  The functions of many networked systems in physics, biology or engineering
rely on a coordinated or synchronized dynamics of its constituents. In power
grids for example, all generators must synchronize and run at the same
frequency and their phases need to appoximately lock to guarantee a steady
power flow. Here, we analyze the existence and multitude of such phase-locked
states. Focusing on edge and cycle flows instead of the nodal phases we derive
rigorous results on the existence and number of such states. Generally,
multiple phase-locked states coexist in networks with strong edges, long
elementary cycles and a homogeneous distribution of natural frequencies or
power injections, respectively. We offer an algorithm to systematically compute
multiple phase- locked states and demonstrate some surprising dynamical
consequences of multistability.
