On the typical and atypical solutions to the Kuramoto equations Tianran Chen Evgeniia KorchevskaiaandJulia Lindberg Abstract. The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has

2025-05-02 0 0 842.61KB 30 页 10玖币

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On the typical and atypical solutions to the Kuramoto equations∗

Tianran Chen†, Evgeniia Korchevskaia‡,and Julia Lindberg§

Abstract. The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has

been much work to eﬀectively bound the number of equilibria to the Kuramoto model for a given network.

By formulating the Kuramoto equations as a system of algebraic equations, we ﬁrst relate the complex

root count of the Kuramoto equations to the combinatorics of the underlying network by showing that the

complex root count is generically equal to the normalized volume of the corresponding adjacency polytope

of the network. We then give explicit algebraic conditions under which this bound is strict and show that

there are networks where the Kuramoto equations have inﬁnitely many equilibria.

Key words. Kuramoto model, adjacency polytope, Bernshtein-Kushnirenko-Khovanskii bound

AMS subject classiﬁcations. 14Q99, 65H10, 52B20

1. Introduction. The Kuramoto model [24] is a mathematical model that describes the dy-

namics on networks of oscillators. It has applications in neuroscience, biology, chemistry and power

systems [4,16,19,34]. Despite its simplicity, it exhibits interesting emergent behaviors. Of inter-

est is the phenomenon of frequency synchronization which is when the oscillators synchronize to a

common frequency. Frequency synchronizations correspond to solutions of the Kuramoto equations

w=wi−

j=0

kij sin(θi−θj) for i= 0, . . . , n,

in the unknowns θ0, . . . , θn. Here, w, w0, . . . , wn, kij are network parameters. This paper aims to

understand the structure of these solutions in “typical” and “atypical” networks.

Earlier work focused on the statistical analysis of inﬁnite networks [24] but more recently,

tools from diﬀerential and algebraic geometry have enabled analysis of synchronizations on ﬁnite

networks. For a ﬁnite network, knowing the total number of synchronization conﬁgurations is

fundamental to understanding this model. From a computational perspective, this knowledge also

plays a critical role in developing numerical methods for ﬁnding synchronization conﬁgurations. For

instance, this number serves as a stopping criterion for monodromy algorithms [28] and allows for the

development of specialized homotopy algorithms [7,9] for ﬁnding all synchronization conﬁgurations.

In 1982, Ballieul and Byrnes introduced root counting techniques from algebraic geometry to

this ﬁeld and showed that a Kuramoto network of Noscillators has at most 2N−2

N−1synchronization

conﬁgurations [2]. It coincides with the bound on the root count for the closely related load-ﬂow

equations discovered by Li, Sauer, and Yorke [26]. Algebraic geometers will recognize this bound as

the bi-homogeneous B´ezout bound for an algebraic version of the Kuramoto equations. This upper

bound can be reached when the network is complete and complex roots are counted. However, for

sparse networks, the root count (even counting complex roots) can be signiﬁcantly lower than this

upper bound [18,31], demonstrating the need for a network-dependent root count.

Guo and Salam initiated one of the ﬁrst algebraic analyses on such sparsity-dependent root

counts [18]. Molzahn, Mehta, and Niemerg provided computational evidence for the connection

∗Submitted to the editors DATE.

Funding: TC and EK are supported by a grant from the Auburn University at Montgomery Research Grant-in-Aid

Program and the National Science Foundation under Grant No. 1923099. TC and JL are supported by the National

Science Foundation under Grant No. 2318837. EK is also supported by the Undergraduate Research Experience program

funded by the Department of Mathematics at Auburn University at Montgomery.

†Department of Mathematics, Auburn University at Montgomery, Montgomery, AL (ti@nranchen.org)

‡School of Mathematics, Georgia Institute of Technology

§Department of Mathematics, University of Texas at Austin, Austin, TX (julia.lindberg@math.utexas.edu)

arXiv:2210.00784v3 [math.AG] 25 Sep 2024

between this root count and network topology [31]. In the special case of rank-one coupling co-

eﬃcients, Coss, Hauenstein, Hong and Molzahn proved this complex root count to be 2N−2,

which is also an asymptotically sharp bound on the real root count [14]. Chen, Davis and Mehta

gave a sharp bound on the complex root count for cycle networks of NN−1

⌊(N−1)/2⌋[11], which is

asymptotically smaller than the bi-homogeneous B´ezout bound discovered by Baillieul and Byrnes,

proving that sparse networks have signiﬁcantly fewer synchronization conﬁgurations. Interestingly,

Lindberg, Zachariah, Boston, and Lesieutre showed that this bound is attainable by real roots [29].

This bound is an instance of the “adjacency polytope bound” [8], which motivates the following.

Question 1.1. For generic choices of network parameters, does the complex root count for the

algebraic Kuramoto equations reach the adjacency polytope bound for all networks?

In the ﬁrst part of this paper, we provide a positive answer to this question and thus establish

the generic root count for the Kuramoto equations derived from a graph Gto be the normalized

volume of the adjacency polytope of G. As a corollary, we show that the Kuramoto equations are

Bernshtein-general. We note that this result is similar to recent work in [5] which shows that the

number of (approximate) complex solutions to the Duﬃng equations is generically the volume of

the Oscillator polytope. We also extend this generic root count result to variations of the Kuramoto

equations, including a special case of the power ﬂow equations from electric engineering.

While the complex root count for the algebraic Kuramoto equations is generically constant and

ﬁnite, there may be network parameters that produce diﬀerent root counts. First, we focus on the

role played by the coupling coeﬃcients in such exceptional situations.

Question 1.2. What are conditions on the coupling coeﬃcients under which the algebraic Ku-

ramoto equations are not Bernshtein-general?

In the second part of this paper, we provide an explicit, combinatorial description for such

exceptional coupling coeﬃcients. In particular, we show that the set of exceptional coupling coef-

ﬁcients can be characterized by “balanced subnetworks”.

For the Kuramoto equations with exceptional coupling coeﬃcients, there are two possibilities.

Either all complex solutions remain isolated but the total number drops below the generic root

count, or non-isolated solution components appear. In the second case, there are inﬁnitely many

solutions, forming curves, surfaces, or geometric structures of even higher dimension.

Ashwin, Bick, and Burylko analyzed non-isolated solutions in complete networks of identical

oscillators [1]. A concrete example of a network of four identical oscillators with uniform coupling

coeﬃcients was described in [14, Example 2.1]. Non-isolated solutions for cycle networks was

discovered by Lindberg, Zachariah, Boston and Lesieutre [29]. Recent work by Sclosa shows that

for every d≥1 there is a Kuramoto network whose stable equilibria form a manifold of dimension d

[35], which, arguably, shows that the study of non-isolated solutions deserves more serious attention.

This is further supported by the recent paper of Harrington, Schenck, and Stillman [20], which shows

that for any 2-connected graph that contains a 3-let, there are parameters for which the Kuramoto

equations have a non-isolated solution set. It is within this context, that the third part of this

paper aims to provide some explicit and constructive answers to the following.

Question 1.3. What are the conditions on the network parameters under which the Kuramoto

equations have inﬁnitely many solutions?

The rest of this paper is structured as follows. Section 2 reviews concepts and notation that

will be used. We then consider the generic root count of the Kuramoto equations in Section 3 and

provide a positive answer to Question 1.1. Next, we turn our attention to non-generic coupling

coeﬃcients in Section 4 and answer Question 1.2 with a combinatorial description of the exceptional

coupling coeﬃcients. Using this description, in Section 5 we answer Question 1.3 by identifying

network parameters where the Kuramoto equations have non-isolated solutions and we construct

explicit parameterizations for these solutions. Finally, we conclude with a few remarks in Section 6.

2. Notation and preliminaries. Column vectors, representing points of a lattice L∼

=Zn, are

denoted by lowercase letters with an arrowhead, e.g., ⃗a. We use boldface letters, e.g., x, for

points in Cn,Rn, or L∨∼

=Zn, and they are written as row vectors. For x= (x1, . . . , xn) and

⃗a = (a1, . . . , an)⊤∈Zn,x⃗a =xa1

1··· xan

nis a Laurent monomial with the convention that x0

i= 1,

for any xi. A linear combination of such monomials f=P⃗a∈Sx⃗a is called a Laurent polynomial, and

its support and Newton polytope are denoted supp(f) = Sand Newt(f) = conv(S), respectively.

With respect to a vector v, the initial form of fis initv(f)(x) := P⃗a∈(S)vc⃗a x⃗a, where (S)v

is the subset of Son which the linear functional ⟨v,•⟩ is minimized. For an integer matrix

A= [⃗a1··· ⃗am], xA:= (x⃗a1,...,x⃗am) deﬁnes a function on the algebraic torus (C∗)n= (C\{0})n

whose group structure is given by (x1, . . . , xn)◦(y1, . . . , yn) := (x1y1, . . . , xnyn). We ﬁx Gto be a

connected graph with vertex set V(G) = {0,1, . . . , n}and edge set E(G). For nodes iand jin V(G),

i∼jindicates their adjacency. Arrowheads will be used to distinguish digraphs from graphs, e.g.,

⃗

Grepresents a digraph and Gan undirected graph. Appendix A has the full list of notation.

2.1. The Kuramoto model. A network of coupled oscillators can be naively thought of as a

swarm of points on the complex plane pulling on one another with varying force while circling around

the origin. It can be used to model a wide variety of seemingly unrelated phenomena ranging from

the ﬁring of neurons and the rhythmic contractions of heart cells, to the oscillations of concentrations

of chemical compounds in a mixture. In this paper, such a network is represented by a connected

graph Gwhose nodes and edges represent the oscillators and their connections, respectively. Each

oscillator has a natural frequency wiand along the edges in G, nonzero constants K={kij }with

kij =kji quantify the coupling strength between oscillators iand j. The data structure (G, K, ⃗w)

encoding this model will simply be called a network. The Kuramoto model describes the nonlinear

interactions among the oscillators by the diﬀerential equations

(2.1) dθi

dt =wi−X

j∼i

kij sin(θi−θj) for i= 0, . . . , n,

where θiis the phase angle of the i-th oscillator [24]. Frequency synchronization conﬁgurations are

deﬁned to be values of (θ0, . . . , θn) at which dθi

dt equals w=1

nPn

i=0 wi. By adopting a rotational

frame, we can assume θ0= 0. Since kij =kji, we can also eliminate one equation. Therefore,

frequency synchronization conﬁgurations are zeroes to the system of ntranscendental functions:

(2.2) (wi−w)−X

j∼i

kij sin(θi−θj) for i= 1, . . . , n.

The problem of counting synchronization conﬁgurations is therefore a root counting problem.

2.2. Algebraic Kuramoto equations. To leverage the power of algebraic geometry, the above

transcendental system can be reformulated into an algebraic system via the change of variables

xi=eiθi. Then sin(θi−θj) = 1

2i(xi

xj−xj

xi), and (2.2) becomes

(2.3) fG,i(x1, . . . , xn) = wi−X

j∼i

aij xi

xj−xj

xifor i= 1, . . . , n,

where aji =aij =kij

2i,wi=wi−w, and x0= 1. The Laurent polynomial system ⃗

fG=

(fG,1, . . . , fG,n)⊤will be called the algebraic Kuramoto system, and it captures all synchro-

nization conﬁgurations in the sense that the real zeros to (2.2) correspond to the complex zeros

of (2.3) on the real torus (S1)n(i.e., |xi|=|eiθ|= 1). Note that fGdepends on K={kij }and ⃗w,

and we will use the notation f(G,K)or f(G,K, ⃗w)when these dependencies are emphasized.

In much of this paper, we relax the root-counting problem by considering all C∗-zeros of (2.3).

One important observation is that we can focus on graphs with no pendant (a.k.a. leaf) nodes.

Lemma 2.1. [27, Theorem 2.5.1] Suppose v̸= 0 is a pendant node of G. Let G′=G− {v}.

Then any C∗-zero of ⃗

fG′extends to two distinct C∗-zeros for ⃗

fG.

2.3. Kuramoto equations with phase delays. In models with phase delays, we may introduce

parameters {δij ∈R|i∼j}, so that along an edge {i, j}, oscillator iresponds not directly to the

phase angle of oscillator jbut its delayed phase θj−δij [36]. Then (2.2) is generalized into:

0 = wi−w−X

j∼i

kij sin(θi−θj+δij),for i= 1, . . . , n.

Letting Cij =eiδij , we can again make this system algebraic giving:

fG,i(x1, . . . , xn) = wi−X

j∼i

aij xiCij

xj−xj

xiCij ,for i= 1, . . . , n.(2.4)

This system diﬀers from (2.3) in the coeﬃcients. Yet, the same set of monomials are involved, and as

we will demonstrate, the algebraic arguments we will develop can be applied to this generalization.

2.4. Power ﬂow equations. The PV power ﬂow system is one important variation of the

Kuramoto system. In it, the graph Gmodels an electric power network where V(G) = {0, . . . , n}

represent buses in the power network. An edge {i, j}∈E(G), representing the connection between

buses iand j, has a known complex admittance g′

ij +ib′

ij. For each bus i, the relationship between

its complex power injection Pi+iQiand the complex voltages is captured by the nonlinear equations

Pi=X

j∼i|Vi||Vj|(g′

ij cos(θi−θj) + b′

ij sin(θi−θj))(2.5)

Qi=X

j∼i|Vi||Vj|(g′

ij sin(θi−θj)−b′

ij cos(θi−θj))(2.6)

where |Vi|is the voltage magnitude at bus iand θiis its phase angle. We ﬁx bus 0 to be the slack

bus with θ0= 0. For a complete treatment of the derivation of the power ﬂow equations see [17].

A node iis a PV node when Qiand θiare unknown while Piand |Vi|are known and it models

a generator bus. As above, with xi=eiθiwe get the PV algebraic power ﬂow equations

fG,i(x1, . . . , xn) = Pi−X

j∼i

gij xi

+xj

xi+bij xi

xj−xj

xi,for i= 1, . . . , n(2.7)

where bij =1

2i|Vi||Vj|b′

ij and gij =1

2|Vi||Vj|g′

ij. When gij = 0, the corresponding power system is

lossless and (2.7) reduces to (2.3). Otherwise, the system is lossy and (2.7) diﬀers from (2.3). This

is, again, a variation of the algebraic Kuramoto system (2.3) that involves the same monomials.

2.5. BKK bound. The root counting arguments in this paper revolve around the Bernshtein-

Kushnirenko-Khovanskii (BKK) bound, especially Bernshtein’s Second Theorem.

Theorem 2.2 (D. Bernshtein 1975 [3]). (A) For a square Laurent system ⃗

f= (f1, . . . , fn), if for

all nonzero vectors ⃗v ∈Rn,init⃗v(⃗

f)has no C∗-zeros, then all C∗-zeros of ⃗

fare isolated, and the

total number, counting multiplicity, is the mixed volume M= MV(Newt(f1),...,Newt(fn)).

(B) If init⃗v(⃗

f)has a C∗-zero for some ⃗v ̸=⃗

0, then the number of isolated C∗-zeros ⃗

fhas,

counting multiplicity, is strictly less than Mif M > 0.

A system for which condition (A) holds is said to be Bernshtein-general. Only the special case of

identical Newton polytopes, i.e., when Newt(f1),...,Newt(fn) are all identical, will be used. This

specialized version strengthens Kushnirenko’s Theorem [25].

2.6. Randomized algebraic Kuramoto system. The analysis of the algebraic Kuramoto sys-

tem (2.3) can be further simpliﬁed through “randomization”. For any nonsingular square matrix

R, the systems ⃗

fGand ⃗

f∗

G:= R⃗

fGhave the same zero set. With a generic choice of R, there will be

no complete cancellation of terms, and ⃗

f∗

Gwill be referred to as the randomized (algebraic) Ku-

ramoto system. This system is unmixed in the sense that f∗

G,1, . . . , f∗

G,n have identical supports

since they involve the same set of monomials, namely, constant terms and {xix−1

j,xjx−1

i}{i,j}∈E(G).

The randomization ⃗

fG7→ ⃗

f∗

Gdoes not alter the zero set but makes a very helpful change to the

tropical structure: There is a mapping between the graph-theoretical features of Gand the tropical

structures of ⃗

f∗

Gthrough which we can gain key insight into the structure of the zeros of ⃗

f∗

2.7. Adjacency polytopes. For each i= 1, . . . , n, supp(f∗

G,i) are all identical and given by

∇G:= {±(⃗ei−⃗ej)|i∼j}∪{⃗

0},

where ⃗eiis the i-th standard basis vector of Rnfor i= 1, . . . , n, and ⃗e0=⃗

0. The conv( ˇ

∇G) is the

adjacency polytope or symmetric edge polytope of G[8,11,13,30], and is related to root polytopes

[33]. It has appeared in number theory and discrete geometry (see the overview provided in [15]).

We will not distinguish ˇ

∇Gfrom conv( ˇ

∇G), e.g., a “face” of ˇ

∇Grefers to a subset F⊆ˇ

∇Gsuch

that conv(F) is a face of conv( ˇ

∇G). The facets and boundary of ˇ

∇Gare denoted F(ˇ

∇G) and ∂ˇ

∇G,

respectively. ˇ

∇Gis centrally symmetric, and both ˇ

∇Gand ∂ˇ

∇Ghave unimodular triangulations.

By Kushnirenko’s Theorem [25], Vol( ˇ

∇G) is an upper bound for the C∗-root count for ⃗

f∗

Gand

⃗

fGand the real root count to (2.2). This is the adjacency polytope bound [8,11].

2.8. Faces and face subgraphs. There is an intimate connection between faces of ˇ

∇Gand

subgraphs of G. Since ⃗

0 is an interior point of ˇ

∇G, every vertex of a proper face Fof ˇ

∇Gis of

the form ⃗ei−⃗ejfor some {i, j}∈E(G). Thus, it is natural to consider the corresponding facial

subgraph GFand facial subdigraph ⃗

GFinvolving a subsets of nodes and edges sets

E(GF) = {{i, j} | ⃗ei−⃗ej∈For ⃗ej−⃗ei∈F}and

E(⃗

GF) = {(i, j)|⃗ei−⃗ej∈F}.

As deﬁned in [7,15], for F∈ F(ˇ

∇G), GFis called a facet subgraph and ⃗

GFis a facet subdigraph.

Higashitani, Jochemko, and Micha lek provided a topological classiﬁcation of face subgraphs [22,

Theorem 3.1] and it was later reinterpreted [10, Theorem 3]. We state the latter here.

Theorem 2.3 (Theorem 3 [10]). Let Hbe a nontrivial connected subgraph of G.

1. His a face subgraph of Gif and only if it is a maximal bipartite subgraph of G[V(H)].

2. His a facet subgraph of Gif and only if it is a maximal bipartite subgraph of G.

Here, G[V] is the subgraph induced by the subset V⊂ V(G). Multiple faces can correspond to

the same facial subgraph. The crisper parameterization is given by the correspondence F7→ ⃗

GF.

Reference [10] describes a necessary balancing conditions for facial subdigraphs.

For a facial subdigraph ⃗

GF, its reduced incidence matrix ˇ

Q(⃗

GF) is the matrix with columns

⃗ei−⃗ejfor (i, j)∈ E(⃗

GF). Its null space can be interpreted as the space of circulations of ⃗

GF.

For a subdigraph ⃗

Hof ⃗

G, the coupling vector k(⃗

H) has entries kij for (i, j)∈⃗

H. Similarly, the

entries of a(⃗

H) are the complexiﬁed coupling coeﬃcients aij =kij

2i. The ordering of the entries is

arbitrary, but when appearing in the same context with ˇ

Q(⃗

H), consistent ordering is implied.

2.9. Facial systems. The vast literature on the facial structure of ˇ

∇Ggives us a shortcut to un-

derstanding the initial systems of ⃗

f∗

G, since they have particularly simple descriptions corresponding

to proper faces of ˇ

∇G. For any 0 ̸=v∈Rn, the initial system initv(⃗

f∗

G) is

(2.8) initv(f∗

G,i)(x) = X

⃗ej−⃗ej′∈F

ci,j,j′x⃗ej−⃗ej′=X

(j,j′)∈E(⃗

GF)

ci,j,j′x⃗ej−⃗ej′for i= 1, . . . , n,

where Fis the face of ˇ

∇Gfor which vis an inner normal vector. We will make frequent use of

this geometric interpretation and therefore it is convenient to slightly abuse the notation and write

initF(⃗

f∗

G) := initv(⃗

f∗

G). It will be called a facial system of ⃗

f∗

G, (or a facet system if F∈ F(ˇ

∇G)).

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OnthetypicalandatypicalsolutionstotheKuramotoequations∗TianranChen†,EvgeniiaKorchevskaia‡,andJuliaLindberg§Abstract.TheKuramotomodelisadynamicalsystemthatmodelstheinteractionofcoupledoscillators.TherehasbeenmuchworktoeffectivelyboundthenumberofequilibriatotheKuramotomodelforagivennetwork.Byformulati...

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On the typical and atypical solutions to the Kuramoto equations Tianran Chen Evgeniia KorchevskaiaandJulia Lindberg Abstract. The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has

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