Dynamics of Stochastically Blinking Systems. Part I: Finite Time Properties

We consider dynamical systems whose parameters are switched within a discrete set of values at equal time intervals. Similar to the blinking of the eye, switching is fast and occurs stochastically and independently for different time intervals. There are two time scales present in such systems, namely the time scale of the dynamical system and the time scale of the stochastic process. If the stochastic process is much faster, we expect the blinking system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. We prove that, with high probability, the trajectories of the two systems stick together for a certain period of time. We give explicit bounds that relate the probability, the switching frequency, the precision, and the length of the time interval to each other. We discover the apparent presence of a soft upper bound for the time interval, beyond which it is almost impossible to keep the two trajectories together. This comes as a surprise in view of the known perturbation analysis results. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system.