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The Kuramoto Model on the Sierpinski Gasket II: Twisted States
Journal article   Open access   Peer reviewed

The Kuramoto Model on the Sierpinski Gasket II: Twisted States

Georgi Medvedev and Matthew S Mizuhara
Journal of Nonlinear Science, v 36(4), 70
15 Jun 2026
DOI: https://doi.org/10.1007/s00332-026-10285-5
Featured in Collection :   Research Supported by Drexel Libraries' OA Programs
url
https://doi.org/10.1007/s00332-026-10285-5View
Published, Version of Record (VoR) Open Access via Drexel Libraries Read and Publish Program 2026 Open CC BY V4.0

Abstract

Kuramoto model <italic>XY</italic> spin model Fractal Sierpinski gasket Gamma\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}-convergence Twisted state
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.

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