The purpose of this work is to develop a unified approach to study the dynamics of a single degree of freedom system excited by both periodic and random perturbations. We consider the noisy Duffing-van der Pol-Mathieu equation as a prototypical single degree of freedom system and achieve a reduction by developing rigorous methods to replace, in some limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

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