Exploring the behavior of systems under random disturbances, this research presents analytical expressions for the time-dependent and stationary probability distributions corresponding to a stochastically perturbed one-dimensional flow with critical points. The study considers two physically relevant situations: delayed evolution, where the flow alternates with a quiescent state, and interrupted evolution, where the variable resets to a random value. For delayed evolution, the impact of delay on first passage time statistics is analyzed. For interrupted evolution, conditions for an extended stationary distribution due to the competition between an attractor and random resetting are examined. The role of the normalization condition in eliminating singularities from unstable critical points is elucidated with examples. The stationary distribution is physically interpreted, offering insights into dichotomous flows.
This article, published in Stochastics and Dynamics, a journal covering stochastic processes and dynamical systems, fits perfectly within the journal's focus. It advances the understanding of how random perturbations impact the behavior of dynamical systems, providing valuable analytical tools for researchers in this area.