Dynamics of macrospin under periodic field and spin transfer torque

The dynamics of a macrospin model for a single magnetic domain is investigated in two cases: (i) under the action of a periodic magnetic field and (ii) under the external field plus a spin transfer torque from spin-polarized current or spin current. It has been proved that (i) without spin transfer torque, the trajectory of magnetization (hysteresis) is always a closed curve oscillating between two stable points and following the same path each time; (ii) under the action of a constant field (or with small periodic perturbation) and a constant spin transfer torque, the spin would always turn to the stable solution finally, which is independent of the parameters or initial conditions; (iii) considering a periodic oscillating field plus a spin transfer torque, the system can also have a stable solution around a limit circle, but on some special points on the attractor, the system may be sensitive to the initial conditions and have no limit circle.