[Paper Review] Counting equilibria of the Kuramoto model using birationally invariant intersection index

Tianran Chen, Robert Davis+1 more|arXiv (Cornell University)|Aug 30, 2017

Nonlinear Dynamics and Pattern Formation33 references3 citations

TL;DR

This paper introduces a novel algebraic geometry approach to count equilibria in the Kuramoto model of coupled oscillators by leveraging the birationally invariant intersection index on toric varieties. For tree and cycle graphs, it establishes tight upper bounds on the number of synchronization configurations, proving that the adjacency polytope bound matches the birationally invariant intersection index for cycle graphs, yielding $ N \binom{N-1}{\lfloor (N-1)/2 \rfloor} $ equilibria.

ABSTRACT

Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.

Motivation & Objective

• To determine the maximum number of equilibrium configurations in the Kuramoto model for networks of coupled oscillators.
• To develop a method that leverages algebraic geometry tools—specifically birationally invariant intersection indices—to bound the number of solutions to the Kuramoto equations.
• To show that for cycle graphs, the adjacency polytope bound precisely matches the generic number of complex solutions, providing a tight upper bound.
• To extend the applicability of intersection theory beyond monomial-generated spaces by proving that the intersection index agrees with the BKK bound even when the function spaces are not monomial-generated.

Proposed method

• Formulate the Kuramoto model’s equilibrium equations as a system of rational functions on the complex torus $(\mathbb{C}^*)^n$ using the substitution $x_i = e^{i\theta_i}$.
• Define for each node $i$ a vector space $L_{G,i}$ of rational functions spanned by $1$ and terms $x_i x_j^{-1} - x_i^{-1} x_j$ corresponding to edges.
• Apply the birationally invariant intersection index $[L_{G,1}, \dots, L_{G,n}]$ to compute the generic number of isolated solutions in $(\mathbb{C}^*)^n$, which bounds the number of real synchronization configurations.
• Use Newton-Okounkov bodies and mixed volumes to relate the intersection index to the mixed volume of polytopes, generalizing the BKK bound.
• For cycle graphs, prove that despite algebraic dependencies among coefficients, the initial systems of the rational functions have independent coefficients, ensuring the BKK bound applies.
• Establish that the adjacency polytope bound equals the birationally invariant intersection index for cycle graphs by analyzing the structure of Newton polytopes and initial forms.

Experimental results

Research questions

• RQ1 What is the maximum number of isolated complex solutions (in $(\mathbb{C}^*)^n$) for the Kuramoto system on a given network topology?
• RQ2 Can the birationally invariant intersection index be computed for rational function systems not generated by monomials, such as those in the Kuramoto model?
• RQ3 For cycle graphs, does the adjacency polytope bound coincide with the true number of generic solutions, and if so, why?
• RQ4 Under what conditions does the BKK bound agree with the birationally invariant intersection index when the function spaces are not monomial-generated?

Key findings

• For a cycle graph with $N = n+1$ vertices, the number of isolated complex solutions of the Kuramoto system is exactly $N \binom{N-1}{\lfloor (N-1)/2 \rfloor}$.
• The adjacency polytope bound for cycle graphs matches the birationally invariant intersection index, providing a tight upper bound on the number of equilibria.
• The intersection index $[L_{C_N,1}, \dots, L_{C_N,n}]$ equals the BKK bound for cycle graphs, despite the function spaces not being monomial-generated.
• The proof relies on showing that for any nonzero weight vector $\mathbf{v}$, the initial systems of the Kuramoto equations have independent coefficients, so no spurious solutions arise from coefficient dependencies.
• The result confirms that the BKK bound is sharp for cycle graphs, even when the system is not in a monomial basis.