Global attractors and the difficulty of synchronizing serial spin-torque oscillators

Previously, synchronization of spin-torque oscillator (STO) has usually been analyzed by phase oscillator model. Here we show that STO is more precisely described as a perturbed heteroclinic cycle where the frequency is sensitive to the distance between the cycle and a saddle. In the presence of external signal or coupling, the frequency changes mainly due to the modification on this distance rather than on the phase directly. Multiple attractors coexist and synchronization depends sensitively on the initial conditions when a STO is driven by an external ac. We reveal that these properties underlay the mechanisms why synchronization region in two serially connected STOs is quite small and why time delay can enhance synchronization. When more STOs are added, the parameter region for a globally attracting synchronization state may disappear. Our analysis suggests that initial conditions have to be controlled or different designs have to be fabricated in order to obtain robust synchronization of a large number of STOs for the purpose of applications.