Topological invariants, mulitvalued maps and computer assisted proofs in dynamics

Computers and Mathematics with Applications

Volumn 32, Issue 4, 1996, Pages 83-104

  Mrozek, M ^a,b  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b JAGIELLONIAN UNIVERSITY   (Poland)

Author keywords

Computer assisted proofs; Conley index theory; Interval arithmetic; Isolating neighborhoods; Mulitivalued maps

Indexed keywords

APPROXIMATION THEORY; CHAOS THEORY; COMPUTATIONAL COMPLEXITY; COMPUTER AIDED ANALYSIS; DIFFERENCE EQUATIONS; DIGITAL ARITHMETIC; DYNAMICS; NONLINEAR EQUATIONS; THEOREM PROVING;

COMPUTER ASSISTED PROOFS; CONLEY INDEX THEORY; INTERVAL ARITHMETIC; LORENZ EQUATIONS; MULTIVALUED APPROXIMATIONS; MULTIVALUED MAPS; TOPOLOGICAL METHODS;

TOPOLOGY;

EID: 0030213967     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/0898-1221(96)00127-7     Document Type: Article

References (31)

1. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32, 66-72 (1995).
2. A.M. Stuart, Numerical analysis of dynamical systems, In Acta Numerica, (Edited by A. Iserles), pp. 467-572, Cambridge Univ. Press, Cambridge, (1994).
3. K. Appel and W. Haken, Every planar graph is four colorable, Part I: Discharging, Illinois J. of Mathematics 21, 429-190 (1977).
4. K. Appel and W. Haken, Every planar graph is four colorable, Part II: Reducibility, Illinois J. of Mathematics 21, 491-567 (1977).
5. D. Zeilberger, Theorems for a price: Tomorrow's semi-rigorous mathematical culture. The Mathematical Intelligencer 16, 11-14, 76 (1994).
6. O. Aberth, Precise Numerical Analysis, William C. Brown, Dubuque, IA, (1988).
7. S.P. Hastings and W.C. Troy, A shooting approach to the Lorenz equations, Bull. AMS 27, 298-303 (1992).
8. B. Hassard and J. Zhang, Existence of a homoclinic orbit of the Lorenz system by precise shooting, SIAM J. Math. Anal. 25, 179-196 (1994).
9. B. Hassard, S.P. Hastings, W.C. Troy and J. Zhang, A computer proof that the Lorenz equations have "chaotic" solutions, Appl. Math. Lett. 7 (1), 79-84 (1994).
10. H. Koch, A. Schenkel and P. Wittwer, Computer assisted proofs in analysis and programming in logic: A case study, Universite de Geneve, (preprint).
11. J.P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Memoirs of the American Mathematical Society 47, 1-121 (1984).
12. O.E. Lanford, Computer assisted proof of the Feigenbaum Conjectures, Bull. AMS 6, 427-434 (1982).
13. Th. Rage, A. Neumaier and Ch. Schlier, Rigorous verification of chaos in a molecular model, Physical Rev. E 50, 2682-2688 (1994).
14. K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: A computer assisted proof, Part II: Details, (preprint CDSNS95-222).
15. K. Mischaikow, M. Mrozek and A. Szymczak, Chaos in Lorenz equations: A computer assisted proof, Part III: The classic case, (In preparation).
16. A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, (preprint).
17. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. Math., Volume 38, AMS, Providence, RI, (1978).
18. K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, (1987).
19. K. Mischaikow, The structure of isolated invariant sets, In Contemporary Mathematics, (Edited by C. McCord), pp. 269-290, AMS, (1993).
20. K. Mischaikow, The Conley index theory: Some recent developments, In Dynamical Systems, (Edited by R. Johnson), Springer, (1995).
21. P. Zgliczyński, Chaos in the Hénon map, (preprint).
22. P. Zgliczyński, Chaos in the Rössler equation, (in preparation).
23. P. Zgliczyński, Fixed point index for iterations of maps, topological horseshoe and chaos, Top. Meth. in Nonlin. Anal. (to appear).
24. M. Mrozek, Leray functor and the cohomological Conley index for discrete dynamical systems, Transactions of the American Mathematical Society 318, 149-178 (1990).
25. T. Kaczyński and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology & Its Appl. (to appear).
26. K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Jap. J. Ind. Appl. Math. 12, 205-236 (1995).
27. M. Mrozek, An algortihmic approach to the Conley index theory, (in preparation).
28. A. Dold, Lectures on Algebraic Topology, Springer-Verlag, Berlin, (1972).
29. B.M. Garay, Discretization and some qualitative properties of ordinary differential equations about equilibria, Acta Math. Univ. Comeniane 62, 249-275 (1993).
30. B.M. Garay, On structural stability of ordinary differential equations with respect to discretization methods, Numer. Math. 72, 449-479 (1996).
31. A. Szymczak, The Conley index and symbolic dynamics, (preprint).