Periodic stochastic responses of nonlinear systems under combined harmonic and Poisson white noise excitations

This study proposes a path integration framework to investigate the periodic response evolution of nonlinear dynamical systems subjected to combined harmonic and Poisson white noise excitations. To address the problem of sharp transition probability density functions induced by Poisson white noise, the variable substitution and mapping techniques are introduced to enhance the accuracy of the probability density function. For the periodic response analysis, a decomposition strategy is developed to reconstruct the multi-step transition probability density functions within a full period, deviating from the traditional single-step approach. These functions are subsequently incorporated into the Chapman–Kolmogorov equation for numerical iteration, enabling the derivation of time-dependent probability density functions for different period phases. The methodology is validated through two representative stochastic systems: one under external harmonic excitation and the other under parametric harmonic excitation. The underlying mechanism of two different excitation modes on the system response is discussed, and the correctness of the results is verified by comparing them with Monte Carlo simulation results.