Algebraic transition matrices in the conley index theory

Robert Franzosa, Konstantin Mischaikow

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

Original languageEnglish (US)
Pages (from-to)889-912
Number of pages24
JournalTransactions of the American Mathematical Society
Volume350
Issue number3
StatePublished - Dec 1 1998

Fingerprint

Conley Index
Index Theory
Transition Matrix
Partial Order
Morse Decomposition
Connection Matrix
Decomposition
Global Bifurcation
Conjugation
Existence Theorem
Triangular
Isomorphism
Series
Zero
Concepts
Form

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Bistable attractor
  • Conley index
  • Connection matrix
  • Transition matrix
  • Travelling waves

Cite this

@article{c3f5223bcda446edb6ad6cad9793b78b,
title = "Algebraic transition matrices in the conley index theory",
abstract = "We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.",
keywords = "Bistable attractor, Conley index, Connection matrix, Transition matrix, Travelling waves",
author = "Robert Franzosa and Konstantin Mischaikow",
year = "1998",
month = "12",
day = "1",
language = "English (US)",
volume = "350",
pages = "889--912",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "3",

}

Algebraic transition matrices in the conley index theory. / Franzosa, Robert; Mischaikow, Konstantin.

In: Transactions of the American Mathematical Society, Vol. 350, No. 3, 01.12.1998, p. 889-912.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Algebraic transition matrices in the conley index theory

AU - Franzosa, Robert

AU - Mischaikow, Konstantin

PY - 1998/12/1

Y1 - 1998/12/1

N2 - We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

AB - We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

KW - Bistable attractor

KW - Conley index

KW - Connection matrix

KW - Transition matrix

KW - Travelling waves

UR - http://www.scopus.com/inward/record.url?scp=21944447709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21944447709&partnerID=8YFLogxK

M3 - Article

VL - 350

SP - 889

EP - 912

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -