Reliability of dynamical systems with combined Gaussian and Poisson white noise via path integral method

Analyzing the reliability of a stochastic dynamical system represents one of the most challenging problems in stochastic dynamics. In this paper, the reliability of dynamical systems excited by combined Gaussian and Poisson white noise is investigated by adapting the path integral (PI) method. We derive the PI solutions for the reliability based on the reliability density function (RDF), where the short-time transition probability density function (TPDF) for the PI method is derived based on the Fourier transformation. And the impact of the combined noise on the reliability is analyzed theoretically for the one-dimensional case. We prove that the reliability for the system excited by combined noise will approach that for the system excited by Gaussian white noise when the mean arrival rate goes to infinite and jump amplitude for the Poisson white noise satisfies certain conditions. Then, one-dimensional and single-degree-of-freedom (SDOF) dynamical systems excited by combined Gaussian and Poisson white noise are worked out to show the application of the PI method on the reliability. Reliability under different system parameters is calculated and presented. The roles of jump amplitude with different distributions on the reliability are calculated and compared, which verifies the theoretical results. Moreover, it is discovered that the first passage of the system is more easily caused by the Gaussian white noise compared with the combined Gaussian and Poisson white noise when the noise intensity is the same.