title: Noise-induced switching and extinction in systems with delay.
abstract: We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.
authors: US Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section, Plasma Physics Division, Washington, DC 20375, USA.
authors: Ira B
authors: IB
authors: Schwartz
authors: Department of Mathematical Sciences, Montclair State University, Montclair, New Jersey 07043, USA.
authors: Lora
authors: L
authors: Billings
authors: Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA.
authors: Thomas W
authors: TW
authors: Carr
authors: Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA.
authors: M I
authors: MI
authors: Dykman