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The aim of this paper is to explain critical features of the human primary generalized epilepsies by investigating the dynamical bifurcations of a nonlinear model of the brain's mean field dynamics. The model treats the cortex as a medium for the propagation of waves of electrical activity, incorporating key physiological processes such as propagation delays, membrane physiology, and corticothalamic feedback. Previous analyses have demonstrated its descriptive validity in a wide range of healthy states and yielded specific predictions with regards to seizure phenomena. We show that mapping the structure of the nonlinear bifurcation set predicts a number of crucial dynamic processes, including the onset of periodic and chaotic dynamics as well as multistability. Quantitative study of electrophysiological data supports the validity of these predictions. Hence, we argue that the core electrophysiological and …