Exact Solutions of Certain Time Delay Systems: The Car-Following Models

Abstract

In spite of the complex dynamics exhibited by even the simplest of nonlinear time delay systems, there exists a host of coupled nonlinear time delay systems which admit exact solutions. Particularly, certain coupled systems of nonlinear delay differential equations modelling traffic flow [1–3],, called the car following models, possess exact analytic solutions in terms of Jacobian elliptic functions under periodic boundary conditions. However, under open boundary conditions, they admit shock-like solutions, representing the stationary propagation of a traffic jam [2, 3]. We will closely follow here the approach of Tutiya and Kanai [4] in the following discussion just to illustrate how exact solutions can arise in delay systems.