  Periodically driven parametric oscillators offer a convenient way to simulate
classical Ising spins. When many parametric oscillators are coupled
dissipatively, they can be analogous to networks of Ising spins, forming an
effective coherent Ising machine (CIM) that efficiently solves computationally
hard optimization problems. In the companion paper, we studied experimentally
the minimal realization of a CIM, i.e. two coupled parametric oscillators [L.
Bello, M. Calvanese Strinati, E. G. Dalla Torre, and A. Pe'er, Phys. Rev. Lett.
123, 083901 (2019)]. We found that the presence of an energy-conserving
coupling between the oscillators can dramatically change the dynamics, leading
to everlasting beats, which transcend the Ising description. Here, we analyze
this effect theoretically by solving numerically and, when possible,
analytically the equations of motion of two parametric oscillators. Our main
tools include: (i) a Floquet analysis of the linear equations, (ii) a
multi-scale analysis based on a separation of time scales between the
parametric oscillations and the beats, and (iii) the numerical identification
of limit cycles and attractors. Using these tools, we fully determine the phase
boundaries and critical exponents of the model, as a function of the intensity
and the phase of the coupling and of the pump. Our study highlights the
universal character of the phase diagram and its independence on the specific
type of nonlinearity present in the system. Furthermore, we identify new phases
of the model with more than two attractors, possibly describing a larger spin
algebra.
