Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Transition matrix theory


Authors: Robert Franzosa and Ewerton R. Vieira
Journal: Trans. Amer. Math. Soc. 369 (2017), 7737-7764
MSC (2010): Primary 37B30, 37D15; Secondary 70K70, 70K50, 55T05
DOI: https://doi.org/10.1090/tran/6915
Published electronically: August 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as verification that each of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore we address how applications of the previous transition matrices to the Conley index theory carry over to the (generalized) transition matrix.


References [Enhancements On Off] (What's this?)

  • [1] C. Conley and P. Fife, Critical manifolds, travelling waves, and an example from population genetics, J. Math. Biol. 14 (1982), no. 2, 159-176. MR 667796, https://doi.org/10.1007/BF01832842
  • [2] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
  • [3] Bernold Fiedler and Konstantin Mischaikow, Dynamics of bifurcations for variational problems with $ {\rm O}(3)$ equivariance: a Conley index approach, Arch. Rational Mech. Anal. 119 (1992), no. 2, 145-196. MR 1176363, https://doi.org/10.1007/BF00375120
  • [4] Robert Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193-213. MR 857439, https://doi.org/10.2307/2000615
  • [5] Robert Franzosa, Ketty A. de Rezende, and Ewerton R. Vieira, Generalized topological transition matrix, Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 183-212, DOI:10.12775/TMNA.2016.046.
  • [6] Robert Franzosa and Konstantin Mischaikow, Algebraic transition matrices in the Conley index theory, Trans. Amer. Math. Soc. 350 (1998), no. 3, 889-912. MR 1360223, https://doi.org/10.1090/S0002-9947-98-01666-3
  • [7] Robert D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), no. 2, 781-803. MR 973177, https://doi.org/10.2307/2000991
  • [8] Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561-592. MR 978368, https://doi.org/10.2307/2001142
  • [9] Tomáš Gedeon, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, and James F. Reineck, The Conley index for fast-slow systems. I. One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999), no. 3, 427-470. MR 1693854, https://doi.org/10.1023/A:1021961819853
  • [10] Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka, Directional transition matrix, Conley index theory (Warsaw, 1997) Banach Center Publ., vol. 47, Polish Acad. Sci., Warsaw, 1999, pp. 133-144. MR 1692367
  • [11] Christopher McCord and Konstantin Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), no. 1, 397-422. MR 1059711, https://doi.org/10.2307/2154116
  • [12] Christopher K. McCord and Konstantin Mischaikow, Equivalence of topological and singular transition matrices in the Conley index theory, Michigan Math. J. 42 (1995), no. 2, 387-414. MR 1342498, https://doi.org/10.1307/mmj/1029005236
  • [13] Konstantin Mischaikow and Marian Mrozek, Conley index, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 393-460. MR 1901060, https://doi.org/10.1016/S1874-575X(02)80030-3
  • [14] James F. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory Dynam. Systems 8$ ^*$ (1988), no. Charles Conley Memorial Issue, 359-374. MR 967644, https://doi.org/10.1017/S0143385700009482
  • [15] Dietmar Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1-41. MR 797044, https://doi.org/10.2307/1999893

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37B30, 37D15, 70K70, 70K50, 55T05

Retrieve articles in all journals with MSC (2010): 37B30, 37D15, 70K70, 70K50, 55T05


Additional Information

Robert Franzosa
Affiliation: Departament of Mathematics and Statistics, University of Maine, Orono, Maine 04469
Email: robert$_$franzosa@umit.maine.edu

Ewerton R. Vieira
Affiliation: Instituto de Matemática e Estátistica, Universidade Federal de Goiás, Goiânia, Goiás, Brazil
Email: ewerton@ufg.br

DOI: https://doi.org/10.1090/tran/6915
Received by editor(s): February 26, 2015
Received by editor(s) in revised form: November 11, 2015
Published electronically: August 15, 2017
Additional Notes: The second author was partially supported by FAPEG under grant 2012/10 26 7000 803 and FAPESP under grant 2010/19230-8
Dedicated: Dedicated to the memory of James Francis Reineck
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society