Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Rigorous high-dimensional shadowing using containment: The general case

Pages: 329 - 342, Volume 14, Issue 2, February 2006      doi:10.3934/dcds.2006.14.329

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Wayne B. Hayes - Dept. of Computer Science, University of California, Irvine, Irvine, California 92697-3425, United States (email)
Kenneth R. Jackson - Computer Science Dept., University of Toronto, Toronto, Ontario M5S 3G4, Canada (email)
Carmen Young - The Fields Institute, 222 College Street, Toronto, ON M5T 3J1, Canada (email)

Abstract: 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:  Nonlinear dynamical systems, shadowing, reliable simulation.
Mathematics Subject Classification:  Primary: 57R19, 65G20; Secondary: 57R50, 65L70.

Received: November 2004;      Revised: June 2005;      Available Online: November 2005.