Algebraic transition matrices in the conley index theory

Transactions of the American Mathematical Society

Volumn 350, Issue 3, 1998, Pages 889-912

  Franzosa, Robert ^a,b   Mischaikow, Konstantin ^a  

^a University of Maine   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Bistable attractor; Conley index; Connection matrix; Transition matrix; Travelling waves

Indexed keywords

EID: 21944447709     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-98-01666-3     Document Type: Article

References (31)

1. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Lecture Note 38 A.M.S. Providence, RI, 1978. MR 80f:58023
2. C. Conley and P. Fife, Critical manifolds, travelling waves, and an example from population genetics, J. Math. Bio. 14 (1982), 159-176. MR 83m:92044
3. M. Eidenschink and K. Mischaikow, A numerical algorithm for finding isolating neighborhoods, in progress.
4. B. Fiedler and K. Mischaikow, Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach, Arch. Rational Mech. Anal. 119 (1992), 145-196. MR 93h:58110
5. R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), 193-213. MR 88a:58121
6. _, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), 781-803. MR 90g:58111
7. _, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561-592. MR 90a:58149
8. R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J.D.B., 71 (1988), 270-287. MR 89c:54078
9. T. Gedeon and K. Mischaikow, Structure of the global attractor of cyclic feedback systems, Dyn. Diff. Eqs. (to appear).
10. H. Kokubu, K. Mischaikow and H. Oka, Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation, Nonlinearity 9 (1996), 1263-1280. CMP 97:03
11. J. Mallet-Paret, Morse decompositions for delay-differential equations, JDE 72 (1988), 270-315. MR 89m:58182
12. C. McCord, The connection map for attractor-repeller pairs, Trans. Amer. Math. Soc. 307 (1988), 195-203. MR 89f:58086
13. C. McCord and K. Mischaikow, Connected simple systems, transition matrices and hele.roclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), 397-422. MR 92k:58193
14. _, On thé global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc. 9 (1996), 1095-1133. MR 9Gm:34139
15. _, Singular and Topological Transition Matrices in thé Conley Index Theory, preprint.
16. K. Mischaikow, Existence of generalized homoclinic orbits for one parameter families of flows, Proc. AMS 103 (1988), 59-68. MR 89k:58147
17. _, Transition Systems, Proc. Roy. Soc. Edin. 112A (1989), 155-175. MR 91b:58215
18. _, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal., to appear.
19. K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal. 24 (1993), 987-1008. MR 94m:92014
20. K. Mischaikow and Y. Morita, Dynamics on the Global Attractor o} a Gradient Flow Arising from the Ginzburg-Landau Equation, JJIAM 11 (1994), 185-202. MR 95h:58082
21. R. Moeckel, Morse decompositions and connection matrices, Erg. Th. & Dyn. Sys. 8 (1988), 227-250. MR 89k:58249
22. K. Mischaikow and M. MRozek, Chaos in the Lorenz Equations: a Computer-Assisted Proof, Bull. A.M.S. (N.S.) 32 (1995), 66-72. MR 95e:58121
23. _, Chaos in the Lorenz Equations: the details, in preparation.
24. _, Singular Index Pairs, preprint.
25. K. Mischaikow and J. Reineck, Travelling waves in predator-prey systems, SIAM J. Math. Anal. 24 (1993), 1179-1214. MR 94m:92014
26. J. Reineck, Connecting orbits in one-parameter families of flows, Erg. Th. & Dyn. Sys. 8 (1988), 359-374. MR 89i:58128
27. _, The connection matrix in Morse-Smale flows, Trans. AMS 322 (1990), 523-544. MR91c:58066
28. I. Rival, Stories about order and the letter N (en), Contemporary Math., 57, A.M.S., Providence, RI, 1986. MR 87k:06006
29. K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987. MR 89d:58025
30. D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. MR 87e:58182
31. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. MR 84d:35002