Transition-event duration in one-dimensional systems under correlated noise

Transition-event duration (TED) is the spent time of a particle which enters a neighborhood of one state to reach another one without ever returning back to the state where it starts. In this paper, an approximate theoretical method is proposed to obtain the transition-event duration distribution (TEDD) of one-dimensional bi-stable strongly nonlinear systems under correlated noises. It strongly extends the existing theoretical methods for obtaining the TEDD. Actually, an approximate theoretical solution of the TEDD is determined by the Fokker–Planck equation (FPE) with two absorbing boundary conditions. Here, we set each absorbing boundary condition at stable and unstable points of a bi-stable potential function. Firstly, a solution to the FPE is presented by the C-type Gram–Charlier expansion. Subsequently, based on the weighted residual scheme, the FPE is reduced to a series of ordinary differential equations (ODEs) with unknown coefficient. Then, an analytical solution to TEDD is derived by solving this series of ODEs. Especially, the concerning stochastic excitations of the system can be divided into five kinds of situations: Gaussian white noise, colored Gaussian noise, two Gaussian white noises with white cross-correlation, two Gaussian white noises with exponential correlation and two colored Gaussian noises with exponential correlation. We mainly discuss the influences of correlation and the parameters of noise on the TEDD. Our corresponding analytic results are compared with Monte Carlo simulations, and good agreements are found.