Order and Chaos in a Discrete Duffing Oscillator: Implications on Numerical Integration

[+] Author and Article Information
P. G. Reinhall, D. W. Storti

Department of Mechanical Engineering FU-10, University of Washington, Seattle, WA 98195

T. K. Caughey

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125

J. Appl. Mech 56(1), 162-167 (Mar 01, 1989) (6 pages) doi:10.1115/1.3176039 History: Received June 03, 1987; Revised May 01, 1988; Online July 21, 2009


In this paper, we study the dynamics of some two-dimensional mappings which arise when standard numerical integration schemes are applied to an unforced oscillator with a cubic stiffness nonlinearity, i.e., the Duffing equation. While the continuous time problem is integrable and is solved analytically in terms of Jacobi elliptic functions, the discrete versions of this simple system arising from standard integration schemes exhibit very complicated dynamics due to the presence of homoclinic tangles. We present an alternative scheme for discretizing the nonlinear term which preserves the integrable dynamics of the continuous system and derive analytic expressions for the orbits and invariant curves of the resulting mapping.

Copyright © 1989 by ASME
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