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We derive an analytical approximation to the probability distributionfunction (pdf) for the response of Mathieu's equation under parametricexcitation by a random process with a spectrum peaked at the main resonantfrequency, motivated by the problem of large amplitude ship roll resonance inrandom seas. The inclusion of random stochastic excitation renders theotherwise straightforward response to a system undergoing intermittentresonances: randomly occurring large amplitude bursts. Intermittent resonanceoccurs precisely when the random parametric excitation pushes the system intothe instability region, causing an extreme magnitude response. As a result, thestatistics are characterized by heavy-tails. We apply a recently developedmathematical technique, the probabilistic decomposition-synthesis method, toderive an analytical approximation to the non-Gaussian pdf of the response. Weillustrate the validity of this analytic approximation through comparisons withMonte-Carlo simulations that demonstrate our result accurately captures thestrong non-Gaussianity of the response.

Probabilistic response and rare events in Mathieu's equation under correlated parametric excitation