The purpose of this work is to develop a unified approach to study the dynamics of single degree of freedom systems excited by both periodic and random perturbations. The near resonant motion of such systems is not well understood. We will study this problem in depth with the aim of discovering a common geometric structure in the phase space, and to determine the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. Depending on the strength of the noise, reduced Markov process takes its values on a line or on graph with certain gluing conditions at the vertex of the graph. The reduced model will provide a framework for computing standard statistical measures of dynamics and stability, namely, mean exit times, probability density functions, and stochastic bifurcations. This work will also explain a counter-intuitive phenomena of stochastic resonance, in which a weak periodic force in a nonlinear system can be enhanced by the addition of external noise.

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