The solar-wind-driven magnetosphere-ionosphere exhibits a variety of dynamical states including low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionospheric known broadly as substorms, and states of continuous strong unloading. The WINDMI model [J. P. Smith et al., J. Geophys. Res. 105, 12 983 (2000)] is a six-dimensional substorm model that uses a set of ordinary differential equations to describe the energy flow through the solar wind-magnetosphere-ionosphere system. This model has six major energy components, with conservation of energy and charge described by the coupling coefficients. The six-dimensional model is investigated by introducing reductions to derive a new minimal three-dimensional model for deterministic chaos. The reduced model is of the class of chaotic equations studied earlier [J. C. Sprott, Am. J. Phys. 68, 758 (2000)]. The bifurcation diagram remains similar, and the limited prediction time, which is in the range of three to five hours, occurs in the chaotic regime for both models. Determining all three Lyapunov exponents for the three-dimensional model allows one to determine the dimension of the chaotic attractor for the system. .