  Previous studies have shown that, in a diverge-merge network with two
intermediate links (the DM network), the kinematic wave model always admits
stationary solutions under constant boundary conditions, but periodic
oscillations can develop from empty initial conditions. Such contradictory
observations suggest that the stationary states be unstable. In this study we
develop a new approach to investigate the stability property of traffic flow in
this and other networks. Based on the observation that kinematic waves
propagate in a circular path when only one of the two intermediate links is
congested, we derive a one-dimensional, discrete Poincar\'e map in the out-flux
at a Poincar\'e section. We then prove that the fixed points of the Poincar\'e
map correspond to stationary flow-rates on the two links. With Lyapunov's first
method, we demonstrate that the Poincar\'e map can be finite-time stable,
asymptotically stable, or unstable. When unstable, the map is found to have
periodical points of period two, but no chaotic solutions. Comparing the
results with those in existing studies, we conclude that the Poincar\'e map can
be used to represent network-wide dynamics in the kinematic wave model. We
further analyze the bifurcation in the stability of the Poincar\'e map caused
by varying route choice proportions. We further apply the Poincar\'e map
approach to analyzing traffic patterns in more general $(DM)^n$ and beltway
networks, which are sufficient and necessary structures for network-induced
unstable traffic and gridlock, respectively. This study demonstrates that the
Poincar\'e map approach can be efficiently applied to analyze traffic dynamics
in any road networks with circular information propagation and provides new
insights into unstable traffic dynamics caused by interactions among network
bottlenecks.

Dynamical Systems