Anticipation in discrete-time networked systems

Abstract

We investigate networks of coupled dynamical systems in discrete time where the units anticipate the states of their neighbors and try to align their states accord ingly. Anticipation is done using past state information and hence introduces a memory e↵ect in the form of a time delay. We show that, under specific conditions depending on the network structure, the system converges to consensus faster un der anticipation. On the other hand, the delays introduced by anticipation can also break consensus altogether under certain other conditions. One of the main results of this study is the derivation of necessary and su cient conditions on the parameters of the anticipation algorithm for faster consensus. We apply the results to coupled nonlinear systems and analyze the stability of synchronized states under anticipation. We show that anticipation can induce synchronization in networks of chaotic maps, which may otherwise be di cult to synchronize.