On the calculation of Lyapunov and moment Lyapunov exponents for noise driven coupled oscillators

The primary concern in the analysis of dynamical systems is the determination and prediction of steady-state motions and their corresponding stability. The stability analysis of a dynamical system becomes more difficult when the system depends on a random process. Since most practical dynamical systems are subject to some form of noise, the determination of conditions under which the resulting stochastic system is stable (in some sense) is of great interest. in the study of stability of solutions of random dynamical systems, both almost-sure stability and p(th) moment stability have been widely used. Stability in the almost-sure sense is determined by the sign of the maximal Lyapunov exponent whereas p(th) moment stability is determined by the moment Lyapunov exponent. The dynamical system we consider consists of two coupled oscillators driven by real noise: (1) q double over dot(i) + w(i)(2)q(i) + 2 epsilon(2)zeta w(i) (q) over dot(i) + epsilon Sigma(j=1)(2) k(ij)q(j)f (xi(t)) = 0 i,j = 1,2, where the q(i)'s are generalized coordinates, w(i) is the ith natural frequency, epsilon(2)zeta represents a small viscous damping coefficient and the stochastic term epsilon xi(t) is a small-intensity, real-noise process. Two perturbative techniques for the evaluation of the Lyapunov exponent in the presence of a small intensity noise have been developed by the authors [1, 2]: a forward method and a backward method. However, since the backward method has the advantage that it also allows for the determination of the moment Lyapunov exponent, it is this latter method that will be described in this paper. Before proceeding with the details of the backward method, a brief summary of some of the theoretical results required is given.