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

• Analytical derivation of the probability distribution for the Mathieu׳s equation under stochastic correlated excitation.
• Quantification of rare and extreme responses due to intermittent periods of parametric resonance.
• Decomposition of the probabilistic response into a main core, describing background statistics, and a heavy tailed region associated with instabilities.
• Analytical approximation of the heavy tailed structure of the probability distribution many standard deviations away from the mean.

We derive an analytical approximation to the probability distribution function (pdf) for the response of Mathieu׳s equation under parametric excitation by a random process with a spectrum peaked at the main resonant frequency, motivated by the problem of large amplitude ship roll resonance in random seas. The inclusion of random stochastic excitation renders the otherwise straightforward response to a system undergoing intermittent resonances: randomly occurring large amplitude bursts. Intermittent resonance occurs precisely when the random parametric excitation pushes the system into the instability region, causing an extreme magnitude response. As a result, the statistics are characterized by heavy-tails. We apply a recently developed mathematical technique, the probabilistic decomposition-synthesis method, to derive an analytical approximation to the non-Gaussian pdf of the response. We illustrate the validity of this analytical approximation through comparisons with Monte-Carlo simulations that demonstrate our result accurately captures the strong non-Gaussianinty of the response.