Shifts in global network dynamics due to small changes at single nodes

Understanding the sensitivity of a system's behavior with respect to parameter changes is essential for many applications. This sensitivity may be desired—for instance, in the brain, where a large repertoire of different dynamics, particularly different synchronization patterns, is crucial—or may be undesired—for instance, in power grids, where disruptions to synchronization may lead to blackouts. In this paper, we show that networks of coupled phase oscillators with nonlinear interactions can acquire a very large and complicated sensitivity to changes made in either their units' parameters or in their connections. Even modifications made to a parameter of a single unit can radically alter the global dynamics of the network in an unpredictable manner. This occurs over a wide parameter region, around the network's transitions to phase synchronization. We argue that this is a widespread phenomenon that can be expected in real-world systems, extending even beyond networks of oscillators.