The mathematical models used to describe the dynamical behavior of a group of road vehicles traveling in a single lane without overtaking are known as car-following models. These models are widely used in many commercially available microscopic traffic simulation software packages. They attempt to mimic the interactions between individual vehicles that are traveling sufficiently close together for the behavior of each vehicle to be dependent upon the motion of the vehicle immediately in front. In this paper we modify the traditional car-following model by adding a new nonlinear term to take account of the driver attempting to achieve a certain desired intervehicle separation distance as well as the traditional aim of matching the velocity of the vehicle ahead. Numerical solution of the resulting coupled system of nonlinear differential equations is used to analyze the stability of the equilibrium solution to a periodic perturbation. For certain parameter values chaotic oscillations are generated, consisting of a broad spectrum of frequency components. Such chaotic motion produces extremely complicated dynamical behavior that has an inherent lack of predictability associated with it. The results of simulating over a range of parameter values are presented and, where it is present, the degree of chaos is estimated.