Stochastic Bifurcations of the Duffing-Mathieu Equations with Time Delays

In the past three decades, the study of stability of buildings, bridges, beams, columns or shells under the influence of parametric and stochastic excitations has traditionally modelled central deflections by the Duffing and Mathieu equations. The study of stability of solutions of these equations is well established in the literature. Time delays appear to be a natural occurrence in structural systems as a result of the excitations of the feedback mechanism. However, less attention is being paid on the instability induced by the time delays. In this paper, the influence of the stochastic delay version of the Duffing-Mathieu equations is studied. A linearized stability analysis of a transcendental characteristic equation of the nonlinear equations is analyzed. As stability is lost, two types of bifurcations, namely, subcritical and supercritical bifurcations are discussed. Then, conditions ensuring stable and unstable bifurcations due to linearized stochastic perturbation are derived. The Markovian diffusion approximation according to the integral averaging method and Lyapunov exponents are employed to obtain explicit analytical results relating to the stability conditions in the stochastic sense.