Rigorous high-dimensional shadowing using containment: The general case

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April 2006, 14(2): 329-342. doi: 10.3934/dcds.2006.14.329

Rigorous high-dimensional shadowing using containment: The general case

Wayne B. Hayes ^1, , Kenneth R. Jackson ^2, and Carmen Young ^3,

1.	Dept. of Computer Science, University of California, Irvine, Irvine, California 92697-3425, United States
2.	Computer Science Dept., University of Toronto, Toronto, Ontario M5S 3G4, Canada
3.	The Fields Institute, 222 College Street, Toronto, ON M5T 3J1, Canada

Received November 2004 Revised June 2005 Published November 2005

Abstract
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A shadow is an exact solution to an iterated map that remains close to an approximate solution for a long time. An elegant geometric method for proving the existence of shadows is called containment, and it has been proven previously in two and three dimensions, and in some special cases in higher dimensions. This paper presents the general proof using tools from differential and algebraic topology and singular homology.

Keywords: reliable simulation., Nonlinear dynamical systems, shadowing.

Mathematics Subject Classification: Primary: 57R19, 65G20; Secondary: 57R50, 65L7.

Citation: Wayne B. Hayes, Kenneth R. Jackson, Carmen Young. Rigorous high-dimensional shadowing using containment: The general case. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 329-342. doi: 10.3934/dcds.2006.14.329

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