The Kuramoto model on the Sierpinski Gasket II: Twisted states

TL;DR

This paper analyzes the stable states of the Kuramoto model on Sierpinski gasket graphs, revealing a connection between fractal geometry, harmonic maps, and network synchronization patterns.

Contribution

It provides a complete description of stable equilibria for the continuum limit of the Kuramoto model on fractal graphs, extending to post-critically finite fractals.

Findings

Stable equilibria correspond to harmonic maps from the SG to the circle.

Unique stable equilibrium exists in each homotopy class.

Results link self-similar structures to network dynamics.

Abstract

We study the Kuramoto model (KM) of coupled phase oscillators on graphs approximating the Sierpinski gasket (SG). As the size of the graph tends to infinity, the limit points of the sequence of stable equilibria in the KM correspond to the minima of the Dirichlet energy, i.e., to harmonic maps from the SG to the circle. We provide a complete description of the stable equilibria of the continuum limit of the KM on graphs approximating the SG, under both Dirichlet and free boundary conditions. We show that there is a unique stable equilibrium in each homotopy class of continuous functions from the SG to the circle. These equilibria serve as generalizations of the classical twisted states on ring networks. Furthermore, we extend the analysis to the KM on post-critically finite fractals. The results of this work reveal the link between self-similar organization and network dynamics.