明了几种类型的马蹄铁的方法。最后，我们提供了统一的Conley指数理论方法来描述一些Julia集的混沌行为。

著录项

  • 作者

    Gidea, Marian.;

  • 作者单位

    State University of New York at Buffalo.;

  • 授予单位 State University of New York at Buffalo.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 1997
  • 页码 103 p.
  • 总页数 103
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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