  Understanding traffic statics and dynamics in urban networks is critical to
develop effective control and management strategies. In this paper, we provide
a novel approach to study the traffic statics and dynamics in a signalized
double-ring network, which can provide insights into the operation of more
general signalized traffic networks. Under the framework of the link queue
model (LQM) and the assumption of a triangular traffic flow fundamental
diagram, the signalized double-ring network is studied as a switched affine
system. Due to periodic signal regulations, periodic density evolution orbits
are formed and defined as stationary states. A Poincar\'{e} map approach is
introduced to analyze the properties of such stationary states. With short
cycle lengths, closed-form Poincar\'{e} maps are derived. Stationary states and
their stability properties are obtained by finding and analyzing the fixed
points on the Poincar\'{e} maps. It is found that a stationary state can be
asymptotically stable, Lyapunov stable, or unstable. The impacts of retaining
ratios and initial densities on the macroscopic fundamental diagrams (MFDs) and
the gridlock times are analyzed. Multivaluedness and gridlock phenomena as well
as the unstable branch with non-zero average network flow-rates are observed on
the MFDs. With long cycle lengths, fixed points on the Poincar\'{e} maps are
solved numerically, and the obtained stationary states and the MFDs are very
similar to those with short cycle lengths.
  Compared with earlier studies, this paper provides an analytical framework
that can be used to provide complete and closed-form solutions to the statics
and dynamics of double-ring networks. This can lead to a better understanding
of how the combination of signalized intersections and turning maneuvers is
expected to impact network properties, like the MFD.
