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The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered. [DOI: 10.1115/1.1445143]

In a recent paper an asymptotic approximation for the moment Lyapunov exponent, g (p), of two coupled oscillators driven by a small intensity real noise was obtained. The utility of that result is limited by the fact that it was obtained under the assumption that the moment p is small, a limitation which precludes, for example, the determination of the stability index. In this paper that limitation is removed and an asymptotic approximation valid for arbitrary p is obtained. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation. [DOI: 10.1115/1.1387021]

The Lyapunov exponents and moment Lyapunov exponents of a near-nilpotent system under stochastic parametric excitation are studied. The system considered is the linearized system of a two-dimensional nonlinear system exhibiting a pitchfork bifurcation. The effect of stochastic perturbation in the vicinity of static pitchfork bifurcation is investigated. Approximate analytical results of Lyapunov exponent are obtained. The eigenvalue problem for the moment Lyapunov exponent is converted to a two-point boundary value problem, which is solved numerically by the method of relaxation. [DOI: 10.1115/1.1364491]

Stochastic averaging equations have been derived for integrable and nonresonant Hamiltonian system with multi-degree-of-freedom and subjected to light damping and real noise excitations. Khasminskii's method is then used with the averaged equations expression for obtaining the largest Lyapunov exponents of the square root of the Hamiltonian. The stochastic stability and bifurcation of the original systems can then be determined approximately.

The feasibility of developing a Lyapunov function for any classically damped linear system is examined. Tests show that the function's attributes are still retained despite two scenarios. First, a nonlinear term was successfully introduced without creating instability in the trivial solution. Second, perturbation of the linear part did not affect the robustness of the function.

The robustness of nonlinear systems subjected to external random excitation is studied using the Lyapunov exponent concept. The one-dimensional perturbed system is expressed in terms of ordinary differential equations to obtain the function of the Lyapunov exponent. The stationary probability density function is then obtained by solving the Fokker-Planck equation for the system. Actual applications to several example problems reveal that the Lyapunov exponent doesnot reflect the qualitative changes in the steady-state probability density function.