A convolution based path integration method approach to the damped parametric pendulum under different random noise excitations

This paper is devoted to applying the path integration (PI) method to analyze the whole space dynamic behaviors of the damped parametric pendulum under different random noises. The random noises considered in this paper include external Gaussian white noise, external Poisson white noise and internal narrow-band noise, which are three typical approximated noises in Engineering. It's a challenging and variable engineering task, since the system is non-homogenous, unbounded and may even lead to high dimensional cases. The convolution operation is an efficient way to accelerate the PI. However, the PI method with convolution operation is considered to have great limitation in handling the unbounded system under non-Gaussian noises. In order to extend the convolution operation to the unbounded rolling case, several special treatments in the back-stepping Runge-Kutta(RK) iterations, high space interpolations and the convolution operations are proposed. After above modification, a series of probability density functions(PDFs) of the system can be efficiently obtained via the convolution based PI method. Complexity analysis shows that the convolution based PI method is extremely efficient in obtaining the smooth transient PDF after reaching periodicity and 3-D stationary PDFs, while the Monte Carlo simulation(MCS) method is only suitable for calculating the time averaged PDF and high dimensional marginal PDF. With the help of the convolution based PI method, the influences of different noises on the damped parametric pendulum system are discussed comprehensively. The 3-D PDFs are novelly depicted in the form of point cloud map, which shows the convolution based PI a strong grasp of the details and the subtle information in the whole space PDF.