The standard van Neumann computer excels at many things. However, it can be very inefficient in solving optimization problems with a large solution space. For that reason, a novel analog approach, the oscillator-based Ising machine, has been proposed as a better alternative for dealing with such problems. In this work, we review the concept of oscillator-based Ising machines. In particular, we address how optimization problems can be mapped onto such machines when the QUBO formulation is given. Furthermore, we provide an ideal circuit that can be used in combination with the wave digital concept for real-time simulated annealing. The functionality of this circuit is explained on the basis of a Lyapunov stability analysis. The latter also provides an answer for the question: when has the Ising machine solved a mapped problem? At the end, we provide emulation results demonstrating the correlation between functionality and stability of the discussed machine. These results show that mapping a problem onto an Ising machine effectively maps the solution of the problem onto an equilibrium of the phase space.