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Algebraic aspects of homogeneous Kuramoto oscillators

Heather Harrington, Hal Schenck, Mike Stillman

TL;DR

This work frames homogeneous Kuramoto oscillators on graphs as polynomial systems via the Kuramoto ideal $I_K=I_ heta+I_G$, enabling an algebraic study of equilibria. It proves that standard solutions lie on the Segre variety $oldsymbol{Σ}$ and provides criteria for when $I_G$ has positive dimensional components, linking these to graph structure; it also classifies all SCT graphs with up to eight vertices regarding exotic and positive-dimensional solutions. Computational methods, including the Macaulay2 Oscillator package, reveal explicit exotic states and offer constructive graph-building techniques that preserve stability properties. Overall, the paper advances a rigorous algebraic-topological viewpoint on network synchronization and stability, with practical tools for exploring larger networks and richer solution sets.

Abstract

We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states.

Algebraic aspects of homogeneous Kuramoto oscillators

TL;DR

This work frames homogeneous Kuramoto oscillators on graphs as polynomial systems via the Kuramoto ideal , enabling an algebraic study of equilibria. It proves that standard solutions lie on the Segre variety and provides criteria for when has positive dimensional components, linking these to graph structure; it also classifies all SCT graphs with up to eight vertices regarding exotic and positive-dimensional solutions. Computational methods, including the Macaulay2 Oscillator package, reveal explicit exotic states and offer constructive graph-building techniques that preserve stability properties. Overall, the paper advances a rigorous algebraic-topological viewpoint on network synchronization and stability, with practical tools for exploring larger networks and richer solution sets.

Abstract

We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states.
Paper Structure (20 sections, 10 theorems, 72 equations, 6 figures, 1 table)

This paper contains 20 sections, 10 theorems, 72 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on Kuramoto oscillators
  3. Conventions
  4. Recent work
  5. Results of this paper
  6. Encoding the graph $G$ algebraically
  7. Algebra of Kuramoto oscillators: determinantal equations
  8. The Segre variety
  9. Algebraic geometry interlude
  10. Low codimension components of $I_G$
  11. The Segre variety and standard solutions
  12. Case study: the complete graph $K_n$
  13. Graphs with exotic solutions
  14. Exotic solutions on at most 7 vertices
  15. Exotic solutions, 8 vertices
  16. ...and 5 more sections

Key Result

Lemma 2.1

With notation as above, and $I_G$ has $V-1$ minimal generators $($rather than the expected number $V)$.

Figures (6)

  • Figure 1: The graph $G$ and ideal $I_G$
  • Figure 2: Graphs with exotic solutions, $V=5$ or $6$
  • Figure 3: Graphs with exotic solutions, $V=7$
  • Figure 4: The exceptional 8-vertex graph with exotic solutions
  • Figure 5: An 8-vertex graph with exotic solutions
  • ...and 1 more figures

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • Definition 2.5
  • ...and 27 more