Moment Lyapunov exponents of a two-dimensional system under combined harmonic and real noise excitations

The moment Lyapunov exponents and the Lyapunov exponents of a two-dimensional system under combined excitations of harmonic and real noise, which is modelled as an Ornstein–Uhlenbeck process, are studied. The moment Lyapunov exponents and the Lyapunov exponents are important characteristics determining the moment and almost-sure stability of a stochastic dynamical system. The eigenvalue problem governing the moment Lyapunov exponent is established. A regular perturbation method is applied to solve the eigenvalue problem to obtain second-order, weak noise expansions of the moment Lyapunov exponents. The influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated.