Randomly Switched Dynamical Systems: the Odds of Meeting a Ghost
Igor Belykh

This talk focuses on mathematical analysis 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 systems. 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
technological or biological network.