  At present, the Kuramoto model is the standard and widely accepted
theoretical approach for analyzing the synchronization of spin-torque
nano-oscillators (STNOs) coupled by an interaction. Nevertheless, the
oscillatory decaying regime as well as the initial condition (IC)-dependence
(hysteretic) that exist in the synchronization of many types of STNOs cannot be
explained by this model. In order to more precisely elucidate the physical
mechanisms behind the two phenomena, in this paper we develop a generalized
pendulum-like model based on the two common features of non-linear
auto-oscillators: one is the stability of the amplitude/energy of dynamic
states; the other is the non-linear dynamic state energy of oscillators. In
this new model, we find that the Newtonian-like particle with sufficient
kinetic energy can overcome the barrier of phase-locking potential to evolve
into a stable asynchronization (AS) state, leading to the (IC)-dependent
synchronization. Furthermore, due to the presence of the kinetic energy, this
particle can also oscillate around the minima of the phase-locking potential,
leading to the oscillatory decaying regime. Thereby, in this work, we adopt
this new model to analyze the IC-dependent mutual synchronization of
perpendicularto-plane (PERP)-STNO pairs, and then we suggest that the initial
conditions can be controlled to avoid such a phenomenon by using magnetic
dipolar coupling.
