Keywords: moment Lyapunov exponent, stochastic stability, Monte Carlo simulation, eigenvalue problem.

In general, the study of the dynamics of many engineering structures under random loadings leads to a system of stochastic differential equations. The sample or almost-sure stability of the trivial solution is determined by the Lyapunov exponent, which characterizes the average exponential rate of growth or decay of the solutions of system. The trivial solution of the system is stable with probability one if the largest Lyapunov exponent is negative. On the other hand, the stability of the pth moment of the trivial solution of the system is determined by the moment Lyapunov exponent. If the moment Lyapunov exponent is negative, then the pth moment is stable.

The moment Lyapunov exponent has been recognized as an ideal avenue for studying the behaviour of a dynamical system, because it provides not only the information about stability or instability, but also how rapidly the response grows or diminishes with time. The pth moment Lyapunov exponent is the ultimate characteristic number in the study of stochastic dynamical systems, since the largest Lyapunov exponent is equal to the slope of the pth moment Lyapunov exponent and the stability index is the non-trivial zero of the moment Lyapunov exponent.

Although moment Lyapunov exponents are important in the study of dynamic stability of stochastic systems, their actual evaluations are very difficult. Various approximate analytical methods have been devised to actually carry out the computation for a number of engineering structural systems.

The determination of the pth moment Lyapunov exponent is in general very difficult. For most practical engineering structures, numerical approaches have to be applied. Numerical determination of the pth moment Lyapunov exponents is important for at least four reasons.

Numerically accurate results of the moment Lyapunov exponents are essential in assessing the validity and the ranges of applicability of the approximate analytical results.

In many engineering applications, the amplitudes of noise excitations are not small and the approximate analytical methods, such as the method of perturbation or the method of stochastic averaging, cannot be applied. Numerical approaches have to be employed to evaluate the moment Lyapunov exponents.

For systems of large dimensions, it is very difficult, if not impossible, to obtain analytical results. For systems under noise excitations that cannot be described in elegant analytical forms or if only the time series of the response of the system is known, numerical approaches must be the required resort. Two numerical methods for the determination of the pth moment Lyapunov exponents are presented in this paper.

The first method is an analytical-numerical approach, in which the partial differential eigenvalue problems governing the moment Lyapunov exponents are established using the theory of stochastic dynamical systems. The eigenfunctions are expanded in series to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The second method is a Monte Carlo simulation approach.

The methods are illustrated through a two-dimensional system under bounded noise parametric excitation. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

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