A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems

SIAM Journal on Applied Dynamical Systems

Volumn 3, Issue 2, 2004, Pages 117-160

  Day, S ^a,b   Junge, O ^c   Mischaikow, K ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b Cornell University ^*  (United States)

^c UNIVERSITY OF PADERBORN   (Germany)

Author keywords

Conley index; Dynamical system; Infinite dimensional; Numercial method

Indexed keywords

EID: 3042678480     PISSN: 15360040     EISSN: None     Source Type: Journal    
DOI: 10.1137/030600210     Document Type: Article

References (26)

1. G. ARIOLI AND P. ZGLICZYŃSKI, Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level, J. Differential Equations, 171 (2001), pp. 173-202.
2. A. W. BAKER, Lower bounds on entropy via the Conley index with application to time series, Topology Appl., 120 (2002), pp. 333-354.
3. M. DELLNITZ, G. FROYLAND, AND O. JUNGE, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler, ed., Springer-Verlag, Berlin, 2001, pp. 145-174; 805-807.
4. M. DELLNITZ AND A. HOHMANN, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), pp. 293-317.
5. M. DELLNITZ, O. JUNGE, M. RUMPF, AND R. STRZODKA, The computation of an unstable invariant set inside a cylinder containing a knotted flow, in Proceedings of the International Conference on Differential Equations, World Scientific, River Edge, NJ, 2000, pp. 1053-1059.
6. E. W. DIJKSTRA, A note on two problems in connection with graphs, Numer. Math., 1 (1959), pp. 269-271.
7. M. EIDENSCHINK, Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory, Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, 1995.
8. J. FRANKS AND D. RICHESON, Shift equivalence and the Conley index, Trans. Amer. Math. Soc., 352 (2000), pp. 3305-3322.
9. J. K. HALE AND G. RAUGEL, Regularity, determining modes and Galerkin methods, J. Math. Pure Appl. (9), 82 (2003), pp. 1075-1136.
10. O. JUNGE, Rigorous discretization of subdivision techniques, in Proceedings of the International Conference on Differential Equations, World Scientific, River Edge, NJ, 2000, pp. 916-918.
11. T. KACZYNSKI, K. MISCHAIKOW, AND M. MROZEK, Computational Homology, Appl. Math. Sci. 157, Springer-Verlag, New York, 2004.
12. T. KACZYNSKI AND M. MROZEK, Conley index for discrete multivalued dynamical systems, Topology Appl., 65 (1995), pp. 83-96.
13. O. KNÜPPEL, BIAS - Basic Interval Arithmetic Subroutines, Technische Universität Hamburg-Harburg, Hamburg, Germany, 1993.
14. O. KNÜPPEL, PROFIL - Programmer's Runtime Optimized Fast Interval Library, Technische Universität Hamburg-Harburg, Hamburg, Germany, 1993.
15. M. KOT AND W. M. SCHAFFER, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), pp.109-136.
16. K. MISCHAIKOW AND M. MROZEK, Conley index, in Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., North-Holland, Amsterdam, 2002, pp. 393-460.
17. K. MISCHAIKOW, M. MROZEK, AND P. PILARCZYK, Graph Approach to the Computation of the Homology of Continuous Maps, Preprint, 2004.
18. K. MISCHAIKOW, M. MROZEK, AND A. SZYMCZAK, Chaos in the Lorenz equations: A computer assisted proof. III. Classical parameter values, J. Differential Equations, 169 (2001), pp. 17-56.
19. M. MROZEK, Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces, Fund. Math., 145 (1994), pp. 15-37.
20. A. SZYMCZAK, The Conley index for decompositions of isolated invariant sets, Fund. Math., 148 (1995), pp. 71-90.
21. A. SZYMCZAK, The Conley index and symbolic dynamics, Topology, 35 (1996), pp. 287-299.
22. A. SZYMCZAK, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), pp. 1075-1088.
23. A. SZYMCZAK, Index Pairs: From Dynamics to Combinatorics and Back, Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, 1999.
24. W. TUCKER, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), pp.53-117.
25. J. ZEMKE, b4m, A Free Interval Arithmetic Toolbox for Matlab Based on BIAS, Technische Universität Hamburg-Harburg, Hamburg, Germany, 1998, http://www.ti3.tu-harburg.de/zemke/b4m.
26. P. ZGLICZYŃSKI AND K. MISCHAIKOW, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), pp. 255-288.