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Research Papers

# Simulation of Moment Lyapunov Exponents for Linear Homogeneous Stochastic Systems

[+] Author and Article Information
Wei-Chau Xie, Qinghua Huang

Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

J. Appl. Mech 76(3), 031001 (Mar 03, 2009) (10 pages) doi:10.1115/1.3063629 History: Received March 14, 2006; Revised October 02, 2008; Published March 03, 2009

## Abstract

Moment Lyapunov exponents are important characteristic numbers for describing the dynamic stability of a stochastic system. When the $pth$ moment Lyapunov exponent is negative, the $pth$ moment of the solution of the stochastic system is stable. Monte Carlo simulation approaches complement approximate analytical methods in the determination of moment Lyapunov exponents and provides criteria on assessing the accuracy of approximate analytical results. For stochastic dynamical systems described by Itô stochastic differential equations, the solutions are diffusion processes and their variances may increase with time. Due to the large variances of the solutions and round-off errors, bias errors in the simulation of moment Lyapunov exponents are significant in improper numerical algorithms. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented in this paper.

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## Figures

Figure 1

Growth of the solution and normalization

Figure 2

Simulation of moment Lyapunov exponents for a first-order linear system

Figure 3

Moment Lyapunov exponents under white noise excitation (ε=0.1)

Figure 4

Moment Lyapunov exponents under white noise excitation (ε=0.5)

Figure 5

Histograms of logarithm of norm compared with normal density approximations

Figure 6

Moment Lyapunov exponents under real noise excitation for different α

Figure 7

Moment Lyapunov exponents under real noise excitation for different σ

Figure 8

Moment Lyapunov exponents under bounded noise excitation for different σ

Figure 9

Moment Lyapunov exponents under bounded noise excitation for different μ

Figure 10

Histograms of logarithm of norm compared with normal density approximations

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