The Conley index theory: A brief introduction

Konstantin Mischaikow

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 9-19
  • ISSN: 0137-6934

Abstract

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A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.

How to cite

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Mischaikow, Konstantin. "The Conley index theory: A brief introduction." Banach Center Publications 47.1 (1999): 9-19. <http://eudml.org/doc/208946>.

@article{Mischaikow1999,
abstract = {A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.},
author = {Mischaikow, Konstantin},
journal = {Banach Center Publications},
keywords = {isolated invariant set; Conley index; index pair; Morse decomposition},
language = {eng},
number = {1},
pages = {9-19},
title = {The Conley index theory: A brief introduction},
url = {http://eudml.org/doc/208946},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Mischaikow, Konstantin
TI - The Conley index theory: A brief introduction
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 9
EP - 19
AB - A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.
LA - eng
KW - isolated invariant set; Conley index; index pair; Morse decomposition
UR - http://eudml.org/doc/208946
ER -

References

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  1. [1] M. Carbinatto, J. Kwapisz, and K. Mischaikow, Horseshoes and the Conley Index Spectrum, http://www.math.gatech.edu/ mischaik. Zbl0971.37005
  2. [2] M. Carbinatto and K. Mischaikow, Horseshoes and the Conley Index Spectrum (II): The theorem is sharp, http://www.math.gatech.edu/ mischaik. Zbl0954.37009
  3. [3] L. Arnold, C. Jones, K. Mischaikow, and G. Raugel, Dynamical Systems: Montecatini Terme, 1994, R. Johnson, ed. Lect. Notes in Math. 1609, Springer 1995. 
  4. [4] C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Lecture Notes 38, A.M.S. Providence, R.I. 1978. 
  5. [5] C. Conley, A qualitative singular perturbation theorem, in: Global Theory of Dynamical Systems, (eds. Z. Nitecki and C. Robinson), Lecture Notes in Math. 819, Springer, 1980, 65-89. 
  6. [6] C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. AMS 158 (1971), 35-61. Zbl0223.58011
  7. [7] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 867-884. Zbl0201.43301
  8. [8] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, II, Amer. J. Math. 94 (1972), 631-650. Zbl0247.35081
  9. [9] C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367-375. Zbl0284.76080
  10. [10] C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, II, J. Math. Pures et Appl. 54 (1975), 429-444. Zbl0292.76045
  11. [11] M. Degiovanni and M. Mrozek, The Conley index for maps in the absence of compactness, Proc. Roy. Soc. Edin. 123A (1993), 75-94. Zbl0789.58066
  12. [12] A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Erg. Thy. Dyn. Sys. 7 (1987), 93-103. Zbl0633.58040
  13. [13] A. Floer, Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds, Duke Math. J. 53 (1986), 1-32. Zbl0607.58016
  14. [14] J. Franks and D. Richeson, Shift Equivalence and the Conley Index, preprint 1998. 
  15. [15] R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. AMS 298 (1986), 193-213. Zbl0626.58013
  16. [16] R. Franzosa, The connection matrix theory for Morse decompositions, Trans. AMS 311 (1989), 561-592. Zbl0689.58030
  17. [17] R. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. AMS 310, (1988), 781-803. Zbl0708.58021
  18. [18] R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, Journal of Differential Equations 71 (1988), 270-287. Zbl0676.54048
  19. [19] R. Franzosa and K. Mischaikow, Algebraic Transition Matrices in the Conley Index Theory, Trans. AMS 350 (1998), 889-912. Zbl0896.58053
  20. [20] T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, and J. Reineck, The Conley index for fast slow systems I: One dimensional slow variable, J. Dyn. Diff. Eq. (to appear). Zbl0945.34029
  21. [21] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Top. Appl. 65 (1995), 83-96. Zbl0843.54042
  22. [22] T. Kaczynski and M. Mrozek, Stable Index Pairs for Discrete Dynamical Systems, Bul. Can. Math. Soc. (to appear). Zbl0903.54018
  23. [23] C. McCord, Mappings and homological properties in the homology Conley index. Erg. Th. Dyn. Sys. 8* (1988), 175-198. Zbl0633.58019
  24. [24] C. McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. AMS 333 (1992), 379-422. Zbl0763.34028
  25. [25] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Jour. AMS 9 (1996), 1095-1133. Zbl0861.58023
  26. [26] C. McCord, K. Mischaikow, and M. Mrozek. Zeta functions, periodic trajectories, and the Conley index, JDE 121 (1995), 258-292. Zbl0833.34045
  27. [27] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM Math. Anal. 26 (1995), 1199-1224. Zbl0841.35014
  28. [28] K. Mischaikow and Y. Morita, Dynamics on the Global Attractor of a Gradient Flow arising in the Ginzburg-Landau Equation, Jap. J. Ind. Appl. Math. 11 (1994), 185-202. Zbl0807.58029
  29. [29] K. Mischaikow and M. Mrozek, Isolating Neighborhoods and Chaos, Jap. J. Ind. Appl. Math. 12 (1995), 205-236. Zbl0840.58033
  30. [30] K. Mischaikow and M. Mrozek, Chaos in the Lorenz Equations: a Computer Assisted Proof, Bull. AMS 32 (1995), 66-72. Zbl0820.58042
  31. [31] K. Mischaikow and M. Mrozek, Chaos in the Lorenz Equations: a Computer Assisted Proof. Part II, Details, Math. Comp. 67 (1998), 1023-1046. Zbl0913.58038
  32. [32] K. Mischaikow, M. Mrozek, and J. Reineck, Singular Index Pairs, J. Dyn. Diff. Eq. (to appear). Zbl0943.34028
  33. [33] K. Mischaikow, M. Mrozek, J. Reiss, and A. Szymczak, Construction of Symbolic Dynamics from Experimental Time Series, http://www.math.gatech.edu/ mischaik. 
  34. [34] K. Mischaikow, M. Mrozek, A. Szymczak, and J. Reiss, From Time Series to Symbolic Dynamics: An Algebraic Topological Approach, http://www.math.gatech.edu/ mischaik. 
  35. [35] M. Mrozek, Leray functor and cohomological index for discrete dynamical systems, Trans. A. M. S. 318 (1990), 149-178. Zbl0686.58034
  36. [36] M. Mrozek, Topological invariants, multivalued maps, and computer assisted proofs, Comp. Math. 32 (1996), 83-104. Zbl0861.58027
  37. [37] M. Mrozek and K. P. Rybakowski, A cohomological Conley index for discrete dynamical systems, J. D. E. 90 (1991), 143-171. Zbl0721.58040
  38. [38] J. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory and Dynamical Systems 8* (1988), 359-374. Zbl0675.58034
  39. [39] D. Richeson, Connection Matrix Pairs for the Discrete Conley Index, preprint 1997. 
  40. [40] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory and Dynamical Systems 8* (1988), 375-393. Zbl0682.58040
  41. [41] J. W. Robbin and D. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergod. Th. Dyn. Sys. 12 (1992), 153-183. Zbl0737.58033
  42. [42] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag 1987. Zbl0628.58006
  43. [43] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. A.M.S. 291 (1985), 1-41. Zbl0573.58020
  44. [44] A. Szymczak, The Conley Index for Discrete Semidynamical Systems, Top. App. 66 (1995), 215-240. Zbl0840.34043
  45. [45] A. Szymczak, The Conley Index and Symbolic Dynamics, Topology 35 (1996), 287-299. Zbl0855.58023
  46. [46] A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edin., to appear. Zbl0882.54032
  47. [47] T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Pol. Mat. 29 (1947), 279-313. Zbl0032.35001

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