Physical systems open to a flow of energy can exhibit spontaneous symmetry breaking and self-organization. These nonequilibrium self-organized systems are known as dissipative structures. We study the oscillatory mode of an electrically driven dissipative structure. Our system consists of aluminum beads in shallow oil, which, when subjected to a high voltage, self-organize into connected 'tree' structures. The tree structures serve as pathways for the conduction of charge to ground. This system shows a variety of spatio-temporal behaviors, such as oscillating movement of the tree structures. Utilizing a dynamical systems model of the electromagnetic phenomena, we explore a potential mechanism underlying the system's behavior and use the model to make additional empirical predictions. The model reproduces the oscillatory behavior observed in the real system, and the behavior of the real system is consistent with predictions from the model under various constraints. From the empirical results and the mathematical model, we observe a tendency for the system to select modes of behavior with increased dissipation, or higher rates of entropy production, in accord with the proposed Maximum Entropy Production (MEP) Principle.