ArticleOriginal scientific text

Title

From the theorem of Ważewski to computer assisted proofs in dynamics

Authors 1

Affiliations

  1. Instytut Informatyki, Uniwersytet Jagielloński, Nawojki 11, 30-072 Kraków, Poland

Bibliography

  1. O. Aberth, Precise Numerical Analysis, William C. Brown Publishers, Dubuque, Iowa, 1988.
  2. V. Benci, A Generalization of the Conley-Index Theory, Rend. Istit. Mat. Trieste 18 (1986), 16-39.
  3. R. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330-352.
  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.
  5. C. C. Conley, Isolated invariant sets and the Morse index, CBMS no. 38, A.M.S., Providence, R.I., 1978.
  6. C. Conley, R. Easton, Isolated Invariant Sets and Isolating Blocks, in: Advances in Differential and Integral Equations, ed. J. Nohel, Studies in Applied Mathematics 5. SIAM Publications, Philadelphia (1969), 97-104.
  7. C. Conley, R. Easton, Isolated Invariant Sets and Isolating Blocks, Trans. Amer. Math. Soc. 158 (1971), 35-61.
  8. M. Degiovanni and M. Mrozek, The Conley index for maps in absence of compactness, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 75-94.
  9. R. Franzosa, Index Filtrations and the Homology Index Braid for Partially Ordered Morse Decompositions, Trans. Amer. Math. Soc. 298 (1986), 193-213.
  10. R. Franzosa, The Connection Matrix Theory for Morse Decompositions, Trans. Amer. Math. Soc. 311 (1989) 561-592.
  11. R. Franzosa, The Continuation Theory for Morse Decompositions and Connection Matrices, Trans. Amer. Math. Soc. 310 (1988), 781-803.
  12. L. Górniewicz, Topological Degree of Morphisms and its Applications to Differential Inclusions, Raccolta di Seminari del Dipartimento di Matematica dell'Universita degli Studi della Calabria, No. 5, 1983.
  13. L. Górniewicz, Homological Methods in Fixed Point Theory of Multi-valued Maps, Dissertationes Math. 129, PWN, Warszawa, 1976.
  14. A. Iserles, A. T. Peplow, A. M. Stuart, A unified approach to spurious solutions introduced by time discretization. Part I: Basic theory, SIAM J. Numer. Anal. 28 (1991), 1723-1751.
  15. T. Kaczyński and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology Appl., accepted.
  16. H. L. Kurland, The Morse Index of an Isolated Invariant Set is a Connected Simple System, J. Differential Equations 42 (1981), 234-259.
  17. H. L. Kurland, Following Homology in Singularly Perturbed Systems, J. Differential Equations 62 (1986), 1-72.
  18. Ch. McCord, K. Mischaikow and M. Mrozek, Zeta Functions, Periodic Trajectories and the Conley Index, J. Differential Equations, accepted.
  19. K. Mischaikow and M. Mrozek, Isolating neighbourhoods and Chaos, Jap. J. Ind. & Appl. Math., accepted.
  20. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc., in print.
  21. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Part II: details, preprint.
  22. J. T. Montgomery, Cohomology of Isolated Invariant Sets under Perturbation, J. Differential Equations 13 (1973), 257-299.
  23. M. Mrozek, Index pairs and the Fixed Point Index for Semidynamical Systems with Discrete Time, Fund. Math. 133 (1989), 179-194.
  24. M. Mrozek, A Cohomological Index of Conley Type for Multi-valued Admissible Flows, J. Differential Equations 84 (1990), 15-51.
  25. M. Mrozek, Leray Functor and the Cohomological Conley Index for Discrete Dynamical Systems, Trans. Amer. Math. Soc. 318 (1990), 149-178.
  26. M. Mrozek, Open index pairs, the fixed point index and rationality of zeta functions, Ergodic Theory Dynamical Systems 10 (1990), 555-564.
  27. M. Mrozek, The Morse Equation in Conley's Index Theory for Homeomorphisms, Topology Appl. 38 (1991), 45-60.
  28. M. Mrozek, The Conley index on compact ANR's is of finite type, Results Math. 18 (1990), 306-313.
  29. M. Mrozek, Shape Index and Other Indices of Conley Type for Continuous Maps on Locally Compact Metric Spaces, Fund. Math., 145 (1994), 15-37.
  30. M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, in preparation.
  31. M. Mrozek and K. P. Rybakowski, A cohomological Conley index for maps on metric spaces, J. Differential Equations 90.1 (1991), 143-171.
  32. M. Mrozek and K. P. Rybakowski, Discretized ordinary differential equations and the Conley index, J. Dynamics Differential Equations 4 (1992), 57-63.
  33. J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8 (1988), 375-393.
  34. K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin Heidelberg 1987.
  35. T. Ważewski, Une méthode topologique de l'examen du phénomène asymptotique relativement aux équations différentielles ordinaires, Rend. Accad. Nazionale dei Lincei, Cl. Sci. fisiche, mat. e naturali, Ser. VIII, vol. III (1947), 210-215.
  36. T. Ważewski, Sur un principe topologique pour l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math. 20 (1947), 279-313.
  37. T. Ważewski, Sur un méthode topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles, Proceedings of the International Congress of Mathematicians 1954, 3 (1955), 5-14.
  38. H. C. Yee, P. K. Sweby and D. F. Griffiths, Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. 1. The Dynamics of Time Discretization and Its Implications for Algorithm Development in Computational Fluid Dynamics, J. Comput. Phys. 97 (1991), 249-310.
Pages:
105-120
Main language of publication
English
Published
1995
Exact and natural sciences