Transition matrix theory
HTML articles powered by AMS MathViewer
- by Robert Franzosa and Ewerton R. Vieira PDF
- Trans. Amer. Math. Soc. 369 (2017), 7737-7764 Request permission
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
- 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, DOI 10.1007/BF01832842
- 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
- Bernold Fiedler and Konstantin Mischaikow, Dynamics of bifurcations for variational problems with $\textrm {O}(3)$ equivariance: a Conley index approach, Arch. Rational Mech. Anal. 119 (1992), no. 2, 145–196. MR 1176363, DOI 10.1007/BF00375120
- 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, DOI 10.1090/S0002-9947-1986-0857439-7
- 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.
- 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, DOI 10.1090/S0002-9947-98-01666-3
- Robert D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), no. 2, 781–803. MR 973177, DOI 10.1090/S0002-9947-1988-0973177-6
- Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. MR 978368, DOI 10.1090/S0002-9947-1989-0978368-7
- 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, DOI 10.1023/A:1021961819853
- Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka, Directional transition matrix, Conley index theory (Warsaw, 1997) Banach Center Publ., vol. 47, Polish Acad. Sci. Inst. Math., Warsaw, 1999, pp. 133–144. MR 1692367
- 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, DOI 10.1090/S0002-9947-1992-1059711-X
- 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, DOI 10.1307/mmj/1029005236
- Konstantin Mischaikow and Marian Mrozek, Conley index, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 393–460. MR 1901060, DOI 10.1016/S1874-575X(02)80030-3
- 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, DOI 10.1017/S0143385700009482
- 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, DOI 10.1090/S0002-9947-1985-0797044-3
Additional Information
- Robert Franzosa
- Affiliation: Departament of Mathematics and Statistics, University of Maine, Orono, Maine 04469
- MR Author ID: 68895
- 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
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3695843
Dedicated: Dedicated to the memory of James Francis Reineck