This paper explores the dynamical range of the WINDMI model (Horton and Doxas, 1998>. Such low-dimensional models provide us with the tools to understand the relationships of simple physical quantities within the magnetosphere (such as energy deposition, macroscopic currents, cross-tail voltage, etc.) without the necessity of coping with the more complete but unwieldy models (MHD or particle codes, for example). The model is highly versatile: certain regions of the parameter space support stable fixed points, while others contain periodic states that exhibit period doubling, and sometimes chaos. States in each of these regimes (stable, periodic, chaotic) are investigated for their ability to accurately describe the observed properties of the magnetosphere-ionosphere system. A brief discussion of applications of this model to current space physics problems is included. ¿ 2000 American Geophysical Union