# First passage problem: Duffing equation

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The first-passage problem of a zero-start random process with respect to single-sided, double-sided, and energy type barriers is investigated via large scale Monte Carlo simulation. The random process is sampled from the response of a Duffing oscillator excited by zero-mean Gaussian white noise. The theoretical justification of the Monte Carlo method is given, and the error bound of the method is derived. The exact stationary second and fourth moments of the response displacement are solved from the FPK equation.

A computer code written in C programming language is implemented on the VAX 8650 super-minicomputer. A fourth order Runge-Kutta scheme is used to numerically integrate the Duffing equation to the white noise input. The response is monitored, and the time of first-passage crossing is recorded. The simulation is repeated up to 200,000 times.

This data is used to generate the first-passage survival probability, first-passage probability density function, and stationary decay-rate ratio.' Also, the non-stationary moment responses are collected from the simulation to reveal the characteristics of the random response of the oscillator. They are also applied to the Gram-Charlier expansion to obtain the response displacement distribution. Finally, from the first-passage probability density function, a curve fitting model is given. Some trends in this function are observed, and a relation regarding the first-passage probability density function is postulated.