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The Conley Index and Rigorous Numerics

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Non Linear Analysis and Boundary Value Problems for Ordinary Differential Equations

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 371))

Abstract

The Conley index is a topological invariant of isolated invariant sets, invented by C. Conley and his students in early 70’s. The theory originated from the paper on existence of smooth isolating blocks by Conley and Easton [5]. The first concept of the index appeared in a conference announcement by Conley [3], in which main ideas of the homotopy and cohomological index were presented. The basic idea of the Conley index is to generalize the Morse index so as to make it defined also in the degenerate case and to give it features similar to the fixed point index (homotopy property, additivity property etc.).

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References

  1. V. Benci, A Generalization of the Conley-Index Theory, Rendiconti Istituto Matematico di Trieste, 18 (1986), 16–39.

    MATH  MathSciNet  Google Scholar 

  2. R. Churchill, Isolated Invariant Sets in Compact Metric Spaces, J. Diff Equ. 12 (1972), 330–352.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Conley, On a Generalization of the Morse Index, Ordinary Differential Equations, 1971 NRL-MRC Conference, Ed. L.Weiss. Academic Press, New York (1972), 133–146.

    Google Scholar 

  4. C. Conley, On a generalization of the Morse index, in Ordinary Differential Equations, 1971 NRL-MRC Conference, ed. L. Weiss, Academic Press, New York (1972), 27–33.

    Google Scholar 

  5. C. Conley, R. Easton, Isolated Invariant Sets and Isolating Blocks, Trans. AMS 158 (1971), 35–61.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Conley, R. Gardner, An Application of the Generalized Morse Index to Travelling Wave Solutions of a Competitive Reaction-Diffusion Model, Indiana Univ. Math. Journ. 33 (1984), 319–343.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Conley, J. Smoller, Shock Waves as Limits of Progressive Wave Solutions of Higher-Order Equations I,II, Comm. Pure Appl. Math. 24(1971), 459–472, 25 (1972), 131–146.

    Google Scholar 

  8. C. Conley, J. Smoller, On the Structure of Magneto-Hydrodynamic Shock Waves, Comm. Pure Appl. Math. 28 (1974), 367–375.

    Article  MathSciNet  Google Scholar 

  9. C. Conley, J. Smoller, On the Structure of Magneto-Hydrodynamic Shock Waves II, J. Math Pures et Appl. 54 (1975), 429–444.

    MathSciNet  Google Scholar 

  10. C. Conley, J. Smoller, Isolated Invariant Sets of Parameterized Systems of Differential Equations in: The Structure of Attractors in Dynamical Systems, (N.G. Markley, J. C. Martin, W. Perizzo, Eds.), Lect. Notes in Math. 668, Springer Verlag, Berlin, 1978.

    Chapter  Google Scholar 

  11. C. Conley, J. Smoller, Topological Techniques in Reaction-Diffusion Equations, Springer Lecture Notes in Biomath. 38 (1980), 473–483.

    Article  MathSciNet  Google Scholar 

  12. C. Conley, J. Smoller, Bifurcations and Stability of Stationary Solutions of the Fritz-Hugh-Nagumo Equations, J. Diff. Equ. 63 (1986), 389–405.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Conley, J. Smoller, Remarks on Travelling Wave Solutions of Non-Linear Diffusion Equations, in: Lect. Notes in Math. 525 (ed. P. Hilton) (1975), 77–89.

    Google Scholar 

  14. C. Conley, E. Zehnder, The Birkhoff-Lewis Fixed Point Theorem and a Conjecture by V.I. Arnold, Inv. Math. 73 (1983), 33–49.

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Conley, E. Zehnder, Morse Type Index Theory for Flows and Periodic Solutions for Hamiltonian Equations, Comm. on Pure and Appl. Math. 37 (1984), 207–253.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Degiovanni, M. Mrozek, The Conley index for maps in absence of compactness, Proc. Royal Soc. Edinburgh 123A (1993), 75–94.

    Article  MATH  MathSciNet  Google Scholar 

  17. J.P Eckmann, H. Koch, P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Memoirs of the American Mathematical Society 47, 1984, 1–121.

    Article  MathSciNet  Google Scholar 

  18. A. Floer, A Refinement of the Conley Index and an Application to the Stability of Hyperbolic Invariant Sets, Ergod. Th. Dyn. Syst. 7 (1987), 93–103.

    MATH  MathSciNet  Google Scholar 

  19. A. Floer, E. Zehnder, The Equivariant Conley Index and Bifurcations of Periodic Solutions of Hamiltonian Systems, Ergod. Th. Dyn. Sys. 8 (1988), 87–97.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Franzosa, Index Filtrations and the Homology Index Braid for Partially Ordered Morse Decomposition, Trans. AMS 298. 1 (1986), 193–213.

    Article  MathSciNet  Google Scholar 

  21. R. Franzosa, The Continuation Theory for Morse Decompositions and Connection Matrices, Trans. AMS. 310. 2 (1988), 781–803.

    Article  MathSciNet  Google Scholar 

  22. R. Franzosa, The Connection Matrix Theory for Morse Decompositions, Trans. AMS 311. 2 (1989), 561–592.

    Article  MathSciNet  Google Scholar 

  23. R. Gardner, Existence of Multidimensional Travelling Wave Solutions of an Initial-Boundary Value Problem, J. Diff. Equ. 61 (1986), 335–379.

    Article  MATH  Google Scholar 

  24. R. Gardner, J. Smoller, The Existence of Periodic Travelling Waves for Singularly Perturbed Predator-Prey Equations via the Conley Index, J. Diff. Equ. 47 (1983), 133–161.

    Article  MATH  MathSciNet  Google Scholar 

  25. B. Hassard and J. Zhang, Existence of a homoclinic orbit of the Lorenz system by precise shooting, SIAM J. Math. Anal., 25, 1994, 179–196.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Hassard, S.P. Hastings, W.C. Troy, J. Zhang, A computer proof that the Lorenz equations have “chaotic” solutions, Appl. Math. Letter,to appear.

    Google Scholar 

  27. S.P. Hastings, W.C. Troy, A shooting approach to the Lorenz equations, Bull. AMS (N.S.) 27, 1992, 298–303.

    Article  MATH  MathSciNet  Google Scholar 

  28. H.Hattori, K.Mischaikow, A Dynamical System Approach to a Phase Transition Problem, J. Diff. Equ.,to appear.

    Google Scholar 

  29. V. Hutson, K. Mischaikow, Travelling Waves for Competing Species, in preparation.

    Google Scholar 

  30. V. Hutson, K. Mischaikow, Travelling Waves for Competitive and Mixed Systems, in preparation.

    Google Scholar 

  31. T. Kaczyríski and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology amp its Appl,accepted.

    Google Scholar 

  32. H. Koch, A. Schenkel, P. Wittwer, Computer assisted proofs in analysis and programming in logic: a case study, Universite de Geneve, preprint.

    Google Scholar 

  33. H.L. Kurland, The Morse Index of an Isolated Invariant Set is a Connected Simple System, J. Diff. Equ. 42 (1981), 234–259.

    Article  MATH  MathSciNet  Google Scholar 

  34. H.L. Kurland, Homotopy Invariants of a Repeller-Attractor Pair: I. The Puppe Sequence of an R-A Pair, J. Diff. Equ. 46(1982), 1–31, II. Continuation of R-A pairs, J. Diff. Equ. 49 (1983), 281–329.

    Google Scholar 

  35. H.L. Kurland, Following Homology in Singularly Perturbed Systems, J. Dif. Equ., 62 (1986) 1–72.

    Article  MATH  MathSciNet  Google Scholar 

  36. O.E. Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull. AMS (N.S.) 6, 1982, 427–434.

    Article  MATH  MathSciNet  Google Scholar 

  37. O.E. Lanford, Computer assisted proofs in analysis, Physica A 124, 1984, 465–470.

    Article  MATH  MathSciNet  Google Scholar 

  38. W.S. Massey, Homology and Cohomology Theory, Marcel Dekker Inc., New York and Basel, 1978.

    MATH  Google Scholar 

  39. Ch. McCord, The Connection Map for Attractor-Repeller Pairs, Trans. AMS 307 (1988), 195–203.

    Article  MATH  MathSciNet  Google Scholar 

  40. Ch. McCord, Mappings and Homological Properties in the Conley Index Theory, Ergod. Th. and Dyn. Sys. 8 (1988), 175–199.

    Article  MATH  MathSciNet  Google Scholar 

  41. Ch. McCord, On the Hopf Index and the Conley Index, Trans. AMS, 313 (1989), 853–860.

    Article  MATH  MathSciNet  Google Scholar 

  42. K. Mischaikow, Existence of Generalized Homoclinic Orbits for One Parameter Families of Flows, Proc. AMS 103 (1988), 59–69.

    Article  MATH  MathSciNet  Google Scholar 

  43. K. Mischaikow, Transition Matrices, Proc. Roy. Soc. Edinburgh 112A (1989), 155–175.

    Article  MATH  MathSciNet  Google Scholar 

  44. K. Mischaikow, Homoclinic Orbits in Hamiltonian Systems and Heteroclinic Orbits in Gradient and Gradient-Like Systems, J. Diff. Equ. 81 (1989), 167–213.

    Article  MATH  MathSciNet  Google Scholar 

  45. K. Mischaikow, Dynamic Phase Transitions: a Connection Matrix Approach, IMA Preprint Series no 584.

    Google Scholar 

  46. K. Mischaikow, Travelling Waves for a Cooperative and a Competitive-Cooperative System, preprint CDSNS90–25.

    Google Scholar 

  47. K. Mischaikow, A C-graph Approach for Studying the Dynamics of a System of Parabolic Equations, preprint.

    Google Scholar 

  48. K. Mischaikow, On the Existence of Connecting Orbits for Scalar Delay Equations, preprint CDSNS90–35.

    Google Scholar 

  49. K. Mischaikow and Y. Morita, Dynamics on the Global Attractor of a Gradient Flow Arising from the Ginzburg-Landau Equation, JJIAM, 11 (1994) 185–202.

    MATH  MathSciNet  Google Scholar 

  50. K. Mischaikow and M. Mrozek, Isolating neighborhoods and Chaos, Jap. J. Ind. Appl. Math., 12, 1995, 205–236.

    Article  MATH  MathSciNet  Google Scholar 

  51. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.), 33 (1995), 66–72.

    Article  MathSciNet  Google Scholar 

  52. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Part II: details, preprint.

    Google Scholar 

  53. K. Mischaikow, M. Mrozek and A. Szymczak, Chaos in Lorenz equations: a computer assisted proof, Part III: the classic case, in preparation.

    Google Scholar 

  54. K. Mischaikow, G. Wolkowicz, Predator Prey with Group Defense, a Connection Matrix Approach, Nonlin. Anal. Th. Meth. and Appl., 14 (1990), 955–969.

    Article  MATH  MathSciNet  Google Scholar 

  55. J.T. Montgomery, Cohomology of Isolated Invariant Sets under Perturbation, J. Diff. Equ. 13 (1973), 257–299.

    Article  MATH  Google Scholar 

  56. M. Mrozek, Index pairs and the Fixed Point Index for Semidynamical Systems with Discrete Time, Fund. Mathematicae, Vol. 133 (1989), 177–192.

    MathSciNet  Google Scholar 

  57. M. Mrozek, The Cohomological Index of Conley Type for Multi-Valued Admissible Flows, J. Diff. Equ. 84. 1 (1990), 15–51.

    Article  MathSciNet  Google Scholar 

  58. M. Mrozek, Leray Functor and the Cohomological Conley Index for Discrete Dynamical Systems, Transactions of the American Mathematical Society 318, 1990, 149–178.

    Article  MATH  MathSciNet  Google Scholar 

  59. M. Mrozek, Open Index Pairs and Rationality of Zeta Functions, Ergod. Th. Dyn. Syst. 10 (1990), 555–564.

    MATH  MathSciNet  Google Scholar 

  60. M. Mrozek, The Morse Equation in Conley’s Index Theory for Homeomorphisms, Topology and its Appl. 38 (1991), 45–60.

    Article  MATH  MathSciNet  Google Scholar 

  61. M. Mrozek, The Conley Index on Compact ANR’s is of Finite Type, Results in Math. 18 (1990), 306–313.

    Article  MATH  MathSciNet  Google Scholar 

  62. M. Mrozek, Shape Index and Other Indices of Conley Type for Continuous Maps on Locally Compact Metric Spaces, Fundamenta Mathematicae, 145 (1994), 15–37.

    MATH  MathSciNet  Google Scholar 

  63. M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Computers amp Mathematics,accepted.

    Google Scholar 

  64. M. Mrozek, An algorithmic approach to the Conley index theory, in preparation.

    Google Scholar 

  65. M. Mrozek, J. Reineck and R. Srzednicki, The Conley index over a base, in preparation

    Google Scholar 

  66. M. Mrozek, J. Reineck and R. Srzednicki, The Conley index over a circle, in preparation.

    Google Scholar 

  67. M. Mrozek, K.P. Rybakowski, Cohomological Conley Index for Continuous Maps on Metric Spaces, J. Diff. Equ. 90. 1 (1991), 143–171.

    Article  MathSciNet  Google Scholar 

  68. M. Mrozek, K.P. Rybakowski, Bounded Solutions of Semilinear Parabolic Equations, Proc. Roy. Soc. Edinburgh, 117A (1991), 305–315.

    Article  MATH  MathSciNet  Google Scholar 

  69. M. Mrozek, K.P. Rybakowski, Conley Index of Difference Equations Approximating Differential Equations, J. of Dynamics and Differential Equations 4 (1992), 57–63.

    Article  MATH  MathSciNet  Google Scholar 

  70. A. Neumaier, Th. Rage, Rigorous chaos verification in discrete dynamical systems, Physica D 67, 1993, 327–346.

    Article  MATH  MathSciNet  Google Scholar 

  71. Th. Rage, A. Neumaier, Ch. Schlier, Rigorous verification of chaos in a molecular model, Physical Rev. E 50, 1994, 2682–2688.

    Article  Google Scholar 

  72. J. Reineck, The Connection Matrix and the Classification of Flows Arising from Ecological Models, Ph. D. Thesis, University of Wisconsin, Madison, 1985.

    Google Scholar 

  73. J. Reineck, Travelling Wave Solutions to a Gradient System, Trans. AMS 307 (1988), 535–544.

    Article  MATH  MathSciNet  Google Scholar 

  74. J. Reineck, Connecting Orbits in One-Parameter Families of Flows, Ergod. Th. Dyn. Sys. 8 (1988), 359–374.

    Article  MATH  MathSciNet  Google Scholar 

  75. J. Reineck, The Connection Matrix in Morse-Smale Flows I,II, Trans. AMS,to appear.

    Google Scholar 

  76. J. Reineck, Continuation to Gradient Flows and the Conley Index, Ergod. Th. Dyn. Sys.,to appear.

    Google Scholar 

  77. J.W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Erg. Th. and Dynam. Sys. 8 (1988), 375–393.

    Article  MATH  MathSciNet  Google Scholar 

  78. K.P. Rybakowski, On the Homotopy Index for Infinite Dimensional Semiflows, Trans. AMS 269 (1982), 351–382.

    Article  MATH  MathSciNet  Google Scholar 

  79. K.P. Rybakowski, The Morse Index, Repeller-Attractor Pairs and the Connection Index for Semiflows on Noncompact Spaces, J. Dif. Equ. 47 (1983), 66–98.

    Article  MATH  MathSciNet  Google Scholar 

  80. K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin Heidelberg 1987.

    Book  MATH  Google Scholar 

  81. K.P. Rybakowski, Nontrivial Solutions of Elliptic Boundary Value Problems with Resonance at Zero, Annali Mat. Pura ed Appl. (IV),CXXXIX(1985), 237–278.

    Google Scholar 

  82. K.P. Rybakowski, An Index Product Formula for the Study of Elliptic Resonance Problems, J. Diff. Equ. 56 (1985), 408–425.

    Article  MATH  MathSciNet  Google Scholar 

  83. K.P. Rybakowski, On a Relation between the Brouwer Degree and the Conley Index for Gradient Flows, Bull. Soc. Math. Belg. (B), 37 (II) (1985), 87–96.

    MATH  MathSciNet  Google Scholar 

  84. K.P. Rybakowski, A Homotopy Index Continuation Method and Periodic Solutions of Second Order Gradient Systems, J. Diff. Equ. 65 (1986), 203–212.

    Article  MATH  MathSciNet  Google Scholar 

  85. K.P. Rybakowski, Some Recent Results in the Homotopy Index Theory in Infinite Dimensions, Rend. Ist. Matern. di Trieste XVIII (1986), 83–92.

    MathSciNet  Google Scholar 

  86. K.P. Rybakowski, On Critical Groups and the Homotopy Index in Morse Theory on Hilbert Manifolds, Rend. Ist. Matern. Univ. di Trieste XVIII (1986), 163–176.

    MathSciNet  Google Scholar 

  87. K.P. Rybakowski, E. Zehnder, A Morse Equation in Conley’s Index Theory for Semiflows on Metric Spaces, Ergod. Th. Dyn. Syst. 5 (1985), 123–143.

    MATH  MathSciNet  Google Scholar 

  88. D. Salamon, Connected Simple Systems and the Conley Index of Isolated Invariant Sets, Trans. AMS 291 (1985), 1–41.

    Article  MATH  MathSciNet  Google Scholar 

  89. D. Salamon, E. Zehnder, Flows on Vector Bundles and Hyperbolic Sets, Trans. AMS 306 (1988), 623–649.

    Article  MATH  MathSciNet  Google Scholar 

  90. E.H. Spanier, Algebraic Topology, McGraw-Hill Book Company, New York, 1966.

    MATH  Google Scholar 

  91. A. Szymczak, The Conley index for discrete semidynamical systems, Topol. Appl.,submitted.

    Google Scholar 

  92. A. Szymczak, The Conley index for decompositions of isolated invariant sets, Fund. Math.,to appear.

    Google Scholar 

  93. W.C. Troy, The Existence of Steady State Solutions of the Kuramoto- Shivashinsky Equation, J. Diff. Equ. 82 (1989), 269–313.

    Article  MATH  MathSciNet  Google Scholar 

  94. J.R. Ward, Jr., Conley Index and Non-Autonomous Ordinary Differential Equations, Results in Math. 14 (1988), 191–210.

    Article  MATH  Google Scholar 

  95. J.R. Ward, Jr., Averaging, Homotopy, and Bounded Solutions of Ordinary Differential Equations, Diff Int. Equ. 3.6 (1990), 1093–1100.

    Google Scholar 

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  1. Jagiellonian University, Cracow, Poland

    M. Mrozek

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  1. M. Mrozek
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  1. University of Udine, Italy

    F. Zanolin

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Mrozek, M. (1996). The Conley Index and Rigorous Numerics. In: Zanolin, F. (eds) Non Linear Analysis and Boundary Value Problems for Ordinary Differential Equations. International Centre for Mechanical Sciences, vol 371. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2680-6_5

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  • DOI: https://doi.org/10.1007/978-3-7091-2680-6_5

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