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

## TL;DR

This paper develops an analytical approximation for the probability distribution of Mathieu's equation response under correlated stochastic excitation, capturing heavy-tailed, non-Gaussian behaviors relevant to ship roll resonance in random seas.

## Contribution

It introduces a novel application of the probabilistic decomposition-synthesis method to derive response distributions for Mathieu's equation with correlated parametric noise, addressing intermittent resonance phenomena.

## Key findings

- Analytical approximation accurately matches Monte-Carlo simulations.
- Response distribution exhibits heavy tails due to intermittent resonances.
- Method effectively captures non-Gaussian response characteristics.

## Abstract

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 analytic approximation through comparisons with Monte-Carlo simulations that demonstrate our result accurately captures the strong non-Gaussianity of the response.