Maximal Lyapunov Exponent and Almost-Sure Stability for Coupled Two-Degree-of-Freedom Stochastic Systems

[+] Author and Article Information
N. Sri Namachchivaya, S. Talwar

Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

H. J. Van Roessel

Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada

J. Appl. Mech 61(2), 446-452 (Jun 01, 1994) (7 pages) doi:10.1115/1.2901465 History: Received October 16, 1992; Revised April 27, 1993; Online March 31, 2008


In this paper, a perturbation approach is used to calculate the asymptotic growth rate of stochastically coupled two-degree-of-freedom systems. The noise is assumed to be white and of small intensity in order to calculate the explicit asymptotic formulas for the maximum Lyapunov exponent, The Lyapunov exponents and rotation number for each degree-of-freedom are obtained in the Appendix. The almost-sure stability or instability of the four-dimensional stochastic system depends on the sign of the maximum Lyapunov exponent. The results presented here match those presented by the first author and others using the method of stochastic averaging, where approximate Itô equations in amplitudes and phase are obtained in the sense of weak convergence.

Copyright © 1994 by The American Society of Mechanical Engineers
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