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Chaos in the Lorenz equations: A computer assisted proof. II: Details. (English) Zbl 0913.58038

Details of a computer assisted proof that the Lorenz equations contain chaotic dynamics for a prescribed open set of parameter values are presented. The difficulty in the rigorous numerical treatment of chaos in the Lorenz equations lies in the fact that the chaotic set is located very close to the stable manifold of the origin, where the Poincaré map is discontinuous. To overcome this problem, the authors introduce the technique of intermediate sections, similar to the multiple shooting used in the numerical treatment of boundary value problems. It combines abstract existence results based on the Conley index theory with rigorous computer assisted computations, being in this way applicable to a wide range of problems in (not necessarily chaotic) dynamics.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65L99 Numerical methods for ordinary differential equations
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
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[1] Uri M. Ascher, Robert M. M. Mattheij, and Robert D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1988. · Zbl 0671.65063
[2] Xinfu Chen, Lorenz Equations, Part III: Existence of Hyperbolic Sets, preprint 1995.
[3] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. · Zbl 0397.34056
[4] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. · Zbl 0638.65058
[5] B. Hassard, J. Zhang, S. P. Hastings, and W. C. Troy, A computer proof that the Lorenz equations have ”chaotic” solutions, Appl. Math. Lett. 7 (1994), no. 1, 79 – 83. · Zbl 0792.65050 · doi:10.1016/0893-9659(94)90058-2
[6] S. P. Hastings and W. C. Troy, A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 298 – 303. · Zbl 0764.58023
[7] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), no. 1, 83 – 96. · Zbl 0843.54042 · doi:10.1016/0166-8641(94)00088-K
[8] 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, I. Gladwell Eds., Clarendon Press, Oxford, 1992. CMP 96:12 · Zbl 0767.65069
[9] J. Łukasiewicz, O logice trójwartościowej (On three-valued logic), Ruch Filozoficzny 5(1920), 169-170.
[10] Konstantin Mischaikow, The structure of isolated invariant sets and the Conley index, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 269 – 290. · Zbl 0802.34051 · doi:10.1090/conm/152/01328
[11] K. Mischaikow, The Conley index theory: some recent developments, CIME Lectures, preprint.
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