Transition matrix theory

Franzosa, Robert; Vieira, Ewerton

doi:10.1090/tran/6915

Transition matrix theory
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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