19,850 research outputs found
Stability and bifurcation in network traffic flow: A Poincar\'e map approach
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 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.Comment: 31 pages, 10 figures, 2 table
Continuous formulations and analytical properties of the link transmission model
The link transmission model (LTM) has great potential for simulating traffic
flow in large-scale networks since it is much more efficient and accurate than
the Cell Transmission Model (CTM). However, there lack general continuous
formulations of LTM, and there has been no systematic study on its analytical
properties such as stationary states and stability of network traffic flow. In
this study we attempt to fill the gaps. First we apply the Hopf-Lax formula to
derive Newell's simplified kinematic wave model with given boundary cumulative
flows and the triangular fundamental diagram. We then apply the Hopf-Lax
formula to define link demand and supply functions, as well as link queue and
vacancy functions, and present two continuous formulations of LTM, by
incorporating boundary demands and supplies as well as invariant macroscopic
junction models. With continuous LTM, we define and solve the stationary states
in a road network. We also apply LTM to directly derive a Poincar\'e map to
analyze the stability of stationary states in a diverge-merge network. Finally
we present an example to show that LTM is not well-defined with non-invariant
junction models. We can see that Newell's model and LTM complement each other
and provide an alternative formulation of the network kinematic wave model.
This study paves the way for further extensions, analyses, and applications of
LTM in the future.Comment: 27 pages, 5 figure
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