Cellular Automata Model of Synchronization in Coupled Oscillators

TL;DR

This paper introduces a cellular automata model for coupled oscillators that captures various synchronization states, phase transitions, and potential chimera states, providing insights into collective dynamics with simple integer-based rules.

Contribution

The paper presents a novel cellular automata framework for modeling synchronization in coupled oscillators, including analysis of phase diagrams and chimera state criteria.

Findings

Identified asynchronous, synchronized, and intermediate states.

Discovered power law distribution of cluster sizes with parameter-independent exponent.

Developed analytical criterion for chimera state persistence.

Abstract

We have developed a simple cellular automata model for nonlinearly coupled phase oscillators which can exhibit many important collective dynamical states found in other synchronizing systems. The state of our system is specified by a set of integers chosen from a finite set and defined on a lattice with periodic boundary conditions. The integers undergo coupled dynamics over discrete time steps. Depending on the values of coupling strength and range of coupling, we observed interesting collective dynamical phases namely: asynchronous, where all the integers oscillate incoherently; synchronized, where all integers oscillate coherently and also other states of intermediate and time-dependent ordering. We have adapted conventional order parameters used in coupled oscillator systems to measure the amount of synchrony in our system. We have plotted phase diagrams of these order parameters in the plane of strength of coupling and the radius of coupling. The phase diagrams reveal interesting properties about the nature of the synchronizing transition. There are partially ordered states, where there are synchronized clusters which are shown to have a power law distribution of their sizes. The power law exponent is found to be independent of the system parameters. We also discuss the possibility of chimera states in this model. A criterion of persistence of chimera is developed analytically and compared with numerical simulation.