# Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling

*2022, Volume 27, Number 6, pp. 697-712*

Author(s):

**Kuptsov P. V.**

A spin-transfer oscillator is a nanoscale device demonstrating self-sustained precession
of its magnetization vector whose length is preserved. Thus, the phase space of this
dynamical system is limited by a three-dimensional sphere. A generic oscillator is described
by the Landau – Lifshitz – Gilbert – Slonczewski equation, and we consider a particular case of
uniaxial symmetry when the equation yet experimentally relevant is reduced to a dramatically
simple form. The established regime of a single oscillator is a purely sinusoidal limit cycle
coinciding with a circle of sphere latitude (assuming that points where the symmetry axis
passes through the sphere are the poles). On the limit cycle the governing equations become
linear in two oscillating magnetization vector components orthogonal to the axis, while the
third one along the axis remains constant. In this paper we analyze how this effective linearity
manifests itself when two such oscillators are mutually coupled via their magnetic fields. Using
the phase approximation approach, we reveal that the system can exhibit bistability between
synchronized and nonsynchronized oscillations. For the synchronized one the Adler equation
is derived, and the estimates for the boundaries of the bistability area are obtained. The twodimensional
slices of the basins of attraction of the two coexisting solutions are considered. They
are found to be embedded in each other, forming a series of parallel stripes. Charts of regimes
and charts of Lyapunov exponents are computed numerically. Due to the effective linearity the
overall structure of the charts is very simple; no higher-order synchronization tongues except
the main one are observed.

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