We study the enclosing problem for discrete and continuous dynamical systems in the context of computer assisted proofs. We review and compare the existing methods and emphasize the importance of developing a suitable set arithmetic for efficient algorithms solving the enclosing problem.
@article{bwmeta1.element.bwnjournal-article-apmv74z1p237bwm,
author = {Mrozek, Marian and Zgliczy\'nski, Piotr},
title = {Set arithmetic and the enclosing problem in dynamics},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {237-259},
zbl = {0967.65113},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p237bwm}
}
Mrozek, Marian; Zgliczyński, Piotr. Set arithmetic and the enclosing problem in dynamics. Annales Polonici Mathematici, Tome 75 (2000) pp. 237-259. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p237bwm/
[000] [1] R. Anguelov, Wrapping function of the initial value problem for ODE: Applications, Reliab. Comput. 5 (1999), 143-164. | Zbl 0937.65073
[001] [2] R. Anguelov and S. Markov, Wrapping effect and wrapping function, ibid. 4 (1998), 311-330. | Zbl 0919.65047
[002] [3] G. F. Corliss and R. Rihm, Validating an a priori enclosure using high-order Taylor series, in: Scientific Computing and Validated Numerics (Wuppertal, 1995), Math. Res. 90, Akademie-Verlag, Berlin, 1996, 228-238. | Zbl 0851.65054
[003] [4] Z. Galias and P. Zgliczyński, Computer assisted proof of chaos in the Lorenz system, Phys. D 115 (1998) 165-188. | Zbl 0941.37018
[004] [5] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, 1987. | Zbl 0638.65058
[005] [6] R. J. Lohner, Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, in: Computational Ordinary Differential Equations, J. R. Cash and I. Gladwell (eds.), Clarendon Press, Oxford, 1992. | Zbl 0767.65069
[006] [7] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66-72. | Zbl 0820.58042
[007] [8] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer assisted proof. Part II: details, Math. Comput. 67 (1998), 1023-1046. | Zbl 0913.58038
[008] [9] K. Mischaikow, M. Mrozek and A. Szymczak, Chaos in the Lorenz equations: a computer assisted proof. Part III: the classical parameter values, submitted. | Zbl 0979.37015
[009] [10] R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966. | Zbl 0176.13301
[010] [11] M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs, Computers Math. 32 (1996), 83-104. | Zbl 0861.58027
[011] [12] M. Mrozek and M. Żelawski, Heteroclinic connections in the Kuramoto-Sivashin- sky equation, Reliab. Comput. 3 (1997), 277-285. | Zbl 0888.65091
[012] [13] A. Neumaier, The wrapping effect, ellipsoid arithmetic, stability and confidence regions, Computing Suppl. 9 (1993), 175-190. | Zbl 0790.65035
[013] [14] M. Warmus, Calculus of approximations, Bull. Acad. Polon. Sci. 4 (1956), 253-259.
[014] [15] M. Warmus, Approximation and inequalities in the calculus of approximations. Classification of approximate numbers, ibid. 9 (1961), 241-245. | Zbl 0098.31304
[015] [16] P. Zgliczyński, Rigorous verification of chaos in the Rössler equations, in: Scientific Computing and Validated Numerics, G. Alefeld, A. Frommer and B. Lang (eds.), Akademie-Verlag, Berlin, 1996, 287-292. | Zbl 0849.65041
[016] [17] P. Zgliczyński, Computer assisted proof of chaos in the Hénon map and in the Rössler equations, Nonlinearity 10 (1997), 243-252. | Zbl 0907.58048
[017] [18] P. Zgliczyński, Remarks on computer assisted proof of chaotic behavior in ODE's, in preparation.