  We present a new way to make Ising machines, i.e., using networks of coupled
self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a
novel result that establishes that the phase dynamics of coupled oscillator
systems, under the influence of sub-harmonic injection locking, are governed by
a Lyapunov function that is closely related to the Ising Hamiltonian of the
coupling graph. As a result, the dynamics of such oscillator networks evolve
naturally to local minima of the Lyapunov function. Two simple additional steps
(i.e., adding noise, and turning sub-harmonic locking on and off smoothly)
enable the network to find excellent solutions of Ising problems. We
demonstrate our method on Ising versions of the MAX-CUT and graph colouring
problems, showing that it improves on previously published results on several
problems in the G benchmark set. Our scheme, which is amenable to realisation
using many kinds of oscillators from different physical domains, is
particularly well suited for CMOS IC implementation, offering significant
practical advantages over previous techniques for making Ising machines. We
present working hardware prototypes using CMOS electronic oscillators.
