Description
The Spin-Torque Nano Oscillator (STNO) has shown the potential to be a nano-sized voltage oscillator operating in the microwave range. The power output measured in experiments is however, rather small, about 1 nW. To generate a more powerful signal, several groups have proposed to harness the power of a network of coupled STNOs oscillating in synchrony. Achieving synchronization has proven to be rather dicult for small arrays while in larger ones the task of synchronization has eluded theorists and experimentalists altogether. In this work we explore in more detail the nature of the bifurcations that lead into and out of the synchronization state for two series-arrayed STNOs. The bifurcation analysis shows bistability between in-phase and out-of-phase limit cycle oscillations. In fact, there are two distinct pairs of such cycles. But as the input current increases the limit cycles increase their amplitudes until they merge with one another in a gluing bifurcation. More importantly, we show that changing the direction of the applied magnetic eld can, in principle, increase the region of synchronized oscillations. To analyze larger arrays, analytic expressions are produced for loci of the symmetry-preserving Hopf bifurcations that spawn synchronized oscillations in a series-array of STNOs of arbitrary size N. The symmetry of the array is exploited to nd implicit analytic expressions for the Hopf curves that form the boundary of the synchronization state. Through stability analysis, loci are mapped showing where the Hopf is both supercritical and possesses a stable center-manifold. Results illustrate the ideal regions of operation of large arrays, up to N = 1000 nano valves, which should yield considerable higher microwave output. To address the issue of bistability, we use invariant manifold covering techniques to approximate the separatrices between basins of attraction in small arrays. A brief survey of methods critiques several existing methods and evaluates their applicability to higher dimension manifolds. This is followed by development of methods to dene the saddle-type special sets required to initialize the invariant manifold routines. A saddle-type torus is found to be a key characteristic of the separatrix `between' the synchronized and out-of-phase limit-cycles.
