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Generalized splay states in phase oscillator

networks

Cite as: Chaos 31, 073128 (2021); doi: 10.1063/5.0056664

Submitted: 12 May 2021 ·Accepted: 24 June 2021 ·

Published Online: 13 July 2021

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Rico Berner,1,2,a)Serhiy Yanchuk,2Yuri Maistrenko,3,4 and Eckehard Schöll1,5,6

AFFILIATIONS

1Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

2Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

3Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52428 Jülich, Germany

4Institute of Mathematics and Centre for Medical and Biotechnical Research, NAS of Ukraine, Tereshchenkivska St. 3,

01601 Kyiv, Ukraine

5Bernstein Center for Computational Neuroscience Berlin, Humboldt Universität, Philippstraße 13, 10115 Berlin, Germany

6Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473 Potsdam, Germany

Note: This paper is part of the Focus Issue, In Memory of Vadim S. Anishchenko: Statistical Physics and Nonlinear Dynamics of

Complex Systems.

a)Author to whom correspondence should be addressed: rico.berner@physik.tu-berlin.de

ABSTRACT

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we intro-

duce generalized m-splay states constituting a special subclass of phase-locked states with vanishing mth order parameter. Such states

typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class

of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known

Kuramoto–Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parame-

ter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings

to phase oscillators with inertia and adaptively coupled phase oscillator models.

Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0056664

Models of coupled phase oscillators are well-known paradig-

matic systems to understand the mechanism behind the emer-

gence of collective phenomena in complex networks. Due to the

relative simplicity of these models, powerful methods such as

the Watanabe–Strogatz theory or the Ott–Antonsen approach

have been developed to describe certain dynamic states. Nowa-

days, a plethora of generalizations of phase oscillator models are

developed to study biological, technological, or socio-economic

systems. Of particular interest are phase models with inertia

for mechanical rotors in power grids or models with an adap-

tive network structure to model synaptic plasticity mechanisms

in neuronal systems. This paper provides a systematic study of

generalized splay states as a particular class of incoherent phase-

locked solutions playing an important role in shaping the global

dynamics of coupled oscillator systems. In particular, we describe

when a continuum of splay states emerges and a part of it (also

continuum) becomes stable. These splay states are a manifesta-

tion of individual variability as one of the inherent properties of

oscillatory networks.

I. INTRODUCTION

Dynamical networks of phase oscillators are a well-known

paradigm for studying the collective behavior of interacting

agents.1,2The importance of such network models relies, in par-

ticular, on the fact that any system of weakly interacting nonlinear

oscillators can be generally reduced to a phase oscillator network.2–5

Extensive reviews have highlighted the importance of phase oscilla-

tor models and reduction techniques.2,6Recent studies also aim at

increasing the range of applicability of phase oscillators by general-

izing the conditions under which reduction techniques are valid.7–9

Chaos 31, 073128 (2021); doi: 10.1063/5.0056664 31, 073128-1

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Chaos ARTICLE scitation.org/journal/cha

A famous representative of the class of phase oscillator mod-

els, the Kuramoto model, in which all oscillators are coupled in the

”all-to-all” manner, has attracted much attention due to its simple

form and mathematical tractability.10,11 The Kuramoto model and

its extensions have gained additional popularity through applica-

tions to real-world problems,2,12–14 including neuroscience15–20 and

power grids.21–26 Despite the simple structure, the Kuramoto model

can exhibit many different dynamical regimes,27–29 and sophisticated

methods have been developed for their analysis. In particular, it

was shown that sinusoidally and globally coupled phase oscillators

are partially integrable. The Watanabe–Strogatz theory allows for a

reduction to only three dimensions, which can also be applied to

even more general classes of phase oscillator models.30–33 A remark-

able observation in the work of Watanabe and Strogatz is the role

of incoherent states34 in the foliation of the phase space. They have

shown that sheets of the foliation can be parameterized by the fam-

ily of incoherent states. In our work, we analyze these incoherent

but phase-locked and frequency-synchronized states for a large class

of phase oscillator models and shed new light on their dynamical

properties. Moreover, we generalize the notion of incoherent states

by introducing generalized splay states.

Another approach developed to understand coupled oscilla-

tors in the continuum limit is the Ott–Antonson ansatz.35 In the

case of an infinite number of oscillators, the Ott–Antonson the-

ory allows for a reduction to a two-dimensional dynamical system

and has been successfully applied to describe the emergence of par-

tially synchronized patterns.28,36–39 Remarkably, for both reduction

techniques, the Watanabe–Strogatz and the Ott–Antonson theory,

the reduced systems possess a clear physical interpretation, and

both approaches are closely related.32,40 In fact, this direct rela-

tionship between the two approaches makes the study of incoher-

ent states (generalized splay states) important for mean field and

other reduction techniques41–43 as well as for the search for future

generalizations.44–48 Moreover, splay states play a role in networks

with non-global coupling. These states have also been found in non-

locally coupled ring networks,49–51 and the concept of local splay

states (or local incoherent states) has been introduced to relate them

to the incoherent states in globally coupled networks.52 Further-

more, splay and incoherent states have been discussed for more

complex coupled systems such as Stuart–Landau oscillators,53,54 sys-

tems with delay53,55 and pulse-coupling56,57 where for the latter a link

to the Watanabe–Strogatz theory has been proved.58

Various generalization have been proposed beyond the clas-

sical Kuramoto model. Starting from the generalization to com-

plex networks,59,60 the theory of phase oscillators has been further

developed to study phenomena of phase transitions,61–63 network

symmetries,64 the impact of inertia21,27,65–73 and other forms of

frequency adaptation,74 delayed coupling,75 or the effect of time-

dependent parameters,76 to name just a few.

Another generalization that has gained much attention in

recent years concerns the phenomena in networks of phase oscilla-

tors with adaptive coupling. Several models have been proposed and

studied to gain insights into the interplay between collective dynam-

ics and adaptivity.16,18,19,77–85 Many of them were inspired by recent

findings in neuroscience related to synaptic plasticity.

In this work, we provide a general analytic study of the local

properties, existence and stability, of incoherent phase-locked states.

For this, we introduce the class of phase oscillator models in Sec. II

and define the notion of the generalized m-splay state. In Sec. III,

we describe manifolds of the splay states and provide explicit con-

ditions for their linear stability. In Secs. IV and V, we generalize the

results for the stability of m-splay states to phase oscillator models

with inertia and adaptivity, respectively. In Sec. VI, we give a geo-

metrical perspective. The results are discussed in Sec. VII. To make

the main text more accessible, we have moved some proofs to the

Appendixes.

II. COUPLED PHASE OSCILLATOR MODELS AND

GENERALIZED SPLAY STATES

We consider systems of Ncoupled phase oscillators,

dtφ=ω1+F(φ), (1)

where φ=(φ1,...,φN)T, and each oscillator is represented by

a dynamical variable φi(t)∈[0, 2π),i=1, ...,N. All oscilla-

tors possess the same common natural frequency ω. Moreover,

F=(f1(φ),...,fN(φ))Tis the coupling vector field with coupling

functions fi, and 1=(1, ..., 1)T.

To measure the phase coherence, we define the mth moment of

the complex mean field, m∈N, as

Zm(φ)=1

j=1

eimφj=Rm(φ)eiρm(φ),

where i is the imaginary unit, Rmdenotes the mth moment of the

(Kuramoto–Daido) order parameter, and ρmis the collective phase

of the mth moment of the mean field.10,86

Definition 1. A solution of the phase oscillator system (1) is

called phase-locked state if

φi(t)=t+ϑi,i=1, ...,N,

with collective frequency ∈Rand fixed relative phases

ϑi∈[0, 2π) of the individual oscillators.

Phase-locked states for oscillator models have been studied

extensively in the past, see, e.g., Refs. 78 and 87. In this paper, we

restrict our attention to a special subclass of phase-locked states for

which little is known about their role in the case of finite ensembles

of oscillators, as follows.

Definition 2. A phase-locked state with φi(t)=t+ϑiis an

m-splay state if it satisfies

Rm(ϑ)=0. (2)

We call m∈Nthe moment of the splay state and Eq. (2) the m-splay

condition.

This definition generalizes the “classical” notion of splay

states53 that are defined by phase distributions with equidistant

phase relation ϑj=kj2π/Nwith k=0, ...,N−1 and form an m-

splay state if (mk mod N)6= 0. These states are also referred to as

twisted states49,50,88 or rotating waves,51,89 and are often related to

certain network symmetries.90–92 In Fig. 1, we illustrate one- and

two-splay states for ensembles of N=2, 3, and 4 oscillators. The

relation Rm(ϑ)=R1(mϑ)holds between the splay states with dif-

ferent moments. In particular, we observe that any two-splay state

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FIG. 1. Illustrations of (a)–(c) one- and (d)–(f) two-splay states for N=2 (left

column), N=3 (center column), and N=4 (right column) coupled phase oscil-

lators represented on the unit circle. The green and blue angles depict fixed and

parameterized (variable) phase relations, respectively.

in Figs. 1(d)–1(f) corresponds to a one-splay state in Figs. 1(a)–1(c)

by doubling the relative angles between the oscillators.

With definition 2 of splay states, we may consider a whole

family that fulfills the m-splay condition (2).

Definition 3. The set

SMm=ϑ∈[0, 2π)N:Rm(ϑ)=0

is called the m-splay manifold.

The fact that SMmforms a N−2 dimensional manifold has

been proven in Ref. 91. Note that the splay manifold has a shift sym-

metry, i.e., if ϑ∈SMm, then ϑ+ψ1∈SMmfor any ψ∈[0, 2π).

In this paper, we derive linear stability conditions for m-splay

states for generic phase oscillator models (1) that possess phase-shift

symmetry and leave the m-splay manifold invariant, and each point

of this manifold corresponds to an m-splay solution.

More specifically, we assume that for some m, the following

hypotheses are fulfilled:

Hypothesis 1. For all ϑ∈SMm, the system of coupled phase

oscillators possesses m-splay states φ(t)=t+ϑwith collective

frequency .

Hypothesis 2. For any ψ∈R, the nonlinearity Fsatisfies

F(φ+ψ1)=F(φ). This implies that the corresponding system (1)

is equivariant with respect to the phase-shift transformation.

With both Hypotheses 1 and 2, we guarantee that, first, all ele-

ments of the splay manifold SMmdescribe a phase-locked state and,

second, the phase-locked states may be considered as time inde-

pendent due to the phase-shift symmetry, i.e., we can consider the

co-rotating reference frame φ→φ+t. These restrictions are met

by many phase oscillator models that have been analyzed over the

last decades, including the Kuramoto–Sakaguchi model,93 models

with higher mode coupling94 and with higher order interactions,95

models of coupled phase oscillators under resource constraints,96

and generalized phase oscillator models including systems with

inertia66,71 or adaptive network structure.80 Some of them are dis-

cussed in the subsequent sections. An important class of systems that

fulfill Hypothesis 1 are those that are coupled via mean field.20,40

In Sec. III, we derive conditions for the linear stability of the

m-splay manifold of system (1) under Hypotheses 1 and 2.

III. STABILITY OF GENERALIZED SPLAY STATES IN

PHASE OSCILLATOR MODELS

This section explores the linear stability of m-splay states as

defined in Sec. II. In the first part, we provide a general result for the

generic class of phase oscillator models. This result is then discussed

for the Kuramoto–Sakaguchi model.

A. General result on the stability of splay states

We start with the variational equation around an arbitrary

m-splay state of system (1), which satisfies Hypotheses 1 and 2. This

variational equation reads

dδφ(t)

dt=L(ϑ)δφ, (3)

where L(ϑ)=DFdenotes the Jacobian of the coupling field F. We

note that due to the shift symmetry of system (1), the Jacobian is time

independent and has zero row sum for each row and thus possesses a

zero eigenvalue corresponding to the eigenvector 1, i.e., L(ϑ)1=0,

for any ϑ. This eigenvector 1acts along the symmetry action, see

Hypothesis 2. The nondiagonal entries of the N×Nmatrix Lare

lij =∂fi

∂φj

(ϑ). (4)

Due to the zero row-sum condition, the diagonal elements lii are

given as

lii = −

j=1,j6=i

∂fi

∂φj

(ϑ). (5)

The linear stability of the m-splay states is determined by the

eigenvalues of the Jacobian matrix L. More precisely, we are inter-

ested in the real parts of these eigenvalues. The following lemma

provides useful insights into the spectral structure of a special class

of matrices Lthat are of major importance subsequently.

We denote a polynomial of degree r∈N0over the com-

plex field Cwith complex argument λ∈Cas pr(λ), i.e., pr(λ)

=Pr

k=0akλk. The characteristic polynomial of an N×Nmatrix is

denoted by pN(L,λ), i.e., pN(L,λ) =det(L−λIN).

Lemma 4. Suppose an N ×N (N >1) matrix L possesses

a zero eigenvalue with multiplicity N −2. Then,the characteristic

polynomial pN(L,λ) is given by

det (L−λIN)=(−1)Nλ(N−2)λ2+a(N−1)λ+a(N−2),

with the coefficients

a(N−1)= −

j=1

ljj = −Tr(L),

a(N−2)=

i=i

j>iliiljj −lijlji=1

2Tr(L)2−Tr(L2).

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This result is a consequence of a general theorem on the coef-

ficients of the characteristic polynomial.97 However, we provide a

direct proof of Lemma 4 in Appendix A for the interested reader.

With Lemma 4, we are able to write explicit stability conditions for

any m-splay state depending on the two explicit characteristics of the

Jacobian L: its trace Tr (L)and the trace of its square Tr (L2).

Proposition 5. Suppose L(ϑ)is the Jacobian from (3),whose

entries are given in (4) and (5),with ϑcorresponding to an m-splay

state of the coupled phase oscillator system (1). Then,we distinguish

the following two cases:

(i) If 2Tr(L2)≤Tr(L)2,then the m-splay state is linearly stable if

and only if

Tr(L) < 0.

(ii) If 2Tr(L2) > Tr(L)2,then the m-splay state is linearly stable if

and only if

Tr(L) < 0and Tr(L2) < Tr(L)2.

Proof. Since ϑcorresponds to an m-splay state, there are

N−2 neutral perturbation directions along the splay manifold.

These perturbations are determined by the condition

j=1

eimϑjδφj=0,

which follows from δZm(φ)=0. In particular, the perturbation

δφ=1constitutes one of the dimensions of the neutral subspace.

Thus, Lpossesses a zero eigenvalue with multiplicity N−2, and

we can apply Lemma 4. We obtain a quadratic equation with real-

valued coefficients depending on Tr(L)and Tr(L2)whose solution

is given by

λ1,2 =Tr(L)

2±1

2q2Tr(L2)−Tr(L)2. (6)

The two cases follow immediately by considering the real parts of

the solutions λ1,2.

Proposition 5 provides a very general linear stability condition

that depends on two features of the Jacobian Lonly. In particular, it

depends on the sum of all eigenvalues λiof L, i.e, Tr(L)=PN

i=1λi,

and on the sum of all squares of eigenvalues, i.e., Tr(L2)=PN

i=1λ2

Figure 2 shows the stability region for an arbitrary m-splay

state in the [Tr(L), Tr(L2)]-plane. In particular, the solid dashed

lines indicate the transition for which the real part of at least one

eigenvalue crosses zero. An analytic expressions for theses lines is

derived as follows. Assume that one eigenvalue λpossesses zero real

parts, i.e., λ=ivwith v∈R. Substituting this assumption into the

quadratic expression of Lemma 4, we obtain

−v2−iTr(L)v+1

2(Tr(L)2−Tr(L2)) =0.

The latter equation can either be fulfilled with v=0 and Tr(L2)

=Tr(L)2or with Tr(L)=0 and Tr(L2)= −v2for all v∈R. Both

conditions agree with the solid black lines in Fig. 2.

In Sec. III B, we apply the results of Lemma 4 and Proposition 5

in order to describe the stability of splay states for a particular model.

FIG. 2. Diagram showing the local properties of m-splay states in dependence of

the values Tr(L)and Tr(L2). The solid black curves indicate transitions between

different stability features of the m-splay state, i.e., stable, saddle, and repelling.

The dashed lines indicate transition between nodes and foci. The shaded parts

of the diagram correspond to linear stability.

B. Kuramoto–Sakaguchi model

In this section, we study the linear stability of splay states

in a globally coupled network of Kuramoto–Sakaguchi phase

oscillators10,93 given by

φi=ω−1

j=1

sin(φi−φj+α), (7)

where αis the phase-lag parameter. This system satisfies Hypothesis

2 of phase-shift invariance. We note that the coupling function of

(7) and its derivative can be written as

fi= −Im Z1eiαeiφi,

∂fi

∂φj=1

NRe ei(φi−φj+α).

With this, we immediately see that any phase distribution ϑthat

fulfills the one-splay condition Z1(ϑ)=0 corresponds to the one-

splay solution φi(t)=ωt+ϑiof (7). Therefore, system (7) satisfies

Hypothesis 1. To derive the stability condition with Proposition

5, we determine the entries of the Jacobian matrix L(ϑ)of the

variational system for (7) around the one-splay states. The entries

read

lij =(1

Ncos(α) if j=i,

Ncos(ϑi−ϑj+α) otherwise. (8)

With these preliminaries, we obtain the following.

Corollary 6. The one-splay state φ=ω1+ϑof Kuramoto–

Sakaguchi system (7) is linearly stable if and only if

cos α < 0.

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Proof. In order to prove this result, we use Lemma 4 and deter-

mine the coefficients of the quadratic polynomial. We obtain the

following:

a(N−1)= −Tr(L)= −cos α,

and

a(N−2)=cos2(α)

2−1

4N2

i,j=1cos(2(ϑi−ϑj)) +cos(2α)

4−1

j=1

R2(ϑ)Re ei(ϑj−ρ2(ϑ))

41−R2

2(ϑ).

Since 1 −R2

2(ϑ)≥0 for any ϑ, the stability condition for the one-

splay state is a(N−1)>0, which is equivalent to cos(α) < 0. 

Corollary 6 implies that a one-splay state is linearly stable

for (7) as long as cos α < 0. Thus, the stability does not depend

on the particular shape of the splay state, i.e., the particular ele-

ment ϑof the manifold SM1. However, the bifurcation occurring at

cos α=0 might be different depending on the distribution of

the phases. Due to Proposition 5, the eigenvalues of the Jaco-

bian (8) may have imaginary parts if 2Tr(L2) < Tr(L)2, i.e., cos2α

< (1−R2

2(ϑ)). In this case, the imaginary parts are given by

Im λ1,2 = ±1

2qsin2α−R2

2(ϑ).

We further note that the second moment of the order parameter

R2characterizes the whole one-splay manifold with respect to the

transverse stability.

In Secs. IV and V, we extend the previous findings to more

complex models of coupled phase oscillators. More precisely, we

consider phase oscillator models with inertia and models of adap-

tively coupled phase oscillators.

IV. PHASE OSCILLATOR MODELS WITH INERTIA

As a first extension of our results from Sec. III, we study the

linear stability of generalized splay states in models of Ncoupled

phase oscillators with inertia. Similar to Sec. III, we first provide the

general result and then consider a more specific class of models.

A. Stability of

-splay states

We consider the following class of phase oscillator models with

inertia

Md2

dt2φ+γd

dtφ=p1+F(φ), (9)

where p∈Rcorresponds to the nondimensionalized power gener-

ation and consumption in power grid models and is related to the

natural frequency in Eq. (1) by ω=p/γ , the parameter Mis the

inertia, and γ > 0 is the damping constant, which extends Eq. (1).

Note that, for identical oscillators, the number of parameters can be

reduced to two. Therefore, we assume M=1 in the following.

Here, we also assume that (9) possesses a set of m-splay states

φ=t+ϑfor all ϑ∈SMm, i.e., Hypothesis 1 holds. We may write

(9) as a set of first order differential equations

dtφ=ψ,

dtψ= −γψ+p1+F(φ),

(10)

where we introduce the new dynamical variable ψ∈RN.

In order to determine the linear stability of m-splay states,

we consider the variational equation for system (10) around an

arbitrary m-splay state, which reads

dtδφ

δψ=0IN

L(ϑ)−γINδφ

δψ=J(ϑ)δφ

δψ(11)

where Jdenotes the Jacobian, and the entries of the N×Nmatrix L

are given as in (4) and (5).

The linear stability of the m-splay states is determined by the

eigenvalues of the Jacobian matrix J. The following Lemma provides

a useful tool to find these eigenvalues.

Lemma 7. The 2N eigenvalues of the matrix h0m1IN

L m2INiwith

m1,m2∈Care given by the solutions of the N quadratic equations

µ2−m2µ−m1λi=0, i=1, ...,N,

where λ1,...,λNare the eigenvalues of L.

The proof of Lemma 7 can be found in Ref. 22. With this lemma

and Lemma 4, the following conditions for the local stability of the

m-splay states of (9) are derived.

Proposition 8. Suppose J(ϑ)is the Jacobian of (11) and L(ϑ)

possesses the entries as given in (4) and (5), where ϑcorresponds to

an m-splay state which solves (9). Then the m-splay state is linearly

stable if and only if γ > 0and Re(µ1,2,3,4) < 0, where

µ1,2,3,4 = −γ

2±rγ

22+λ1,2,

λ1,2 =Tr(L)

2±1

2q2Tr(L2)−Tr(L)2.

(12)

It is interesting to note that λ1,2 equals the eigenvalues (6) of the phase

oscillator model without inertia.

Proof. Due to Proposition 5, the eigenvalues of Lare given by

λ3= ··· = λN=0 and

λ1,2 =Tr(L)

2±1

2q2Tr(L2)−Tr(L)2.

Using Lemma 7, the eigenvalues of the Jacobian Jare given by the

solutions of

µ2+γ µ −λi=0.

The N−2 zero eigenvalues λ3,...,N=0 lead to µ=0 and µ= −γ,

each with the multiplicity N−2. Moreover, there are roots

µi,1,2 = −γ

2±rγ

22+λi,i=1, 2,

which yield the result. 

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Note that the characteristic polynomial of the Jacobian in (11)

can be rewritten in a form that agrees with the findings in Ref. 73 for

the case of N=4 oscillators. In particular, the eigenvalues (12) can

be also expressed as solutions of the equation

µ4+2γ µ3+(γ 2−Tr(L))µ2

−γTr(L)µ +Tr(L)2−Tr(L2)

2=0. (13)

We conclude that the stability properties of m-splay states

depend only on the parameters γ, Tr(L), and Tr(L2). The quantities

Tr(L)and Tr(L2)contain also information about the specific splay

states. The values Tr(L)and Tr(L2)provide a foliation of the splay

manifold so that each sheet of this foliation, with the same values

of Tr(L)and Tr(L2), possesses the same transverse local dynamics.

In order to find the boundary of the stability region, we substitute

µ=ivinto (13) and find that

v4−(γ 2−Tr(L))v2+Tr(L)2−Tr(L2)

−iγv2v2+Tr(L)=0. (14)

The latter equation is solved when either one of the following

conditions is fulfilled:

(i) v=0 and Tr(L2)=Tr(L)2for all γ > 0,

(ii) −2v2=Tr(L)and Tr(L)2/2+γTr(L)−Tr(L2)=0 for all

v∈R.

In Fig. 3(a), we display both conditions as surfaces in

[Tr(L), Tr(L2),γ]-space. In panels (b)–(e) of Fig. 3, we show two-

parameter cross sections with fixed values of γ, where the stable

regions are shaded. We note that the area corresponding to stable

dynamics increases with increasing γ.

B. Application to the Kuramoto–Sakaguchi model

with inertia

In the following, we study the linear stability of generalized

splay states in a globally coupled network of Ncoupled phase

oscillators with inertia22,23,25,26,70,71,73 of the form

M¨

φi+γ˙

φi=p−σ

j=1

sin(φi−φj+α), (15)

where φi∈[0, 2π) represents the phase of the ith rotator. The

parameter Mis the inertia constant, γ > 0 is the damping constant,

and σis the coupling constant. The parameter αcan be regarded as

a phase-lag of the interaction.93

As for the Kuramoto–Sakaguchi model (see Sec. III B), the

coupling functions and their derivatives can be written as

fi= −σIm Z1eiαeiφi,

∂fi

∂φj=σ

NRe ei(φi−φj+α).

With this, we immediately see that any one-splay state is a solution of

(15) since Z1=0 and hence fi=0 for all ϑ∈SM1and i=1, ...,N.

FIG. 3. Phase diagram showing the local stability of the m-splay states for system

(9) in dependence on γ, Tr(L), and Tr(L2). In panel (a), the surfaces separating

stable from unstable regimes are depicted in orange and green corresponding

to the conditions (i) and (ii) for Eq. (14), respectively. In panels (b)–(e), sec-

tions for fixed values γ=0.1, 0.5, and 1,5 are shown, respectively. Stable regions

are shaded. The line colors indicate the cross sections with the corresponding

surfaces.

Proposition 8 leads to the following criteria for the stability of

one-splay states.

Corollary 9. The one-splay state of system (15) is linearly

stable if and only if γ > 0and Re(µ1,2,3,4) < 0, where

µ1,2,3,4 = −γ

2±rγ

22+λ1,2,

λ1,2 =σ

2cos α+qR2

2(ϑ)−sin2α.

(16)

Note that the characteristic polynomial of the Jacobian in (11)

can be rewritten in a form that agrees with the findings in Ref. 73 for

the N=4 case. In particular, the eigenvalues (16) can be expressed

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Chaos ARTICLE scitation.org/journal/cha

as solutions of the equation

µ4+2γ µ3+(γ 2−σcos(α))µ2

−γ σ cos(α)µ +σ2

4(1−R2

2(ϑ)) =0. (17)

Similar to the general case above, we illustrate the stability

properties of the one-splay state depending on γ,σ,α, and R2that

characterizes the splay manifold with respect to the linear stability.

For this, note first that the parameters γand σcan be reduced to one

parameter γ0=√σγ with respect to the stability of the one-splay

state. In particular, the mapping γ7→ √σ γ and µ7→ √σµ leaves

the stability features invariant but renders (17) independent of σ.

Hence, we may consider (17) for σ=1 without loss of generality.

FIG. 4. Phase diagram showing the local stability of the one-splay states for sys-

tem (15) in dependence of the phase-lag parameter α, the second moment of

the order parameter R2(ϑ), and damping γ. In panel (a), the surface separating

stable from unstable regimes is depicted in green. The surface corresponds to

condition (ii) for Eq. (18). In panels (b)–(e), sections for fixed values of γ=0.1,

0.5, 1, and 3 are depicted, respectively. Stable regions are shaded.

FIG. 5. Illustrations of one-splay states (R1=0) for system sizes of (a)

N=2, (b) N=3, (c) N=4, (d) N=5, and (e) and (f) N=6. The phases

of each phase oscillator are represented on the unit circle by ϑi7→ exp(iϑi). The

green and blues angles depict fixed and parameterized (variable) phase relations,

respectively.

As in previous cases, we look for the boundary of the stability

region by substituting µ=ivinto (17),

v4−(γ 2−cos α)v2+1

4(1−R2

2(ϑ))

−iγv2v2+cos α=0. (18)

The obtained equation is solved when either one of the following

conditions is fulfilled:

(i) v=0 and R2

2(ϑ)=1 for all γ > 0,

(ii) −2v2=cos αand sin2α−R2

2(ϑ)+2γcos α=0 for all v∈R.

The first condition corresponds to a singular point on the

splay manifold with R2=1 and is not of general relevance. Hence,

in Fig. 4(a), we display the second condition as a surface in

[α,R2

2(ϑ),γ]-space. In panels (b)–(e) of Fig. 4, we depict a two-

dimensional cross section with fixed values of γ,where the stable

regions are indicated by shading. We note that the area correspond-

ing to stable dynamics increases for increasing γ. Moreover, we

observe that for any γthe stability intervals in αdecrease from

[−π/2, π/2] for R2=1 to a smaller interval for R2=0, respectively.

In Fig. 5, we illustrate different one-splay states and their

corresponding second order parameter R2.

V. ADAPTIVE PHASE OSCILLATOR MODELS

In this section, we study the linear stability of generalized splay

states in the following general class of coupled phase oscillators with

adaptation:

dtφ=ω1+F(φ,κ), (19)

dtκ= −κ+G(φ), (20)

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where ωis the common natural frequency of the phase oscil-

lators and F(φ,κ)=(f1(φ,κ),...,fN(φ,κ))Tis the coupling vec-

tor field with coupling functions fi. The adaptivity variables are

given by κ=(κ1,...,κK)T∈RK. Their dynamics is determined

by the dissipation parameter and the adaptation vector field

G(φ)=(g1(φ),...,gK(φ))Tand adaptation functions gl,

l=1, ...,K.

Due to the existence of adaptivity variables κ, we have to gen-

eralize the definition of a splay state. A phase-locked state with

φi(t)=t+ϑiand κl(ϑ)=gl(ϑ)/ (l=1, ...,K) is said to form

a generalized m-splay state if Rm(ϑ)=0. In order guarantee the

existence of m-splay states, we extend Hypothesis 1 accordingly. In

particular, we assume

Hypothesis 3. For all ϑ∈SMm, a system of coupled phase oscil-

lators (19) and (20) possesses m-splay states φ(t)=t+ϑ,κl(ϑ)

=gl(ϑ)/ with collective frequency .

Additionally, we assume the phase-shift symmetry.

Hypothesis 4. For any ψ∈R, the nonlinearities Fand Gsatisfy

F(φ+ψ1,κ)=F(φ,κ)and G(φ+ψ1)=G(φ). This implies that

the corresponding system (19) and (20) is equivariant with respect

to the phase-shift transformation.

A. Stability of

-splay states

To study the linear stability of m-splay states, we consider the

variational equations (19) and (20) around an arbitrary m-splay

state, which reads

dtδφ

δκ=L B

C−IMδφ

δκ=Jδφ

δκ, (21)

where J=J(ϑ)denotes the Jacobian. Due to the phase-shift sym-

metry, (δφT,δκT)=(1T, 0, ..., 0)is an eigenvector of Jwith zero

eigenvalue and hence L1=0. The entries of the N×Kmatrix

B=B(ϑ), the K×Kmatrix C=C(ϑ), and the N×Nmatrix

L=L(ϑ)are given as follows. The nondiagonal entries of Lare

given by

lij =∂fi

∂φj

(ϑ,κ(ϑ)). (22)

The diagonal elements lii are given such that the row sums vanish,

i.e., PN

j=1lij =0 for all i=1, ...,N. We have

lii = −

j=1,j6=i

∂fi

∂φj

(ϑ,κ(ϑ)). (23)

The entries of Band Care given by

bil =∂fi

∂κl

(ϑ,κ(ϑ)) (24)

and

cli =∂gl

∂φi

(ϑ), (25)

respectively.

In order to understand the linear stability of the phase-locked

states, we have to determine the eigenvalues of the Jacobian matrix J.

The following lemma provides a useful tool to find these eigenvalues.

In the following, we use the superscript Hto indicate the Hermitian

conjugate.

Lemma 10. Let L,B,and C be any complex N ×N,N×K, and

K×N matrices,respectively. For the N +K eigenvalues of the matrix

J=L B

C−IK,the following statements hold true.

(i) If K <N,the eigenvalues of J are given by the solutions of

( +µ)K−Ndet (µ +) (µIN−L)−BC=0.

(ii) If K ≥N,J possesses K −N eigenvalues −. The 2N remaining

eigenvalues are given by

det (µ +) (µIN−L)−BC=0.

Proof. Using the Schur complement,98 we write

det L−µINB

C−( +µ)IK

= −( +µ)Kdet (L−µIN)+( +µ)−1BC

= −( +µ)K−Ndet (µ +) (µIN−L)−BC.

(i) Suppose K<N, then the K×Nmatrix Chas at least an

N−K-dimensional kernel. Hence, there exists an N−Kdimen-

sional vector space Vsuch that for all v∈V,

(µ +) (µIN−L)−BCv=0

for µ= −. Therefore, the polynomial

det (µ +) (µIN−L)−BC=0

possesses at least N−Kroots −.

(ii) Suppose K≥N, we find

det (J−µIN+K)

= −( +λ)K−Ndet (µ +) (µIN−L)−BC,

and Jpossesses K−Neigenvalues −. All other eigenval-

ues are given by the coupled set of Nquadratic equations

det [(µ+)(µIN−L)−BC ]=0. 

With this lemma and Lemma 4, the following conditions for

the local stability of the m-splay states are derived. Note that Land

BC do not necessarily commute.

Proposition 11. Suppose J is the Jacobian of (21) and L,B,

and C possess the entries as given in (22) and (23),(24),and (25),

respectively, where [ϑ,κ(ϑ)]corresponds to an m-splay state which

solves (19) and (20). Let us further write ˜

L=BC. Then the m-splay

state is linearly stable if and only if  > 0, and for all solutions µ1,2,3,4

of the quartic equation,

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Chaos ARTICLE scitation.org/journal/cha

µ4+(2−Tr(L))µ3+2−2Tr(L)+Tr(L)2−Tr(L2)

2−Tr(˜

L)µ2

+Tr(L)Tr(˜

L)−Tr(L˜

L)+(Tr(L)2−Tr(L2)−Tr(˜

L)) −2Tr(L)µ

+Tr(˜

L)2−Tr(˜

L2)+2(Tr(L)Tr(˜

L)−Tr(L˜

L)) +2(Tr(L)2−Tr(L2))

2=0, (26)

we have Re(µ1,2,3,4) < 0.

The proof of this proposition is given in Appendix B. With this

result, the stability of an m-splay state depends on the traces of L,

L2,˜

L=BC,˜

L2, and L˜

Lexplicitly. We note that, as it has been shown

in Ref. 26, the phase oscillator models with inertia are a subclass of

phase oscillator models with adaptivity. In particular, considering

L=0 in (26) completely resembles the finding for phase oscillator

models with inertia in (13).

VI. GEOMETRIC PERSPECTIVE

This short section aims at giving a qualitative geometric view

on the obtained results. In fact, the m-splay states are a particular

class of incoherent states satisfying a special but important condition

Zm=0, i.e., the mth order parameter vanishes. Due to surprisingly

frequently arising symmetries or special coupling configurations

in dynamical networks (Kuramoto–Sakaguchi, for example), these

states appear and form high-dimensional manifolds SMm, with the

dimension D= (dimension of the phase space) −2, i.e., the number

of real-valued conditions from Zm=0. Due to the high dimension-

ality, such states and their stable/unstable manifolds play a crucial

role in the global dynamics.

Here, we show that the manifold SMmof the splay states is

foliated by the two parameters Tr (L)and Tr (L2), such that each

(D−2)-dimensional sheet of this foliation has the same local stabil-

ity properties. One part of this foliation can be stable, another part

is unstable, and the corresponding eigenvalues are given explicitly.

Moving along the manifold [changing Tr (L)and Tr (L2)], one can

observe classical local bifurcations.

Our generalizations on phase oscillators with inertia or with

adaptation show that the above general geometric picture is pre-

served, but with different dimensions and some more parameters

of the foliation. For adaptive networks, for example, the foliation

parameters are the traces of L,L2,˜

L=BC,˜

L2, and L˜

VII. CONCLUSIONS

In this article, we have introduced generalized m-splay states

as a concept for incoherent phase-locked states in ensembles of a

finite number of phase oscillators. We have provided a description

of their shape and illustrated these states for a small number of oscil-

lators in Sec. II. Additionally, we have considered each splay state as

part of an N−2 dimensional manifold called the splay manifold. In

Sec. III, we have described the local dynamical properties of gener-

alized m-splay states and have given explicit stability conditions for

their stability. Here, we have identified two specific properties of the

Jacobian matrix Lto be of relevance for the stability. In particular,

we have shown that the traces of Land L2describe the stability for

any splay state.

In order to illustrate these abstract results from Sec. III A, we

have applied our findings in Sec. III B to the Kuramoto–Sakaguchi

model that possesses one-splay states. We have found that the sta-

bility for all splay states is determined by the phase-lag parameter

αalone. However, it is notable that the local dynamics around each

one-splay state is determined by the second moment of the order

parameter R2. Depending on R2, a one-splay state is either a node or

a focus.

In Sec. IV, the results have been transferred to phase oscil-

lator models with inertia. In Sec. IV A, we have generalized the

findings for the stability of generalized splay states and have demon-

strated the stability in dependence on the damping constant γand

the traces of Land L2. We have further described analytically the

two-dimensional surfaces that separate stable regions from unsta-

ble regions in [Tr(L), Tr(L2),γ]-space. As before, we have applied

the general results to a specific model. Here, we have considered

the Kuramoto–Sakaguchi model with inertia which possesses one-

splay states. Due to our previous findings, we have derived the

shape of the two-dimensional surface explicitly that separates sta-

ble from unstable regions in (α,R2,γ )-space. In contrast to the pure

Kuramoto–Sakaguchi model, the stability of the one-splay states

depends explicitly on R2for the model with inertia. Thus, the splay

manifold consists of stable and unstable regions. The phase oscilla-

tor model with inertia that has been considered in this article can

also be interpreted as a phase oscillator model with adaptivity.26

As the last part of this article, we have shown the generic stabil-

ity condition of any m-splay state for a very generic class of adaptive

phase oscillator models. Here, we have observed that the stability is

not determined by the traces of Land L2alone. It turns out that yet

another Laplacian matrix ˜

Ldescribing the interaction of the phases

with the adaptive variables is needed to understand the stability

properties. Hence, the bifurcation scenarios can be more complex.

In summary, in this article, we have developed a general frame-

work to study the local dynamical features of generalized splay

states. These states generalize certain concepts of incoherent states

as they have been studied previously.34 In contrast to Ref. 34, the

findings in this article are valid for ensembles of finite size as

well. For the particular class of Kuramoto–Sakaguchi models, we

have also pinpointed the important characteristics that describe

the local dynamics transverse to the splay manifold even beyond

pure phase oscillator models. Due to the intimate relation between

partial integrability and the splay manifold as proposed by the

Watanabe–Strogatz approach,31 we believe that the present findings

provide important insights for future development of generalized

dimension reduction techniques.

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In the field of chimera states, splay states play an important role

for both transient and asymptotic dynamics. Multiple coexistence

of splay states with chimera states gives typically rise to riddled and

intermingled basins of attraction causing the extreme sensitivity and

unpredictability of the global network dynamics.73

Another field for application of our results lies in the research

on epileptic seizures. It was shown that a drop of the degree of syn-

chronization may occur just before the onset of a seizure.99–101 In

particular, this drop of synchronization, where the order parame-

ter tends to zero, hints at the dynamical importance of splay states

(incoherence) for the emergence of seizures. Moreover, modern

approaches to treat tinnitus102,103 and Parkinson’s disease104 make

active use of incoherent states. In this regard, our findings may offer

new insights since these methods essentially rely on the stability of

incoherent states.

DEDICATION

We dedicate this paper to the memory of Vadim S.

Anishchenko.

ACKNOWLEDGMENTS

This work was supported by the German Research Foundation

DFG (Project Nos. 411803875 and 440145547).

APPENDIX A: PROOF OF LEMMA 4

Let us prove the result by complete induction. Consider the case

N=2. By direct calculation, we find that the statement of the lemma

holds true. Now, assume the result holds for any N. Consider the

characteristic polynomial of the following (N+1)×(N+1)matrix

and assume that it possesses N−1 roots at zero:

L(N+1)(N+1)=





l(N+1)(N+1)l(N+1)1··· l(N+1)N

l1(N+1)

lN(N+1)







where

LN=





l11 ··· l1N

.....

lN1··· lNN





.

We use the Laplace expansion of det(L(N+1)(N+1)−λI(N+1))with

respect to the first column. We get

det(L(N+1)(N+1)−λI(N+1))

=l(N+1)(N+1)−λdet (LN−λIN)+

i=1

(−1)ili(N+1)det ˆ

LN,i

with ˆ

LN,igiven by







l(N+1)1l(N+1)2··· ··· ··· l(N+1)N

l11 −λl12 ··· ··· ··· l1N

.....

l(i−1)1··· ··· l(i−1)(i−1)−λ··· l(i−1)N

l(i+1)1··· ··· l(i+1)(i+1)−λ··· l(i+1)N

.....

lN1··· ··· ··· ··· lNN −λ







Remember, by assumption, while evaluating the characteristic poly-

nomial p(N+1)(λ) of L(N+1)(N+1), we only have to consider contribu-

tions to the coefficients aN+1and aN. Note that det ˆ

LN,iis already a

polynomial of degree N; thus, it can contribute to aNonly. Consider

an additional Laplacian expansion of det ˆ

LN,iwith respect to the first

row. Let (ˆ

LN,i)ibe the matrix where we cut off the first row and the

ith column of ˆ

LN,i. We find that the term −(−1)il(N+1)idet ˆ

LN,i,iof

the Laplacian expansion of det ˆ

LN,icontributes to a polynomial of

degree N. Any other term results in a polynomial in λwith degree

lower than N. Apply the induction ansatz that the statement in the

lemma holds for any K≤N, we find

det(L(N+1)(N+1)−λI(N+1))

= −(−1)Nλ(N−1)λ2+¯

a(N−1)λ+¯

a(N−2)

+(−1)Nl(N+1)(N+1)λ(N−1)λ+¯

a(N−1)

−(−1)N−1

i=1

li(N+1)l(N+1)iλN−1,

where ¯

akare the coefficients of p(LN,λ). Reorganizing the last

equation yields the proof.

APPENDIX B: PROOF OF PROPOSITION 11

Due to Lemma 10, the eigenvalues of the Jacobian Jare deter-

mined by the solutions of

det (µ +) (µIN−L)−BC=0.

By assumption, the m-splay states form an (N−2)-dimensional

manifold SMm. Consider the N−2 perturbation directions along

this family δˆ

φ=(δφT, 0, ..., 0)T, where the components are given

by PN

j=1eimϑjδφj=0. From Jδˆ

φ=0, we get Lδφ=0 and BCδφ

=B0=0. Hence, the matrix ¯

L(µ) =(µ +)L+BC needs to have

at least N−2 zero eigenvalues for any µand thus Lemma 4 applies.

The eigenvalues of ¯

Lare 0 with algebraic multiplicity N−2 and the

solutions of λ2+a(N−1)λ+a(N−2)=0 where the coefficients read

a(N−1)= −(µ +)Tr(L)−Tr(BC),

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Chaos ARTICLE scitation.org/journal/cha

and

2a(N−2)=(µ +)2(Tr(L)2−Tr(L2))

+2(µ +)(Tr(L)Tr(BC)−Tr(LBC))

+Tr(BC)2−Tr((BC)2),

where we have used well-known relations for the trace. Knowing the

eigenvalues of ¯

L, we can introduce a Hermitian transformation Q

such that QH¯

LQ is upper triangular, see Schur form of a matrix in

Ref. 105 for a proof of the existence of Q. With this, the polynomial

equation

det (µ +)µIN−QH¯

L(µ)Q=0

possesses N−2 solutions µ=0 and correspondingly N−2

solutions µ= −. The four other solutions are given by the

two quadratic equations µ2+µ −λ1,2(µ) =0, where λ1,2 solve

λ2+a(N−1)λ+a(N−2)=0 with a(N−1)and a(N−2)as above. The quar-

tic form in (26) follows directly by elementary algebraic transforma-

tions.

DATA AVAILABILITY

The data that support the findings of this study are available

within the article.

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