Local attractors, degeneracy and analyticity: symmetry effects on the locally coupled Kuramoto model

TL;DR

This paper analyzes the symmetry effects in the locally coupled Kuramoto model, revealing analytical solutions, local attractors, and complex phenomena like chaos and bifurcations, with implications for understanding synchronization.

Contribution

It introduces symmetry-based analytical solutions for the Kuramoto model and uncovers new features like degeneracy and bifurcations in the system.

Findings

Existence of local attractors and chaotic phase slips.

Analytical expressions for critical coupling in specific configurations.

Stable fixed points can be achieved with limited solutions.

Abstract

In this work we study the local coupled Kuramoto model with periodic boundary conditions. Our main objective is to show how analytical solutions may be obtained from symmetry assumptions, and while we proceed on our endeavor we show apart from the existence of local attractors, some unexpected features resulting from the symmetry properties, such as intermittent and chaotic period phase slips, degeneracy of stable solutions and double bifurcation composition. As a result of our analysis, we show that stable fixed points in the synchronized region may be obtained with just a small amount of the existent solutions, and for a class of natural frequencies configuration we show analytical expressions for the critical synchronization coupling as a function of the number of oscillators, both exact and asymptotic.