Density of instantaneous frequencies in the Kuramoto-Sakaguchi model Julio D. da Fonseca1 Edson D. Leonel1 and Rene O. Medrano-T1 2_2

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Density of instantaneous frequencies in the

Kuramoto-Sakaguchi model

Julio D. da Fonseca1∗

, Edson D. Leonel1†

, and Rene O. Medrano-T1 2‡

December 13, 2022

1Departamento de Física, Universidade Estadual Paulista, Bela Vista, 13506-900 Rio Claro,

SP, Brazil

2Departamento de Física, Universidade Federal de São Paulo, UNIFESP, 09913-030, Campus

Diadema, São Paulo, Brasil

Abstract

We obtain a formula for the statistical distribution of instantaneous frequencies in the

Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi’s theory of

globally coupled phase oscillators, which we review in full detail by discussing its assump-

tions and showing all steps behind the derivation of its main results. Our formula is a

stationary probability density function with a complex mathematical structure, is consis-

tent with numerical simulations and gives a description of the stationary collective states

of the Kuramoto-Sakaguchi model.

1 Introduction

Synchronization is the process by which interacting oscillatory systems adjust their frequencies

in order to display the same common value [1]. Power grids [2], semiconductor laser arrays

[3], cardiac pacemaker cells [4], and neurosciences [5] are just a few examples in a multitude of

domains where synchronization is an active research subject.

∗jcddafonseca@gmail.com

†edson-denis.leonel@unesp.br

‡rene.medrano@unifesp.br

arXiv:2210.05011v4 [nlin.AO] 12 Dec 2022

The works of A. Winfree and Y. Kuramoto brought seminal contributions to the study of

synchronization [4, 6, 7, 8]. Inspired by Winfree’s pioneering ideas [9], Kuramoto formulated a

model of coupled phase oscillators today known as Kuramoto model. The Kuramoto model was

introduced in Ref. [7], and its ﬁrst and more detailed analysis by Kuramoto himself, published

in Ref. [8]. Since then, many studies about the Kuramoto model and its variants appeared

in the literature. (Reviews about the Kuramoto model can be found in Refs. [10, 11]; see

Refs. [12, 13, 14, 15] for later studies related to variants of the Kuramoto model and their

applications.)

The Kuramoto model consists of an ensemble of oscillators with a mean-ﬁeld coupling and

randomly distributed natural (or intrinsic) frequencies. An oscillator is characterized by its

phase, and the ﬁrst-order time derivative of the oscillator’s phase, which here we call instan-

taneous frequency, is deﬁned by an autonomous ﬁrst-order ordinary diﬀerential equation. The

theoretical analysis of the Kuramoto model [8, 11] evinces a transition between two station-

ary collective states: an incoherent state and a synchronization one. In the incoherent state,

instantaneous and natural frequencies have the same statistical distribution. In the synchro-

nization state, some oscillators have instantaneous frequencies sharing the same value. The

number of synchronized oscillators depends on the model’s parameter called coupling-strength,

and synchronization only occurs for a coupling strength above a critical value [8, 11]. In a

simpliﬁed version of the Kuramoto model, identical (with the same natural frequencies) and

symmetrically-coupled oscillators show multiple regular attractors [16], and the synchronization

state is the most probable one[17, 18].

H. Sakaguchi and Y. Kuramoto created a generalization of the Kuramoto model [19] intro-

ducing into the coupling function a phase shift, also called phase-lag. The Kuramoto-Sakaguchi

model and its variants appear in the study of a wide range subjects such as chimera states

[20, 21], chaotic transients [22], pulse-coupled oscillators [23], and Josephson-junction arrays

[24]. In addition, the coupling function with a phase-lag can be interpreted as an approximate

model of interactions with time-delayed phases [25]. The Kuramoto-Sakaguchi model exhibits

the same stationary collective states as the original Kuramoto model [19].

Collective states of Kuramoto-like models are commonly characterized by means of an order

parameter, which is zero in the incoherent state and takes ﬁnite values in the synchronization

state. In this work, we follow a diﬀerent approach from the usual order-parameter analysis

uncovering how instantaneous frequencies are statistically distributed in the stationary collec-

tive states of the Kuramoto-Sakaguchi model. Instantaneous frequencies collectively reﬂect

the occurence of synchronized behavior [26], and they are also relevant in the study of other

phenomena (e.g. frequency spirals [32]).

We will show how to obtain a formula for the statistical distribution of instantaneous fre-

quencies. The formula is deﬁned by a stationary probability density function, which we refer

to as density of instantaneous frequencies. Our goal is similar to the one pursued in Ref. [26]

for the Kuramoto model, but here we will show how to obtain a more general result in a more

straightforward way. A related (but still rather a diﬀerent) problem was addressed by Sakaguchi

and Kuramoto in Ref. [19], where they analyzed the statistical distribution of coupling-modiﬁed

frequencies, namely instantaneous frequencies averaged over inﬁnitely large time intervals (see

Refs. [11, 26] for further details).

This work is based on the Kuramoto-Sakaguchi theory, described, as far as we know, only

in Ref. [19]. We will discuss the fundamental assumptions of the Kuramoto-Sakaguchi theory

and detail how its main results can be derived. Our opinion is that an explicit presentation of

the Kuramoto-Sakaguchi theory is still absent.

We organized this paper as follows. In Section 2, we present the Kuramoto-Sakaguchi the-

ory and state diagrams pointing out the transition between the incoherent and synchronization

states. In Section 3, we extend the Kuramoto-Sakaguchi theory by providing additional ana-

lytical results and obtaining the formula of the density of instantaneous frequencies. In Section

4, we discuss the properties of our formula in a speciﬁc application example, in which natural

frequencies have a Gaussian statistical distribution. In Section 5, we check the consistency of

our formula with numerical simulation data. Conclusions and an outlook on possible research

directions are given in Section 6.

2 Kuramoto-Sakaguchi theory

The Kuramoto-Sakaguchi (KS) model[19] consists of an inﬁnitely large number Nof all-to-all

coupled oscillators. The state of an oscillator of index i= 1...N is characterized by its phase

θi, which changes in time according to

θi=ωi−K

j=1

sin(θi−θj+α),(1)

where ˙

θiis the ﬁrst-order time-derivative of θi,ωiis a random number with a prescribed density1,

and Kand αare real constant parameters. We refer to ˙

θias the instantaneous frequency, ωi

is called natural frequency and K, the coupling strength. The oscillator of index jcan be

represented by the complex number exp(iθj). Oscillators are then points in a complex plane

moving over a unit-radius circle centered at the origin.

A valuable concept for the analysis of collective behavior in the KS model is that of mean

ﬁeld, deﬁned by

Z=1

j=1

exp(iθj),(2)

1For the sake of simplicity, hereafter we always use the term “density” to refer to a probability density

function.

which can be interpreted as the average oscillator state. The mean ﬁeld can be written as a

complex number

Z=Rexp(iΘ) (3)

where Θdenotes the mean-ﬁeld phase and R, the mean-ﬁeld modulus, referred to as order

parameter. If the oscillators are quasi-aligned, i.e., they have approximately the same phase,

then R'1. Yet, a quasi-uniform scattering of all oscillator-points over the circle results in a

mean-ﬁeld located near the origin, i.e. R'0.

For N−→ ∞, the mean ﬁeld, at a time instant t, is given by

+π

−π

exp(iθ)n(θ, t)dθ, (4)

where n(θ, t)is the density of phases at the same time instant. Two simple scenarios are

assumed concerning the properties of n(θ, t)in the long-time (t−→ ∞) and large-size (N−→

∞) limits. First, n(θ, t) = 1

2πfor −π < θ ≤+π, and n(θ, t)=0, otherwise, i.e. n(θ, t)is a

time-independent and uniform density (the value 1

2πcomes from the normalization condition).

Second, n(θ, t)is a steadily traveling wave with velocity Ω, i.e. n(θ−Ω∆t, t) = n(θ, t + ∆t)for

any time instant tand time interval ∆t. This means that the wave proﬁle does not change in

time, and the wave propagates with constant-in-time velocity Ω. We call the wave-propagation

velocity, Ω,synchronization frequency.

The ﬁrst scenario deﬁnes the incoherent state, and the second, the synchronization state. In

the incoherent state, oscillators are uniformly spread over the unit circle. In the synchronization

state, a bunch of oscillators is synchronized, that is, they change collectively their phase at the

same constant rate Ω.

After inserting a uniform phase density in Eq. (4), we see that Z= 0. So, from Eq.

(3), the order parameter (R) is zero in the incoherent state. Yet, if a KS system exhibits

synchronization, then the assumption of a traveling wave with a stationary and non-uniform

proﬁle, moving with constant velocity Ω, means that Ris ﬁnite and time-independent. Moreover,

Zmoves in the complex plane following a circular path of radius Rand velocity Ω. That is, a

uniform circular motion given by

Z(t) = Rexp [i(Ωt+ Θ0)] ,(5)

where Θ0is the mean-ﬁeld phase at an arbitrary initial time instant.

Let us consider KS oscillators in a diﬀerent complex plane, with the same origin as the

previous one, but with both axis rotating with angular velocity Ω. In the new rotating frame, we

represent the oscillator of index jby the complex number exp(iψj), where ψjis the oscillator’s

phase. The analogous of Eqs. (2), (3), and (4) are

Z0=1

j=1

exp(iψj),(6)

Z0=Rexp(iΨ),(7)

and, for N−→ ∞,

Z0=

+π

−π

exp(iψ)n(ψ)dψ. (8)

The quantities Z0and n(ψ)are representations of the mean ﬁeld and the phase density in

the rotating frame. Comparing Eq. (3) to Eq. (7), we see that Zand Z0have the same length

R. The mean-ﬁeld length is invariant to the change of frames because the phase density proﬁle

is kept unchanged. Note also that, in Eq. (8), we removed the time dependence from the phase

density, since both the rotating frame and the steadily traveling wave move together with the

same velocity Ω. So, both Rand Ψare time-independent, which is the same as stating that

the mean ﬁeld is ﬁxed in the rotating frame.

Some conventions are useful to simplify the analysis of the KS model at a time instant

t > to= 0, with t0denoting the initial time instant. We choose a ﬁxed frame such that its real

axis has the same direction as the mean ﬁeld at time t0. So, Θ0= 0 and Z(0) = R. Another

important convention is deﬁning a rotating frame such that, at the initial time t0, its real axis

is dephased by αfrom the ﬁxed-frame’s real axis (the same parameter αof Eq. (1)). This is

the same as setting Ψ = α.

Thus, from Eqs. (5) and (7), at the time instant t,

Z(t) = Rexp (iΩt)(9)

and

Z0=Rexp iα. (10)

Also, as a consequence of the above conventions, a simple geometric inquiring yields the relations

ψi=˙

θi−Ω(11)

and

ψi=θi−Ωt+α. (12)

In Eq. (11), ˙

ψiis the instantaneous frequency of an oscillator of index iin the rotating frame.

Using Eqs. (11) and (12), we can recast Eq. (1) as

ψi=ωi−Ω−K

j=1

sin(ψi−ψj+α).(13)

Multiplying the right-hand sides of Eqs. (6) and (10) by exp [−i(ψi+α)] and equating their

imaginary parts result in

N−1

j=1

sin(ψi−ψj+α) = Rsin ψi.(14)

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DensityofinstantaneousfrequenciesintheKuramoto-SakaguchimodelJulioD.daFonseca1*,EdsonD.Leonel1,andReneO.Medrano-T12December13,20221DepartamentodeFísica,UniversidadeEstadualPaulista,BelaVista,13506-900RioClaro,SP,Brazil2DepartamentodeFísica,UniversidadeFederaldeSãoPaulo,UNIFESP,09913-030,CampusDiad...

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Density of instantaneous frequencies in the Kuramoto-Sakaguchi model Julio D. da Fonseca1 Edson D. Leonel1 and Rene O. Medrano-T1 2_2

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