Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Chen, Tianran; Davis, Robert; Mehta, Dhagash

doi:10.1137/17m1145665

Cited by 34 publications

(62 citation statements)

References 46 publications

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“…Existing root count results 4,17 dictate that there are no more than six complex synchronization configurations for this Kuramoto network of three oscillators. Our goal is to understand these synchronization configurations by examining simpler subnetworks of this network.…”

Section: A Examplementioning

confidence: 94%

“…The decomposition illustrated above is constructed from the geometric information encoded in a polytope -the "adjacency polytope." 16,17 In this section, we briefly review the definition. A polytope is the convex hull of a finite lists of points in R n (a bounded geometric object with finitely many flat faces).…”

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

“…The explicit formulae for this bound in the case of cycle and tree graphs have also been established. 17 A. Facet subnetworks and subsystems i j The main goal of this paper is to demonstrate that much more information of the original Kuramoto network can be extracted from this polytope. Of particular importance are the facets of the adjacency polytope, which are proper faces of maximum dimension.…”

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

See 2 more Smart Citations

Directed acyclic decomposition of Kuramoto equations

Chen

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divideand-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.The spontaneous synchronization of a network of oscillators is an emergent phenomenon that naturally appears in many seemingly independent complex systems including mechanical, chemical, biological, and even social systems. The Kuramoto model is one of the most widely studied and successful mathematical models for analyzing synchronization behaviors. While much is known about the macro-scale question of whether or not a Kuramoto network can be synchronized, detailed analysis of the possible configurations of the oscillator once it has reached synchronization remains difficult for large networks partly due to the nonlinear interactions involved. In this work, we demonstrate that by dividing the link between two oscillators into two one-way interactions, complex networks can indeed be decomposed into much simpler subnetwork. This is a crucial step toward fully understanding synchronization configurations in large networks. arXiv:1903.04492v2 [math.CO]

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Section: A Examplementioning

confidence: 94%

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

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Directed acyclic decomposition of Kuramoto equations

Chen

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Symmetric edge polytopes are of interest in many fields such as commutative algebra ( [18]), algebraic geometry ( [11]) algebraic combinatorics ( [12,19,20]) and number theory ( [6,15,22]). One of the reasons the symmetric edge polytopes gained much attraction is due to its connection to the Kuramoto model ( [14]), which describes the behavior of interacting oscillators ( [7]).…”

Section: Directed Edge Polytopes and Symmetric Edge Polytopesmentioning

confidence: 99%

Rigid Gorenstein toric Fano varieties arising from directed graphs

2022

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A directed edge polytope $${\mathcal {A}}_G$$ A G is a lattice polytope arising from root system $$A_n$$ A n and a finite directed graph G. If every directed edge of G belongs to a directed cycle in G, then $${\mathcal {A}}_G$$ A G is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety $$X_G$$ X G with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension 2 and $${\mathbb {Q}}$$ Q -factorial in codimension 3 is rigid. In the present paper, we classify all directed graphs G such that $$X_G$$ X G is a toric Fano variety which is smooth in codimension 2 and $${\mathbb {Q}}$$ Q -factorial in codimension 3.

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“…In many cases, an adjacency polytope bound gives an upper bound of the number of possible solutions in the Kuramoto equations [6]. In [7,8,25], explicit formulas of the adjacency polytope bounds, i.e., the normalized volumes of the symmetric edge polytopes of certain classes of graphs are given. Moreover, in [9], another type of adjacency polytope bounds is discussed.…”

Section: Introductionmentioning

confidence: 99%

Symmetric edge polytopes and matching generating polynomials

Ohsugi¹,

Tsuchiya²

2021

Combinatorial Theory

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Symmetric edge polytopes A G of type A are lattice polytopes arising from the root system A n and finite simple graphs G. There is a connection between A G and the Kuramoto synchronization model in physics. In particular, the normalized volume of A G plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph G, we give a formula for the h * -polynomial of A G by using matching generating polynomials, where G is the suspension of G. This gives also a formula for the normalized volume of A G . Moreover, via methods from chemical graph theory, we show that for any cactus graph G, the h * -polynomial of A G is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type B, which are lattice polytopes arising from the root system B n and finite simple graphs.

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Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Cited by 34 publications

References 46 publications

Directed acyclic decomposition of Kuramoto equations

Directed acyclic decomposition of Kuramoto equations

Rigid Gorenstein toric Fano varieties arising from directed graphs

Symmetric edge polytopes and matching generating polynomials

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