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ArticleOriginal scientific text

Title

Connection matrix pairs

Authors 1

Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730, U.S.A.

Abstract

We discuss the ideas of Morse decompositions and index filtrations for isolated invariant sets for both single-valued and multi-valued maps. We introduce the definition of connection matrix pairs and present the theorem of their existence. Connection matrix pair theory for multi-valued maps is used to show that connection matrix pairs obey the continuation property. We conclude by addressing applications to numerical analysis. This paper is primarily an overview of the papers [R1] and [R2].

Keywords

ENG
Conley index, multi-valued maps, connection matrix pairs, Morse decomposition, index filtrations

Bibliography

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  2. [Fr] J. Franks, Homology and Dynamical Systems, CBMS Reg. Conf. Ser. in Math. 49, AMS, Providence, 1982.
  3. [F1] R. Franzosa, The Connection Matrix Theory for Morse Decompositions, Trans. AMS 311 (1989), 561-592.
  4. [F2] R. Franzosa, The Continuation Theory For Morse Decompositions and Connection Matrices, Trans. AMS 310 (1988), 781-803.
  5. [FM] R. Franzosa and K. Mischaikow, The Connection Matrix Theory for Semiflows on (Not Necessarily Locally Compact) Metric Spaces, J. Differential Equations 71 (1988), 270-287.
  6. [G1] L. Górniewicz, Homological Methods in Fixed Point Theory of Multi-valued Maps, Dissertationes Mathematicae 129 (1976).
  7. [G2] L. Górniewicz, Topological Degree of Morphisms and its Applications to Differential Inclusions, Raccolta di Seminari del Dipartimento di Matematica dell'Università degli Studi della Calabria 5 (1983).
  8. [KM1] T. Kaczynski and M. Mrozek, Conley Index for Discrete Multi-valued Dynamical Systems, Top. App. 65 (1995), 83-96.
  9. [KM2] T. Kaczynski and M. Mrozek, Stable Index Pairs for Discrete Dynamical Systems, preprint, 1994.
  10. [R1] D. Richeson, Connection Matrix Pairs for the Discrete Conley Index, preprint.
  11. [R2] D. Richeson, Morse Decompositions and Connection Matrix Pairs For Multi-valued Maps, preprint.
Banach Center Publications
1999/47/1
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Pages:
219-232
Main language of publication
English
Published
1999
Exact and natural sciences