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The Kuramoto model on the Sierpinski Gasket II: Twisted states

Georgi S. Medvedev, Matthew S. Mizuhara

TL;DR

This work develops a Γ-convergence framework to understand the Kuramoto model on graphs approximating fractal domains, notably the Sierpinski gasket. By showing that KM energies converge to Dirichlet energies and constructing covering spaces to handle circle-valued states, the authors prove that stable KM equilibria converge to harmonic maps SG→$\mathbb{T}$, with a unique stable equilibrium per homotopy class. They explicitly construct these harmonic maps via energy minimization on a fundamental domain and its covering, including higher-order winding; they also extend the analysis to post-critically finite fractals. The results illuminate how self-similar network structure governs long-time dynamics and yield a general mechanism for predicting twisted-like states on fractal networks with Dirichlet- and Neumann-type boundary conditions.

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.

The Kuramoto model on the Sierpinski Gasket II: Twisted states

TL;DR

This work develops a Γ-convergence framework to understand the Kuramoto model on graphs approximating fractal domains, notably the Sierpinski gasket. By showing that KM energies converge to Dirichlet energies and constructing covering spaces to handle circle-valued states, the authors prove that stable KM equilibria converge to harmonic maps SG→, with a unique stable equilibrium per homotopy class. They explicitly construct these harmonic maps via energy minimization on a fundamental domain and its covering, including higher-order winding; they also extend the analysis to post-critically finite fractals. The results illuminate how self-similar network structure governs long-time dynamics and yield a general mechanism for predicting twisted-like states on fractal networks with Dirichlet- and Neumann-type boundary conditions.

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.

Paper Structure

This paper contains 14 sections, 12 theorems, 117 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Harmonic functions on SG
  3. $\Gamma$-convergence
  4. The KM on nearest-neighbor graphs
  5. Harmonic maps on $\mathbb{T}$
  6. $\Gamma$-convergence on the covering space
  7. Harmonic maps from SG to the circle
  8. The covering space
  9. Harmonic structure on $\tilde{K}$
  10. Higher order harmonic maps
  11. The proof of Theorem \ref{['thm.main-sg']}
  12. The KM on post-critically finite fractals
  13. Background on p.c.f. fractals
  14. Convergence of KM on p.c.f. fractals

Key Result

Theorem 1.4

(Theorem 2.4 in MM2024) Let $f,g \in C(K,\mathbb{T})$. Then $f\sim g$ if and only if $\bar{\omega}(f)=\bar{\omega}(g)$.

Figures (10)

  • Figure 1: SG.
  • Figure 2: Graphs $\Gamma_n$ approximating SG.
  • Figure 3: Twisted states on SG are equilibria of the KM on SG. The degrees of the equilibria shown above: a)$\bar{\omega}(f)=(1,0,0,\dots)$, and b)$\bar{\omega}(f)=(1,1,1,1,0,0,\dots)$. These plots were obtained by numerically solving the initial value problem for the KM \ref{['KM-intro']} on on $\Gamma_8$ with initial conditions taken as the corresponding harmonic map $u^*$ satisfying \ref{['HM-Lap']}-\ref{['HM-deg']} and restricted to $V_8$.
  • Figure 4: Harmonic extension algorithm for harmonic functions, see \ref{['classical-extension']}.
  • Figure 6: A half-twisted state $u^{(r,n)}$ as described in Remark \ref{['ex.half-twist']}. Here $r=1/2$ and $n=3$.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2: cf. Str06Kozlov1993
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 17 more