Georgia Institute of Technology College of Sciences

In this talk, we present recent results regarding the existence of invariant probability measures for delay equations with (stochastic) negative feedback. No prior knowledge on invariant measures is assumed. Applications include Nicholson's blowflies equation and the Mackey-Glass equations. Just like the dynamics of prime numbers, these systems exhibit "randomness" combined with deep structure. We will prove this both analytically and numerically and focus mainly on intuition. In general, additive noise typically destroys all dynamical properties of the underlying dynamical system. Therefore, we are motivated to study a class of stochastic perturbations that preserve some of the dynamical properties of the negative feedback systems we consider.