Braitenberg vehicles are well-known models of animal behavior used as steering mechanisms in mobile robotics and artificial life. Because of their simplicity, they are mainly used for teaching robotics, while the lack of a quantitative theory has limited their use for research purposes. This article contributes to our formal understanding of Braitenberg vehicle 3a by presenting the convergence properties of its trajectories under parabolic-shaped stimuli. We show previously unreported features of the motion of the vehicle: the conditional stability, the oscillatory behavior, and the existence of periodic trajectories. The mathematical model used provides a theoretical relation between the environment, the internal control mechanism of the vehicle, and some morphological parameters, a link already found in experimental works. This work provides theoretical support for experimental research using Braitenberg vehicle 3a, and paves the way for further research in biology, robotics, and artificial life.