Stochastic dynamics of parametrically excited two d.o.f. systems with symmetry

In this paper we investigate the non-linear dynamics of a two-degree-of-freedom system with symmetries subject to random parametric excitation. The study of this non-linear near-Hamiltonian system is simplified by using the symmetry and separation of scales present in the problem. To this end, we study the equations as a random perturbation of a four-dimensional weakly dissipative Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced process is simply a Markov process on a line. Examination of the reduced Markov process on the line yields many important results, namely, probability density functions, and stochastic bifurcations. The steady state dynamics is computed explicitly. Phenomenological and dynamical bifurcations are investigated. The approach adopted in this paper can in principle, be applied to any four-dimensional integrable system.