A two-state model for noise induced resonance in bistable systems with delay

We discuss a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise, exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one-dimensional parametrized stochastic differential equation with point delay, basic properties whereof we check. With the aim of capturing the effective dynamics and, in particular, the resonance-like behavior of the reference model, we construct a simplified or reduced model, the two-state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance. Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two-state model and the reference model. The resonance characteristics developed for the reduced model can thus be applied to the original model.