### 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 language | English (US) |
---|---|

Pages (from-to) | 889-912 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 350 |

Issue number | 3 |

State | Published - Dec 1 1998 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

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

### Cite this

*Transactions of the American Mathematical Society*,

*350*(3), 889-912.

}

*Transactions of the American Mathematical Society*, vol. 350, no. 3, pp. 889-912.

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

Research output: Contribution to journal › Article

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 -