Synchronization of Spin-Torque Oscillators via Continuation Method

In this article, we study synchronization phenomena of spin-torque oscillators coupled on a ring. Spin-torque oscillators are nanoelectronic devices that promise efficient microwave generation provided they are synchronized in large arrays. Due to their nonlinear and nonisochronous nature, their synchronization properties are difficult to analyze explicitly. To address this challenge, we employ a recently developed continuation method and transform the network of coupled oscillators (each described by an ordinary differential equation) into a single partial differential equation (PDE). We then analyze the synchronization of this PDE in two cases: when all the oscillators are identical and when there are two different types of oscillators. In the case of identical oscillators, we characterize all possible synchronous solutions and provide necessary and sufficient conditions for stability. For nonidentical oscillators, we derive and solve a differential synchronization condition, which allows us to accurately reconstruct the shape of the equilibrium profiles. All the presented results are derived for the PDE and then validated by numerical simulations of the original network of ordinary differential equations (ODEs).