  Nonlinear dynamical systems such as coupled oscillators are being actively
investigated as Ising machines for solving computationally hard problems in
combinatorial optimization. Prior works have established the equivalence
between the global minima of the Lyapunov function (commonly referred to as the
energy function) describing the coupled oscillator system and the ground state
of the Ising Hamiltonian. However, the properties of the oscillator Ising
machine (OIM) from a nonlinear control viewpoint, such as the stability of the
OIM solutions remains unexplored. Therefore, in this work, using nonlinear
control-theoretic analysis, we (i) Identify the conditions required to ensure
the functionality of the coupled oscillators as an Ising machine; (ii) Show
that all globally optimal phase configurations may not always be stable,
resulting in some configurations being more favored over others, and thus,
creating a biased OIM; (c) Elucidate the impact of the stability of locally
optimal phase configurations on the quality of the solution computed by the
system. Our work, fostered through the unique convergence between nonlinear
control theory and analog systems for computing, provides a new toolbox for the
design and implementation of dynamical system-based computing platforms.
