Marginal Instability and Intermittency in Stochastic Systems

Dynamic systems with lumped parameters which experience random temporal variations are considered. The variations may "smear" boundary between the system's states which are dynamically stable and unstable in the classical sense. The system's response within such a "twilight zone" of marginal instability is found to be of an intermittent nature, with 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 for the variations in the vicinity of their peaks together with Krylov-Bogoliubov averaging for the transient response. This results in a solution for the response probability density function (PDF). The analysis is also used to derive on-line identification procedure for the system from its observed response with set of rare outbreaks. Potential examples of applications include ID and 2D short-term galloping of elastically suspended bodies in cross-flow of fluid with random temporal variations of flow speed; bundles of heat exchanger tubes in cross-flow with potential for Flutter-type instability; and rotating shafts. 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. One of potential applications is population dynamics where behaviour of predator-prey (or parasite-host) pair in random environment is studied using extended stochastic Lotka-Volterra model. The analysis provides potential for probabilistic predictions of response outbreaks, in particular for the cases of intermittency (like the notorious case of seven outbreaks in budworms (forest parasites) in eastern Canada since 1710).