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1995 | 34 | 1 | 105-120
Tytuł artykułu

From the theorem of Ważewski to computer assisted proofs in dynamics

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
Słowa kluczowe
Rocznik
Tom
34
Numer
1
Strony
105-120
Opis fizyczny
Daty
wydano
1995
Twórcy
  • Instytut Informatyki, Uniwersytet Jagielloński, Nawojki 11, 30-072 Kraków, Poland
Bibliografia
  • [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.
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Bibliografia
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