# Dynamic analysis in Greenberg's traffic model

## TL;DR

This paper conducts a detailed dynamical analysis of Greenberg's traffic model by varying the critical velocity, revealing complex behaviors such as stability, limit cycles, and chaos through phase-plane maps and simulations.

## Contribution

It introduces a novel nonlinear approach to analyze Greenberg's traffic model, providing new insights into traffic variable behaviors and their qualitative and quantitative interactions.

## Key findings

- Identification of stable, oscillatory, and chaotic trajectories.
- Development of phase-plane maps for density, flow, and velocity.
- Demonstration of complex traffic dynamics through simulations.

## Abstract

Based on the classical traffic model by Greenberg, a linear differential equation, we analyze it by means of varying the critical velocity $v_o$ that appears in it as a parameter. In order to make such analysis we have obtained a solution for such a model and discretized it, obtaining related expressions for density $k$, flow $q$ and velocity $v$ to be treated as paired functions to obtained maps in phase-planes in which it is possible to observe distinct behaviors which span from monotonic and oscillatory stable trajectories, limit cycles of distinct periodicity, and chaotic ones. These behaviors are analyzed from a dynamical approach and then ilustrated with simulations performed in each case. As it is shown in this paper, these analyses are similar to those carried out in similar though simpler expressions (i.e. logistic-type functions), but taking in this case a new and direct approach through a nonlinear expression not used before to perform studies like these presented in this document, with a deep detail in the manner in which traffic variables are involved qualitatively and quantitatively.