Dynamics of Stochastically Blinking

We study stochastically blinking dynamical systems as in the companion paper (Part I). We analyze the asymptotic properties of the blinking system as time goes to infinity. The trajectories of the averaged and blinking system cannot stick together forever but the trajectories of the blinking system may converge to an attractor of the averaged system. There are four distinct classes of blinking dynamical systems. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or non-invariance under the dynamics of the blinking system. In the case of invariance, we prove that the trajectories of the blinking system converge to the attractor(s) of the averaged system with high probability if switching is fast. In the noninvariant single attractor case, the trajectories reach a neighborhood of the attractor rapidly and remain close most of the time with high probability when switching is fast. In the non-invariant multiple attractor case, the trajectory may escape to another attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific example of a blinking dynamical system.