Random Vibration of a Nonlinear Autoparametric System

A noisy autoparametric system exhibiting 1:2 resonance is studied as a random perturbation of a fourdimensional Hamiltonian system. The problem involves three time scales. Nonstandard stochastic averaging technique is rigorously developed, application of which results in a lower-dimensional description of the system. Probability density of the limiting process is obtained using FEM methods. These results are validated using numerical simulation of the original equations. While the numerical simulations take several hours of computer time, FEM solutions take no more than a few minutes. The methods developed here could also be used for other auto-parametric systems such as in capsizing of ships in random seas.