# Absence of chaos in Digital Memcomputing Machines with solutions

## TL;DR

This paper proves that digital memcomputing machines with solutions do not exhibit chaos, as their dynamical properties prevent strange attractors and chaotic behavior, ensuring predictable convergence to solutions.

## Contribution

It demonstrates that DMMs with solutions cannot support chaos due to their topological and dynamical properties, confirming a previous conjecture.

## Key findings

- No strange attractors coexist with solutions in DMMs.
- DMMs with solutions lack topological transitivity and chaos.
- Without solutions, DMMs only have invariant tori or periodic orbits.

## 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).