Computational fluid dynamic simulations are providing new insights into the connections between patient-specific hemodynamic flows and the initiation, progression and treatment of cerebral aneurysms. Recent studies linking aneurysm rupture to the formation of vortices have motivated the need for a more fundamental understanding of swirling blood flow patterns and their evolution during the cardiac cycle. In this talk, I describe how dynamical systems theory is being used to advance our knowledge of vortex dynamics within cerebral aneurysms.