Structural and Symmetry Analysis of Discrete Dynamical Systems

TL;DR

This paper introduces new methods for analyzing various types of discrete dynamical systems, focusing on symmetries, compatibility, and quantum extensions, with applications to cellular automata and lattice models.

Contribution

It develops a framework of discrete relations on simplicial complexes, algorithms for symmetry analysis, and a novel approach to quantization using gauge connections in finite groups.

Findings

Symmetries relate to soliton-like structures in deterministic systems.

Algorithms effectively analyze symmetries and phase transitions.

Quantum models can embed into larger classical systems.

Abstract

To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop algorithms for analysis of compatibility and construction of canonical decompositions of such systems. To illustrate these techniques we describe their application to some cellular automata. Much attention is paid to study symmetries of the systems. In the case of deterministic systems, we reveale some important relations between symmetries and dynamics. We demonstrate that moving soliton-like structures arise inevitably in deterministic dynamical system whose symmetry group splits the set of states into a finite number of group orbits. We develop algorithms and programs exploiting discrete symmetries to study microcanonical ensembles and search phase transitions in mesoscopic lattice models. We propose an approach to quantization of discrete systems based on introduction of gauge connection with values in unitary representations of finite groups --- the elements of the connection are interpreted as amplitudes of quantum transitions. We discuss properties of a quantum description of finite systems. In particular, we demonstrate that a finite quantum system can be embedded into a larger classical system. Computer algebra and computational group theory methods were useful tools in our study.