This paper examines the almost-sure asymptotic stability condition of a linear multiplicative stochastic system, which is a linear part of a co-dimension two-bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by an ergodic real noise. The excitation is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process which is the output of a linear filter system, while both the detailed balance condition and the strong mixing condition are removed. Through a perturbation method and the spectrum representations of the Fokker Planck operator and its adjoint operator of the linear filter system, the explicit asymptotic expressions of the maximal Lyapunov exponent for three case studies, in which different forms of the coefficient matrix included in the noise excitation term are assumed, are obtained.
The Maximal Lyapunov Exponent for a Three-Dimensional Stochastic System
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, October 2, 2003; final revision, April 13, 2004. Associate Editor: N. Sri Namachchivaya. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering, University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Liew, K. M., and Liu, X. B. (November 9, 2004). "The Maximal Lyapunov Exponent for a Three-Dimensional Stochastic System ." ASME. J. Appl. Mech. September 2004; 71(5): 677–690. https://doi.org/10.1115/1.1782648
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