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Understanding traffic statics and dynamics in urban networks is critical todevelop effective control and management strategies. In this paper, we providea novel approach to study the traffic statics and dynamics in a signalizeddouble-ring network, which can provide insights into the operation of moregeneral signalized traffic networks. Under the framework of the link queuemodel (LQM) and the assumption of a triangular traffic flow fundamentaldiagram, the signalized double-ring network is studied as a switched affinesystem. Due to periodic signal regulations, periodic density evolution orbitsare formed and defined as stationary states. A Poincar\'{e} map approach isintroduced to analyze the properties of such stationary states. With shortcycle lengths, closed-form Poincar\'{e} maps are derived. Stationary states andtheir stability properties are obtained by finding and analyzing the fixedpoints on the Poincar\'{e} maps. It is found that a stationary state can beasymptotically stable, Lyapunov stable, or unstable. The impacts of retainingratios and initial densities on the macroscopic fundamental diagrams (MFDs) andthe gridlock times are analyzed. Multivaluedness and gridlock phenomena as wellas the unstable branch with non-zero average network flow-rates are observed onthe MFDs. With long cycle lengths, fixed points on the Poincar\'{e} maps aresolved numerically, and the obtained stationary states and the MFDs are verysimilar to those with short cycle lengths.  Compared with earlier studies, this paper provides an analytical frameworkthat can be used to provide complete and closed-form solutions to the staticsand dynamics of double-ring networks. This can lead to a better understandingof how the combination of signalized intersections and turning maneuvers isexpected to impact network properties, like the MFD.

Analysis of traffic statics and dynamics in a signalized double-ring network: A Poincar\'{e} map approach