Maximal lyapunov exponent and rotation number for stochastically perturbed co-dimension two bifurcations

Almost-sure asymptotic stability of three- and four-dimensional co-dimension two dynamical systems under small intensity stochastic excitations is investigated. The method of stochastic averaging is used to derive a set of approximate Ito equations. These equations, along with their sample properties, are then examined to obtain the almost-sure stability conditions. The sample properties of the process are based on the boundary behavior of the associated scalar diffusion process of the amplitude Ito equations. The maximal Lyapunov exponent is calculated using the ergodic scalar diffusive process, which in turn yields the almost-sure stability conditions. This method is then applied to a linear, gyroscopic problem of a rotating shaft with a random loading.