Nonlinear delay difference equation with bistability as a new traffic flow model

Abstract

We propose an improved version of the Rule-184 fuzzy cellular automaton model for describing the key properties of real-world traffic flow. A crucial aspect of modeling involves incorporating the time delay caused by vehicles’ inertia and reaction time in a nonlinear difference equation. Specifically, the effect of time delay was incorporated into the model’s transition rates as an internal form. We then conducted a bifurcation analysis to investigate whether this model has important properties for representing traffic flow. The analysis results indicate that the proposed model exhibits bistability at a certain density region, in which both free and jamming flows coexist. Furthermore, the uniform solution was stable at all densities depending on the values of the parameters included in the model.