Magnetization dynamics in a three-layered nanopillar Co/Cu/Co structure with one fixed and one free layer driven by external magnetic field and spin-polarized electric current was investigated using methods of the theory of bifurcations. Mathematical model is based on the Landau-Lifshits-Gilbert equation with the current term in the Sloncžewski–Berger form. Orientation of applied magnetic field was considered to be parallel to the anisotropy axis. Physical model included the magnetocrystalline anisotropy field and the demagnetizing field. Because of small size of the structure, the space dependence of magnetization, as usually, was not taken into account. The resulting system of equations has the form of the nonlinear dynamical system with the polynomial right sides. Mathematical simulation of magnetization dynamics for several typical values of field and current was performed. The numerical experiments revealed the features of switching process in more detail and permitted to find new regimes of magnetization dynamics, such as incomplete and accidental switching.