Short-term instability in stochastic systems

Abstract

Dynamic systems with lumped parameters which experience random temporal variations are considered in this paper. These variations may lead to “short-term” dynamic instability that is reflected in the system’s response as alternating periods of zero or almost zero response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, the corresponding response should be analyzed to evaluate the system’s reliability. Results of such analyses are presented separately for cases of slow and rapid parameter variations. Linear models of the systems are studied in the former case using parabolic approximation (PA) for the variations in the vicinity of their peaks together with Krylov–Bogoliubov (KB) averaging for the transient response. This results in a solution for the response probability density function (PDF). The case of rapid broadband parameter variations is studied using theory of Markov processes. The system is assumed to operate beyond its stochastic instability threshold–although only slightly–and its nonlinear model is used accordingly. The analysis is based on solution of the Fokker–Planck–Kolmogorov (FPK) partial differential equation for stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the origin indicating an intermittent nature of the response.