Using nonlinear dynamical systems theory, we analytically studied a spin-torque device in which the magnetization of the polarizer (the fixed layer) is tilted at an arbitrary angle out of the thin-film plane. While the analytical theory can determine the major features of the system, macrospin simulations were employed to demonstrate the unique characteristics of the system, such as the hysteretic switching between bistable states. Material dependencies of the dynamic and static state diagrams were also studied in the framework of the macrospin model. Full-scale micromagnetics simulations were finally performed to reveal more subtle features of the dynamics of such tilted polarizer systems. Both the macrospin and micromagnetics simulations gave quantitatively the same results as our analytical theory.