Georgia Institute of Technology College of Sciences

This talk focuses on mathematical analysis and modeling of dynamical systems and networks whose coupling or internal parameters stochastically evolve over time. We study networks that are composed of oscillatory dynamical systems with connections that switch on and off randomly, and the switching time is fast, with respect to the characteristic time of the individual node dynamics. If the stochastic switching is fast enough, we expect the switching system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. There are four distinct classes of switching dynamical networks. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or non-invariance under the dynamics of the switching system. In the case of invariance, we prove that the trajectories of the switching system converge to the attractor(s) of the averaged system with high probability. In the non-invariant single attractor case, the trajectories rapidly reach a ghost attractor and remain close most of the time with high probability. In the non-invariant multiple attractor case, the trajectory may escape to another ghost 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 technological or biological network.