Abstract
Mathematical rigorous error bounds for the numerical approximation of dynamical systems have long been hindered by the wrapping effect. We present a new method which constructs high order zonotope (special polytopes) enclosures for the orbits of discrete dynamical systems. The wrapping effect can made arbitrarily small by controlling the order of the zonotopes. The method induces in the space of zonotopes a dynamical system of amazing geometrical complexity. We emphasis the visualization of the zonotopes to better understand the involved dynamics.
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References
D. P. Davey And N. F. Stewart, Guaranteed error bounds for the initial value problem using polytope arithmetic, BIT 16 (1976), 257–268.
D. Eppstein, Zonohedra and zonotopes, Tech. report, Dept. of Information & Computer Science U.C. Irvine, CA, 92717, 1995.
K. G. Guderley And C. L. Keller, A basic theorem in the computation of ellipsoidal error bounds, Numer. Math. 19: 3 (1972), 218–229.
W. Kahan, A computable error bound for systems of ordinary differential equations, Abstracts in SIAM Review 8 (1966), 568–569.
W. Kuhn, Rigorous and reasonable error bounds for the numerical solution of dynamical systems, Ph.D. thesis, Georgia Institute of Technology, 1997.
W. Langford, Unfolding of degenerate bifurcations, (P. Fisher And W. Smith, eds. ), Marcel Dekker, 1985, pp. 87–103.
R. Lohner, Einscließung der Lösung gewöhnlicher Anfangs-und Randwertaufgaben und Anwendungen, Ph.D. thesis, Univ. Karlsruhe, 1988.
R. E. Moore, Interval analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
A. Neumaier, Interval methods for systems of equations, publ Cambridge University Press, 1990.
R. Rihm, Interval methods for initial value problems in odes, (J. Herzberger, ed. ), Elsevier Science B.V., 1994.
R. Schneider And W. Weil, Zonoids and related topics, Convexity and its Applications (P. M. Gruber And J. M. Wills, eds.), Birkhäuser Verlag, Basel, 1983, pp. 296–317.
C. Siegel And J. Moser, Lectures on celestial mechanics, Springer-Verlag, 1971.
G. M. Ziegler, Lectures on polytopes, Springer-Verlag, 1995.
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© 1998 Springer-Verlag Berlin Heidelberg
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Kühn, W. (1998). Zonotope Dynamics in Numerical Quality Control. In: Hege, HC., Polthier, K. (eds) Mathematical Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03567-2_10
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DOI: https://doi.org/10.1007/978-3-662-03567-2_10
Publisher Name: Springer, Berlin, Heidelberg
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