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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebraic transition matrices
in the Conley index theory


Authors: Robert Franzosa and Konstantin Mischaikow
Journal: Trans. Amer. Math. Soc. 350 (1998), 889-912
MSC (1991): Primary 58F35; Secondary 58F30, 35K57
MathSciNet review: 1360223
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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.


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Additional Information

Robert Franzosa
Affiliation: Department of Mathematics, University of Maine, Orono, Maine 04469
Email: franzosa@gauss.umemat.maine.edu

Konstantin Mischaikow
Affiliation: Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: mischail@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01666-3
PII: S 0002-9947(98)01666-3
Keywords: Conley index, connection matrix, transition matrix, bistable attractor, travelling waves
Received by editor(s): January 3, 1995
Received by editor(s) in revised form: October 4, 1995
Additional Notes: Research was supported in part by NSF Grant DMS-9101412. Part of this paper was written while the second author was visiting the Instituto de Ciencias Mathematicas de São Carlo of the Universidade de São Paulo. He would like to take this opportunity to thank the members of the institute for their warm hospitality.
Article copyright: © Copyright 1998 American Mathematical Society