Pattern formation in 2D traffic flows
Julien Cividini
Discrete & Continuous Dynamical Systems - S 2014, 7(3): 395-409 doi: 10.3934/dcdss.2014.7.395
We review numerical and analytical results that have been obtained for a general model of intersecting flows of two types of particles propagating towards east ($\varepsilon $) and north ($\mathcal{N} $) on a bidimensional $M \times M$ square lattice. The behaviour of this model can also be reproduced by a system of mean field equations. The low density behaviour of both models is studied, with a focus on pattern formation. Using periodic boundary conditions, particles self-organize into a pattern of alternating diagonal stripes, which corresponds to an instability in the mean-field equations. With open boundary conditions, translational symmetry is broken. One then observes an asymmetry between the organization of the two types of particles, leading to tilted diagonals whose angle of inclination differs from $45^\circ$, both for the particle system and the equations. The angle of inclination of the stripes is measured using two different numerical methods. Finally, simplified theoretical arguments based on a particular mode of propagation give a quantitative estimate for the angle of the stripes. A complementary understanding of the phenomenon in terms of effective interactions between particles is presented.
keywords: self-propelled particles pattern formation granular systems Traffic models cellular automata.

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