Georgia Institute of Technology College of Sciences

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenario.  By studying the asymptotics of the Markov perturbed stationary distributions, probabilistic characterizations of the alternating pattern between synchronization and desynchronization behaviors is given.  More precisely, it is shown that if a random network without intrinsic noise perturbation is synchronized, then after intrinsic noise perturbation high-probability synchronization and low-probability desynchronization can occur intermittently and alternatively in time, and moreover, both the probability of (de)synchronization and the proportion of time spent in (de)synchrony can be explicitly estimated.

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss a characterization of Anosovity based on Reeb flows and its consequences.

The  theory of nonautonomous dynamical systems has undergone  major development during the past 23 years since I talked  about attractors  of nonautonomous difference equations at ICDEA Poznan in 1998. 

Two types of  attractors  consisting of invariant families of  sets   have been defined for  nonautonomous difference equations, one using  pullback convergence with information about the system   in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant  family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and  not essential for determining future behaviour.  

The forward  asymptotic  behaviour can also be described through the  omega-limit set  of the  system.This set  is closely  related to what Vishik  called the uniform attractor although it need not be invariant. It  is  shown to be asymptotically positively invariant  and also, provided  a future uniformity condition holds, also asymptotically positively invariant.  Hence this omega-limit set provides useful information about  the behaviour in current  time during the approach to the future limit. 

In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. No knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (UT Austin, Ben-Gurion U.).