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The paper deals with the bifurcation analysis of a simple mathematical model describing an automobile running on an even surface. Bifurcation analysis is adopted as the proper procedure for an in-depth understanding of the stability of steady-state motion of cars (either cornering or running straight ahead). The aim of the paper is providing the fundamental information for inspiring further studies on vehicle dynamics with or without a human driver. The considered mechanical model of the car has two degrees of freedom, nonlinear tire characteristics are included. A simple driver model is introduced. Experimental validations of the model are produced. As a first step, bifurcation analysis is performed without driver (fixed control). Ten different combinations of front and rear tire characteristics (featuring understeer or oversteer automobiles) are considered. Steering angle and speed are varied. Many different dynamical behaviors of the model are found.