  Dynamical Ising machines are continuous dynamical systems evolving from a
generic initial state to a state strongly related to the ground state of the
classical Ising model on a graph. Reaching the ground state is equivalent to
finding the maximum (weighted) cut of the graph, which presents the Ising
machines as an alternative way to solving and investigating NP-complete
problems. Among the dynamical models driving the Ising machines,
relaxation-based models are especially interesting because of their relations
with guarantees of performance achieved in time scaling polynomially with the
problem size. However, the terminal states of such machines are essentially
non-binary, which necessitates special post-processing relying on disparate
computing. We show that an Ising machine implementing a special dynamical
system (called \mdII{}) solves the rounding problem dynamically. We prove that
the \mdII-machine starting from an arbitrary non-binary state terminates in a
state, which trivially rounds to a binary state with the cut at least as big as
obtained after the optimal rounding of the initial state. Besides showing that
relaxation-based dynamical Ising machines can be made self-contained, our
findings demonstrate that dynamical systems can directly perform complex
information processing tasks.
