Title: Absence of chaos in Digital Memcomputing Machines with solutions,
Abstract: Digital memcomputing machines (DMMs) are non-linear dynamical systems
designed so that their equilibrium points are solutions of the Boolean problem
they solve. In a previous work [Chaos 27, 023107 (2017)] it was argued that
when DMMs support solutions of the associated Boolean problem then strange
attractors cannot coexist with such equilibria. In this work, we demonstrate
such conjecture. In particular, we show that both topological transitivity and
the strongest property of topological mixing are inconsistent with the point
dissipative property of DMMs when equilibrium points are present. This is true
for both the whole phase space and the global attractor. Absence of topological
transitivity is enough to imply absence of chaotic behavior. In a similar vein,
we prove that if DMMs do not have equilibrium points, the only attractors
present are invariant tori/periodic orbits with periods that may possibly
increase with system size (quasi-attractors).