Synchronization in a Kuramoto Mean Field Game Rene CarmonaQuentin CormierH. Mete Soner October 25 2022

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Synchronization in a Kuramoto Mean Field Game

Rene Carmona∗Quentin Cormier†H. Mete Soner‡

October 25, 2022

Abstract

The classical Kuramoto model is studied in the setting of an inﬁnite horizon mean ﬁeld

game. The system is shown to exhibit both synchronization and phase transition. Incoherence

below a critical value of the interaction parameter is demonstrated by the stability of the

uniform distribution. Above this value, the game bifurcates and develops self-organizing time

homogeneous Nash equilibria. As interactions become stronger, these stationary solutions

become fully synchronized. Results are proved by an amalgam of techniques from nonlinear

partial diﬀerential equations, viscosity solutions, stochastic optimal control and stochastic

processes.

Key words: Mean ﬁeld games, Kuramoto model, Synchronization, viscosity solutions.

Mathematics Subject Classiﬁcation: 35Q89, 35D40, 39N80, 91A16, 92B25

1 Introduction

Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto

model [

] has found an amazing range of applications from neuroscience to Josephson

junctions in superconductors, and has become a key mathematical model to describe self

organization in complex systems. These autonomous oscillators are coupled through a

nonlinear interaction term which plays a central role in the long time behavior of the system.

While the system is unsynchronized when this term is not suﬃciently strong, fascinatingly

they exhibit an abrupt transition to self organization above a critical value of the interaction

parameter. Synchronization is an emergent property that occurs in a broad range of complex

systems such as neural signals, heart beats, ﬁre-ﬂy lights and circadian rhythms. Expository

papers [

] and the references therein provide an excellent introduction to the model and

its applications.

The analysis of the coupled Kuramoto oscillators through a mean ﬁeld game formalism is

ﬁrst explored by [

] proving bifurcation from incoherence to coordination by a formal

linearization and a spectral argument. [

] further develops this analysis in their application

∗

Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08540,

USA, email:

rcarmona@princeton.edu

. Research of Carmona was partially supported by AFOSR FA9550-19-1-0291

and ARPA-E DE-AR0001289.

†Inria Saclay, France, email: quentin.cormier@inria.fr.

‡

Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08540,

USA, email:

soner@princeton.edu

. Research of Soner was partially supported by the National Science Foundation

grant DMS 2106462.

arXiv:2210.12912v1 [math.OC] 24 Oct 2022

Carmona, Cormier & Soner Kuramoto Mean Field Game

to a jet-lag recovery model. We follow these pioneering studies and analyze the Kuramoto

model as a discounted inﬁnite horizon stochastic game in the limit when the number of

oscillators goes to inﬁnity. We treat the system of oscillators as an inﬁnite particle system, but

instead of positing the dynamics of the particles, we let the individual particles endogenously

determine their behaviors by minimizing a cost functional and hopefully, settling in a Nash

equilibrium. Once the search for equilibrium is recast in this way, equilibria are given by

solutions of nonlinear systems. Analytically, they are characterized by a backward dynamic

programming equation coupled to a forward Fokker-Planck-Kolmogorov equation, and in

the probabilistic approach, by forward-backward stochastic diﬀerential equations. Stability

analysis of the solutions is delicate because of this forward-backward nature of the solution,

and to the best of our knowledge, it remains a challenging problem. Except possibly in the

ﬁnite horizon potential case (cf. [

] and the references therein) it has not been fully addressed

in the existing literature on the subject. For the stability results of the Kuramoto model in

the classical setting, the interested reader could consult [14, 15] and the references therein.

With ﬁnitely many oscillators, we consider the following version of the model already

introduced in [

]. We ﬁx a large integer

and for

i∈ {

,...,N}

let

θi

be the phase

of the

-th oscillator at time

t≥

0. We assume the phases

θi

are controlled Ito diﬀusion

processes satisfying, d

θi

αi

, where

’s are independent Brownian motions, and

the control processes

αi

are exerted by the individual oscillators so as to simultaneously

minimize their costs given by

αi7→ Ji(α) := EZ∞

e−βt κ L(θi

t,θt) + 1

2(αi

t)2dt,

where

= (

α1,...,αN

)and

θt

= (

θ1

t,...,θN

). The positive constants

σ, β

are respectively,

the common standard deviations of the random shocks aﬀecting the dynamics of the phases,

and the common discounting factor used to compute the present value of the cost. The

centrally important positive constant

models the strength of the interactions between the

oscillators.

In line with the classical literature on Kuramoto’s synchronization theory, we assume

that the running cost function Lis given by

L(θi,θ) = 1

j6=i

2sin (θi−θj)/22=1

j=1

2sin (θi−θj)/22.

The cost

accounts for the cooperation between the

oscillators by incentivizing them to

align their frequencies, while the term (

αi

)

represents a form of kinetic energy which is also

to be minimized. It is convenient to express the above cost functional by using the empirical

distribution measure of the oscillators as follows,

L(θi

t,θt) = c(θi

t,¯µN

t),where c(θ, µ) := Z2sin (θ−θ0)/22µ(dθ0),(1.1)

and the empirical measure ¯µN

tis given by,

¯µN

t= ¯µ(θt) := 1

j=1

δθj

As the ﬁnite particle system is essentially intractable, especially for large values of

, we

follow approach of [

] that is now considered standard, and approximate the

Nash equilibria for the above system of oscillators by letting their number

go to inﬁnity.

Carmona, Cormier & Soner Kuramoto Mean Field Game

Then, for a given ﬂow of probability measures

= (

µt

)

t≥0

, the stochastic optimal control

problem for the representative oscillator is to minimize

α∈ A 7→ EZ∞

e−βt `(t, Xt) + 1

2α2

tdt, (1.2)

where

is the set of all right-continuous and progressively measurable processes, the running

cost

(

t, x

)is equal to

κc

(

x, µt

)with

as in

(1.1)

, and

is the controlled phase of the

representative oscillator given by

0αudu

σBt

, for a Brownian motion

. The

Nash equilibrium, as deﬁned in Deﬁnition 3.1 below, is achieved when the ﬂow µ= (µt)t≥0

is given by the marginal laws of the optimal process

X∗

. By direct methods, Lemma 4.5

proves the existence of such equilibrium ﬂows starting from any initial distribution.

It is immediate that the uniform distribution

) = d

)on the torus gives a

stationary equilibrium ﬂow. Indeed,

(

x, U

)

≡

1and therefore, the optimal control for

the above problem with the constant ﬂow

is identically equal to zero. As the uniform

distribution has no special structure, it represents incoherence among the oscillators, and

when the interaction parameter is small, we show that all the solutions of the Kuramoto mean

ﬁeld game converge to this incoherent state. This global attraction is proved in Lemma 4.3

for κ < βσ2/4. Theorem 4.4 considers all κless than the critical value

κc:= βσ2+σ4/2,(1.3)

and proves that there are that start “close” to the uniform distribution converge to it as

time tends to inﬁnity. Thus, Lemma 4.3 and Theorem 4.4 reveal that incoherence is the

main paradigm in the sub-critical regime

κ < κc

. Theorem 4.1 analyzes the case

κ > κc

, and

proves that there are inﬁnitely many self-organizing stationary solutions for these interaction

parameter values. In particular, these solutions do not converge to the incoherent uniform

distribution and numerically they are stable. Hence,

κc

is a sharp threshold for the stability

of incoherence, and there is a phase transition from total disorder to self organization exactly

at this critical interaction parameter

κc

. Furthermore, Theorem 4.2 shows convergence to

full synchronization as κgets larger.

The classical Kuramoto model with noise has been the object of many studies, and the

mean ﬁeld version is the following McKean-Vlasov stochastic diﬀerential equation

dXt=−κZT

sin(Xt−y)L(Xt)(dy) dt+σdBt,

where

(

)is the law of the random variable

. The uniform distribution is shown in [

]

to be both locally and globally stable when

κ < σ2

. The corresponding ﬁnite particle system

is studied in [

]. There, it is proven that the solutions of the ﬁnite model remain close to

the solution of the above equation for a very long time, on the order of

(

exp

(

)). Similar

results are also proved for the Kuramoto mean ﬁeld game with an ergodic cost in [

], by

using bifurcation theory techniques including the Lyapunov-Schmidt reduction method to

show the existence of non-uniform stationary solutions near the critical value

κ∗

σ4/

Rabinowitz bifurcation theorem and other global techniques are used in [

] for similar results.

The classical Kuramoto model and its mean ﬁeld game versions provide a mechanism for

the analysis of self organization. However, they cannot model synchronization with external

drivers, thus requiring additional terms. Indeed, the jet-lag recovery model of [

] introduce a

cost for misalignment with the exogenously given sunlight frequency, providing an incentive

to be in synch with the environment as well. These studies are clear evidences of the modeling

potential of the mean ﬁeld game formalism in all models when self organization is the salient

feature.

Carmona, Cormier & Soner Kuramoto Mean Field Game

The paper is organized as follows. After a short section on notation, the Kuramoto mean

ﬁeld game is introduced in Section 3, and the main results are stated in Section 4. Section 5

brieﬂy summarizes all control problems used in the paper. Stationary solutions are deﬁned

and a ﬁxed-point characterization is proved in Section 6. The super-critical case is studied in

Section 7 and full synchronization in Section 8. Incoherence is demonstrated in Section 9

by proving the convergence of all solutions to the uniform distribution when the interaction

parameter is small, and local stability of the uniform distribution is established in Section 10

for all

κ<κc

. For completeness, solutions starting from any distribution are constructed in

the Appendix A, and we provide the expected comparison result for a degenerate Eikonal

equation in the Appendix B.

2 Notation

The state-space is the one-dimensional torus

πZ

(

)is the space of all probability

measures on

. For

ν∈ P

(

f∈C

(

), we use the standard notation

(

) :=

RTf

(

)

(

)

We say that a probability measure

ν∈ P

(

)is the law

(

)of

, if

[

(

)] =

(

)for

every f∈C(T). We also use the following space of continuous functions,

C:= {ξ= (γ, η) : [0,∞)7→ R2:continuous and bounded }.

We ﬁx a ﬁltered probability space (Ω

,F,P

)supporting an

-adapted Brownian motion

(

)

t≥0

. We assume that the ﬁltration

{Ft}t≥0

satisﬁes the usual conditions, i.e.

is complete and

is right-continuous. The initial ﬁltration is non-trivial so that for any

probability measure

µ0∈ P

(

), one can construct an

measurable,

valued random

variable

with distribution

µ0

. For

t≥

0, the set

of all progressively measurable

processes α: [t, ∞)→Ris called the admissible controls, and we set A:= A0.

For µ∈ P(T), z ∈Tand a Borel subset B⊂T, we deﬁne the translation of µby,

µ(B;x) := µ({z∈T:x+z∈B}).(2.1)

Finally, we record several elementary trigonometric identities that are used repeatedly.

For µ∈ P(T), let c(x, µ)be as in the Introduction. As 2 (sin (x/2))2= 1 −cos(x),

c(x, µ) = ZT

2 (sin ((x−y)/2))2µ(dy) = 1 −a(µ) cos(x)−b(µ) sin(x),(2.2)

where a(µ) := µ(cos), and b(µ) := µ(sin). In particular, there is z∗∈Tsuch that

a(µ(·;z∗)) = g(µ) := p(a(µ))2+ (b(µ))2,and b(µ(·;z∗)) = 0.(2.3)

3 Kuramoto mean-ﬁeld game

Given a ﬂow of probability measures µ= (µt)t≥0, set

`µ(t, x) := κ[c(x, µt)−1] = −κµt(cos) cos(x)−κµt(sin) sin(x), x ∈T, t ≥0.

Consider the optimal control problem (1.2) with this running cost. Then, the problem is

vµ:= inf

α∈A Jµ,κ(α) := inf

α∈A

EZ∞

e−βt `µ(t, Xα

t) + 1

2α2

tdt, (3.1)

where as in the Introduction,

Xα

0αudu

σBt

, with a Brownian motion (

)

t≥0

and an initial condition X0satisfying L(X0) = µ0.

Carmona, Cormier & Soner Kuramoto Mean Field Game

Deﬁnition 3.1.

We say that

= (

µt

)

t≥0

is a solution to the Kuramoto mean-ﬁeld game

with interaction parameter

starting from initial distribution

µ0

,if there exists

α∗∈ A

such

that Jµ,κ(α∗) = infα∈A Jµ,κ(α)and µt=L(Xα∗

t)for all t≥0.

Example 3.2.

Consider an initial condition

satisfying

Ecos

(

) =

Esin

(

) = 0, and

the ﬂow of probability measures

= (

µt

)

t≥0

with

µt

(

σBt

)

Then, for every

t≥

`µ(t, x) = −κ(µt(cos) cos(x) + µt(sin) sin(x))

=−κ( cos(x)E[cos(X0+σBt)] + sin(x)E[sin(X0+σBt)])

=−κ(cos(x)e−σ2

2tE[cos(X0)] −sin(x)e−σ2

2tE[sin(X0)]) = 0.

Therefore, for any

α∈ A

Jµ,κ

(

) =

ER∞

0e−βt α2

t≥

0 =

Jµ,κ

(0), implying that

α∗≡

is the minimizer of

Jµ,κ

(

), and

is the law of the dynamics controlled by

α∗

. Hence,

is a

solution of the Kuramoto mean-ﬁeld game for every κ.

Now suppose that

(

)is the uniform probability measure on the torus

) = d

As any translation of

is equal to itself,

µt

(

σBt

) =

for all

t≥

0. Thus,

is a

stationary solution.

The uniform distribution represents complete incoherence, and we refer to it as the

incoherent (or uniform) solution. We next introduce the stationary solutions of the Kuramoto

mean-ﬁeld games.

Deﬁnition 3.3.

We call a probability measure

µ∈ P

(

)a stationary solution if the constant

ﬂow

= (

µt

)

t≥0

with

µt

for all

t≥

0is a solution of the Kuramoto mean-ﬁeld game.

We say that

is self-organizing or non-uniform if it is not equal to the uniform measure

We record the following simple result for future reference.

Lemma 3.4.

The uniform probability measure

on the torus is the incoherent stationary

solution of the Kuramoto mean-ﬁeld game. Moreover, a stationary solution

is the uniform

probability measure if and only if µ(cos) = µ(sin) = 0.

Proof.

In Example 3.2, we have shown that

is a stationary solution and that

(

·, U

)

≡

Now suppose that

is a stationary solution with

(

cos

) =

(

sin

) = 0. Then, as in Example 3.2,

we conclude that the optimal solution of the control problem

(1.2)

α∗≡

0, and the optimal

state process satisﬁes d

X∗

. As by stationarity

(

X∗

) =

for every

t≥

0, the

density

solves the Fokker-Plank equation

fxx

(

)=0on the torus. Hence,

is equal

to a constant, and µ=U.

Remark 3.5

(Invariance by translation)

Assume that

is a stationary solution. The

symmetry of the problem implies that the translated measure

(

;

)is also a stationary

solution for every z.

4 Main Results

In this section, we state all the main results of the paper. Recall the critical interaction

parameter

κc

(1.3)

. In Section 7, we study the super-critical case

κ > κc

, and prove the

following result.

Theorem 4.1

(Super-critical interaction: synchronization)

For all interaction parameters

κ > κc, there are non-uniform stationary solutions of the Kuramoto mean ﬁeld game.

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SynchronizationinaKuramotoMeanFieldGameReneCarmona*QuentinCormierH.MeteSonerOctober25,2022AbstractTheclassicalKuramotomodelisstudiedinthesettingofaninnitehorizonmeaneldgame.Thesystemisshowntoexhibitbothsynchronizationandphasetransition.Incoherencebelowacriticalvalueoftheinteractionparameterisdem...

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Synchronization in a Kuramoto Mean Field Game Rene CarmonaQuentin CormierH. Mete Soner October 25 2022

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