Set arithmetic and the enclosing problem in dynamics

• Autorzy: Marian Mrozek , Piotr Zgliczyński
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

dynamical systems; interval arithmetic; rigorous numerical analysis; enclosing problem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 74
• Numer: 1
• Strony: 237-259

Daty

wydano

2000

otrzymano

1999-04-30

Twórcy

autor

Marian Mrozek

• Institute of Computer Science, Jagiellonian University, 30-072 Kraków, Poland

autor

Piotr Zgliczyński

• School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

Bibliografia

• [1] R. Anguelov, Wrapping function of the initial value problem for ODE: Applications, Reliab. Comput. 5 (1999), 143-164.
• [2] R. Anguelov and S. Markov, Wrapping effect and wrapping function, ibid. 4 (1998), 311-330.
• [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.
• [4] Z. Galias and P. Zgliczyński, Computer assisted proof of chaos in the Lorenz system, Phys. D 115 (1998) 165-188.
• [5] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, 1987.
• [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.
• [7] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66-72.
• [8] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer assisted proof. Part II: details, Math. Comput. 67 (1998), 1023-1046.
• [9] K. Mischaikow, M. Mrozek and A. Szymczak, Chaos in the Lorenz equations: a computer assisted proof. Part III: the classical parameter values, submitted.
• [10] R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
• [11] M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs, Computers Math. 32 (1996), 83-104.
• [12] M. Mrozek and M. Żelawski, Heteroclinic connections in the Kuramoto-Sivashin- sky equation, Reliab. Comput. 3 (1997), 277-285.
• [13] A. Neumaier, The wrapping effect, ellipsoid arithmetic, stability and confidence regions, Computing Suppl. 9 (1993), 175-190.
• [14] M. Warmus, Calculus of approximations, Bull. Acad. Polon. Sci. 4 (1956), 253-259.
• [15] M. Warmus, Approximation and inequalities in the calculus of approximations. Classification of approximate numbers, ibid. 9 (1961), 241-245.
• [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.
• [17] P. Zgliczyński, Computer assisted proof of chaos in the Hénon map and in the Rössler equations, Nonlinearity 10 (1997), 243-252.
• [18] P. Zgliczyński, Remarks on computer assisted proof of chaotic behavior in ODE's, in preparation.