This paper develops a unified approach to study the dynamics of nonlinear oscillators excited by both periodic and random perturbations. This study is motivated by problems that range from nonlinear energy harvesting to ship capsizing in random seas. The near resonant dynamics of such systems, in the presence of weak noise, is not well understood. Nonlinear systems driven by sufficiently strong periodic parametric excitation often display a range of phenomena from period doubling to chaos. In

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... e presence of weak noise there are transitions between the domains of attraction of the stable periodic orbits. The effects of noisy perturbations on the passage of trajectories through the resonance zones is studied in depth using the large deviation theory.