Directed cycles and multi-stability of coherent dynamics in systems of coupled nonlinear oscillators

We analyse the dynamics of networks of coupled nonlinear systems in terms of both topology of interconnections as well as the dynamics of individual nodes. Here we focus on two basic and extremal components of any network: chains and cycles. In particular, we investigate the effect of adding a directed feedback from the last element in a directed chain to the first. Our analysis shows that, depending on the network size and internal dynamics of isolated nodes, multiple coherent and orderly dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state an attracting rotating wave solution occurs. The basin of attraction of this solution apparently grows with the number of nodes in the loop. The effect is observed in networks exceeding a certain critical size. Emergence of the attracting rotating wave solution can be viewed as a “topological bifurcation” of network dynamics in which removal or addition of a single connection results in dramatic change of the overall coherent dynamics of the system.