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# Numerical Determination of Moment Lyapunov Exponents of Two-Dimensional Systems

[+] Author and Article Information
Wei-Chau Xie

Solid Mechanics Division, Faculty of Engineering,  University of Waterloo, Waterloo, ON N2L 3G1, Canada

Ronald M. So

Department of Mechanical Engineering,  The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

J. Appl. Mech 73(1), 120-127 (Jun 01, 2005) (8 pages) doi:10.1115/1.2041663 History: Received December 04, 2002; Revised June 01, 2005

## Abstract

The $pth$ moment Lyapunov exponent of an $n$-dimensional linear stochastic system is the principal eigenvalue of a second-order partial differential eigenvalue problem, which can be established using the theory of stochastic dynamical system. An analytical-numerical approach for the determination of the $pth$ moment Lyapunov exponents, for all values of $p$, is presented. The approach is illustrated through a two-dimensional system under bounded noise or real noise parametric excitation. Series expansions of the eigenfunctions using orthogonal functions are employed to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

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## Figures

Figure 1

Λx(t)(p) of system under bounded noise excitation, σ=1.0, ν=2.0, primary resonance

Figure 2

Λx(t)(p) of system under bounded noise excitation, σ=1.0, ν=1.0, secondary resonance

Figure 3

Λx(t)(p) of system under bounded noise excitation, σ=1.0, ν=3.0, no resonance

Figure 4

Λx(t)(p) of system under real noise excitation, α=2.0, σ=1.0

Figure 5

Λx(t)(p) of system under real noise excitation, α=1.0, σ=1.0

Figure 6

Λx(t)(p) of system under real noise excitation, α=0.5, σ=1.0

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