### Time-varying dynamical networks

- Series
- Mathematical Biology Seminar
- Time
- Wednesday, November 28, 2012 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles Bldg Rm.005
- Speaker
- Igor Belykh – Georgia State – belykh@math.gsu.edu

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.