Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

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Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on

complex directed graphs

Di Patti, Francesca ; Fanelli, Duccio; Miele, Filippo; Carletti, Timoteo

Published in:

Communication in Nonlinear Science and Numerical Simulation

DOI:

10.1016/j.cnsns.2017.08.012

Publication date:

2017

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Citation for pulished version (HARVARD):

Di Patti, F, Fanelli, D, Miele, F & Carletti, T 2017, 'Ginzburg-Landau approximation for self-sustained oscillators

weakly coupled on complex directed graphs', Communication in Nonlinear Science and Numerical Simulation,

vol. 56, pp. 447-456. https://doi.org/10.1016/j.cnsns.2017.08.012

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Commun Nonlinear Sci Numer Simulat 56 (2018) 447–456

ContentslistsavailableatScienceDirect

Commun

Nonlinear

Sci

Numer

Simulat

journalhomepage:www.elsevier.com/locate/cnsns

Research

paper

Ginzburg-Landau

approximation

for

self-sustained

oscillators

weakly

coupled

on

complex

directed

graphs

Francesca

Di

Patti

a,b,c,∗

_,

_Duccio

_Fanelli

a,b,c

_,

_Filippo

_Miele

d

_,

_Timoteo

_Carletti

e a Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italia b INFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italia

c CSDC, via G. Sansone 1, 50019 Sesto Fiorentino, Italia

d Spanish National Research Council (IDAEA-CSIC), E-08034 Barcelona, Spain

e Department of Mathematics and Namur Center for Complex Systems - naXys, University of Namur, rempart de la Vierge 8, B 50 0 0 Namur, Belgium

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 7 February 2017 Accepted 15 August 2017 Available online 16 August 2017 Keywords:

Reaction-diffusion model Complex Ginzburg–Landau equation Pattern formation

Synchronization

a

b

s

t

r

a

c

t

Anormalformapproximationfortheevolutionofareaction-diffusionsystemhostedona directedgraphisderived,inthevicinityofasupercriticalHopfbifurcation.Weakdiffusive couplingsareassumedtoholdbetweenadjacentnodes.Underthisworkingassumption, aComplexGinzburg–Landauequation(CGLE) isobtained, whose coefficientsdepend on theparametersofthemodelandthetopologicalcharacteristicsoftheunderlyingnetwork. TheCGLEenablesonetoprobethestabilityofthesynchronousoscillatingsolution,as dis-playedbythereaction-diffusionsystemabove Hopfbifurcation.Morespecifically, condi-tionscanbeworkedoutfortheonsetofthesymmetrybreakinginstabilitythateventually destroystheuniformoscillatorystate.NumericaltestsperformedfortheBrusselatormodel confirmthevalidityoftheproposedtheoreticalscheme.PatternsrecordedfortheCGLE re-semblecloselythoserecovereduponintegrationoftheoriginalBrussellatordynamics.

© 2017ElsevierB.V.Allrightsreserved.

1. Introduction

Manyreal-life phenomenacanbe ultimatelydescribedintermsofmutuallyinteractingentitieswhichcan occasionally giverise tocollectivebehaviors[1,2].The emergingpatternsmayplay importantfunctionalroles,asit isforinstancethe caseforbiochemicalprocesses[3]andecologicalapplications[4–6].Synchronyisamongthemoststrikingexampleof self-organizeddynamics[7].Itisencounteredinawidegalleryofnaturalsystems,thinktothebeatingoftheheart[8]andthe firingofthefirefly[9].Italsoplaysaroleofparamountimportanceforthecorrectfunctioningofman-designedtechnology, ase.g.inpowergrids[10,11].

Synchronizationofself-sustained oscillations isa particularlyrich field ofinvestigation. Mathematically, self-sustained oscillationscorrespond tostablelimitcycles inthestate spaceofanautonomouscontinuous-time dynamicalsystem. Os-cillatorscanbeembeddedincontinuum space,beingsubjecttodiffusivecouplings.Theensuingreaction-diffusionsystem displayssynchronousoscillations, which,underspecific conditions,prove robustto external perturbations. Eachoscillator canalternativelyoccupya node ofacomplexnetwork [12–15],a generalization thatopensup theperspectiveto tacklea largeplethoraofproblemsthatdealwithadiscreteandheterogeneoushostingsupport[15–17].

∗ _{Corresponding author at: Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italia.} E-mail address: [email protected] (F. Di Patti).

http://dx.doi.org/10.1016/j.cnsns.2017.08.012 1007-5704/© 2017 Elsevier B.V. All rights reserved.

Forcontinuous reaction-diffusionsystems nearthe supercritical Hopf bifurcation, one can obtain asimplified, normal formdescription ofthe dynamicsinterms ofan associatedGinzburg–Landau equation (CGLE) [7,18,19].Thisis a reduced picturewhichprovidesanaccuraterepresentationoftheoriginalmodel,whileallowingforanalyticalprogresstobemade. Fromamoregeneralperspective,itisalsoimportanttoclassifyminimal,thougheffectivedescriptiveframeworksthatcould guideinthesearch foruniversaltraitsthat happentobe sharedbymultispeciesmodels,besides theirintrinsicdegreeof inherentspecificity. Simplifiedschemes thatexemplifyan exactunderlyingdynamicscan beeffectivelyemployed toshed lightontheonsetofspatio-temporalchaosandpropagationofnonlinearwaves.Externallyimposednonhomogeneous per-turbationcan, forinstance,breakthe synchronyofthe oscillationsasdisplayedby the originalreaction-diffusion system, orequivalentlyits CGLEanalogue,somaterializinginacolorful densitypatternsthat sustainspatio-temporal propagation. The formal link betweencontinuousreaction diffusion-systemsandthe CGLE wasestablished inthe pioneeringwork by Kuramoto [18],exploitinga multiple-scale perturbative analysis.More recently theanalysis hasbeen extended by Nakao

[20] totherelevantsettingwherethereaction-diffusionsystemismadetoevolveonasymmetric,henceundirected, net-work. As remarked in [21], directionality mattersand can seed the emergence of non trivial collective dynamics which cannotmanifestwhenthescrutinizedsystemismadetoevolveonasymmetricdiscretesupport.Motivatedbythisfinding, weconsideredin[22]thedynamicsofareaction-diffusionsystemdefinedonadirectedgraphwhichdisplaysastablefixed pointandobtainedaneffectivedescriptionfortheevolutionmodetriggeredunstable,justabovethethresholdofcriticality. Theanalysisexploitsamultipletime-scaleanalysisandeventuallyyieldsaStuart–Landaufortheamplitudeoftheunstable mode,whosecomplexcoefficientsreflectthetopologyofthenetwork,thefactualdrivetotheinstability.

Inthispaperwe aimatfollowingthesimilar strategyof[22]toderive anapproximateequation fortheevolutionofa reactiondiffusionsystemonadirected(andbalanced)graph,inthevicinityofasupercriticalHopfbifurcation.Asamatter of fact we willassume weakthe strength of thecoupling that linksadjacent nodes.In doing so,we will generalize the workofNakao[20]totheinterestingsettingwhereasymmetryinthecouplingsneedstobeaccommodatedforand,atthe sametime, reformulate theclassical workof Kuramoto[18],ona discrete spatialbacking.To anticipateourfindings, and atvariancewiththeanalysisreportedin[22],we willfinallyobtain aCGLEasaminimal descriptionforthedynamicsof theself-sustainedoscillators coupledona complexandasymmetricgraph.TheobtainedCGLE enablesone toanalytically probethestabilityofthesynchronousuniformstate,asdisplayedbythereaction-diffusionsystem.Specifically,itallowsto determinetheparameters settingthatinstigatesa symmetrybreakingtransitiontonon-uniformpatterns.Numericaltests madefortheBrusselatormodel,hereassumedasareferencemodelforitspedagogicalinterests,willconfirmtheadequacy oftheproposedapproximatescheme.

2. Diffusiveoscillatorsonnetworks

We will here consider a generic two dimensional reaction-diffusion system and label with xj

(

t

)

=

(

φ

j,

ψ

j

)

T for j=

1_,...,N the two-dimensional real vector of the concentrations.The index j refers to the node of the network to which the selected componentsrefer to. N standsfor thesize of the network,i.e. thetotal number ofnodes. The onlyfurther assumptionthatweshallmakeistheexistenceofself-sustainedoscillationsforthesystemunderscrutinyandforthis rea-son wewillpoint tox_j(t) asto theoscillators’variables.The dynamicsofthesystemishencedescribedbythefollowing differentialequation ˙ x j= F

(

x j,

μ

)

+D N  k=1



jkx k (1)

wherethetwo-dimensionalnonlinearfunctionFspecifiesthereactionterms:itdependsonthelocalconcentrationsofthe speciesxjandon

μ

whichisavectorofarbitrarydimensionthatgathertogethertheparametersofthemodel.Thesecond

termrepresentsthediffusivecoupling:D=Kdiag

(

D_φ,D_ψ

)

denotesthediagonalmatrixofthediffusioncoefficients. Kisa constantparameter thatsetthestrengthofthecoupling.Aswewillmakeclearinthefollowingtheperturbative analysis thatweshalldevelop,holdforK_<<1.InEq.(1),



istheLaplacianmatrixwhoseelementsread



i j=Ai j−

δ

i jki.Herewe

focusondirectednetworks, thustheadjacencymatrixAisnotsymmetric.Usingstandardnotations,Ai j=1ifalinkexists

that goesfromnodei tonodej.Otherwise,Ai j=0.ki isthenumberofoutgoingedges fromnode i,

δ

ijistheKronecker’s

delta.Weassumethatsystem(1)admitsahomogeneousequilibriumpointthatweherelabelx∗=

(

φ

∗_,

_ψ

∗

₎

T_.Thisrequest

impliesdealingwithabalancednetwork,namelyanetworkwheretheoutgoingandincomingconnectivitiesareequal. Weadditionallyrequirethatx∗undergoesaHopfbifurcationfor

μ

₌

μ

₀.Accordingly,theJacobianmatrixassociatedto system(1)hasapairofimaginaryeigenvalues ± i

ω

0.Slightlyabovethesupercritical Hopfbifurcation,x∗becomes

unsta-ble,thereaction-diffusionsystemadmitsan timedependent homogeneoussolution.Thisistheuniformstateobtainedby replicatingoneachnodeofthenetworkandincompletesynchrony,thelimitcycledisplayedbythesysteminitsa-spatial limit(K=0).Thespatialcoupling,sensitivetotinynonhomogeneities,whichconfigureasinjectedperturbation,can even-tually destabilized theuniform synchronous equilibrium. When diffusion is small(K=



2_{<<1), the method ofmultiple}

timescales[18]constitutesaviablestrategytocharacterizethenonlinearevolutionoftheperturbationandhenceelaborate onthestabilityofthetime-dependentuniformperiodicsolution.

To this end, inspired by the analysis carried out in [20], we introduce small inhomogeneous perturbations,

δφ

j and

δψ

j, to the uniform equilibriumpoint, namely

(

φ

j,

ψ

j

)

=

(

φ

∗,

ψ

∗

)

+

(

δφ

j,

δψ

j

)

for j=1,...,N.We then substitute this

F. Di Patti et al. / Commun Nonlinear Sci Numer Simulat 56 (2018) 447–456 449

ansatzintoEq.(1).PerformingaTaylorexpansionoftheresultingsystemandpacking

δφ

_jand

δψ

_jintothecolumnvector

u_j=



δφ

_j,

δψ

_j



T_,oneendsupwiththefollowingequationforthetimeevolutionofu_j:

∂

dtu j= Lu j+D N  k=1



jku k+Mu ju j+Nu ju ju j+... (2)

whereListheJacobianmatrixevaluatedatthesteadystatex∗.Adoptingthesamesymbolicnotationsof[18],Mujujand

Nujujujdenotetwo-dimensionalvectorswhosecomponentsrespectivelyread



Mu ju j



l= 1 2! 2  m,n=1

∂

2_F l

(

x ∗,

μ

)

∂

xn

∂

xm

(

u j

)

m

(

u j

)

n



Nu ju ju l



l= 1 3! 2  m,n,p=1

∂

3_F l

(

x ∗,

μ

)

∂

xn

∂

xm

∂

xp

(

u j

)

m

(

u j

)

n

(

u j

)

p forl₌1_,2.

SettingthemodelabovethesupercriticalHopfbifurcation,translatesintoredefining

μ

bymeansofasmallparameter



as

μ

=

μ

0+



2

μ

1,where

μ

1 isorderone.Tofollowthenonlinearevolutionoftheamplitudeofthemostunstablemode,

itisappropriateto introduceaslowtimevariable

τ

₌



2_{t and,accordingly,torewritethetotalderivative withrespectto}

theoriginaltimet: d dt −→

∂

∂

t +



2

∂

∂τ

. (3)

Asa key assumption,alreadyalluded tointhe precedingdiscussion,we setK=



2_{:inother words we supposethat the}

impactofdiffusionisofthesameorderasthedeviationfromthebifurcationpoint.Moreover,nearcriticality,thematrixL

andtheoperators_Mand_N maybeexpandedinpowersof



2 L = L 0+



2L 1+...

M= M0+



2M1+... N= N0+



2N1+...

(4)

Wefurtherassumethatucanbeexpressedasaseries,functionofbothtand

τ

:

u j

(

t

)

=



u (j1)

(

t,

τ

)

+



2_u (2)

j

(

t,

τ

)

+... (5)

InsertingEqs.(3)–(5)intoEq.(2)weget:



∂

∂

tI2+



2

∂

∂τ

I2− L0−



2L 1− ...



(



u (_j1)+



2_u (2) j +...

)

−



2_D N  k=1



jk

(



u _k(1)+



2u (_k2)+...

)

=



2M0u (j1)u ( 1) j +



3

₍

_2M 0u (j1)u ( 2) j +N0u (j1)u ( 1) j u ( 1) j

)

+O

(



4

₎

Collectingtogethertermsofthesameorderin



returnsthefollowingsetofequations



∂

∂

tI2− L0



u (ν)_j = B (ν)_j (6)

for

ν

=1,2,3....TheexistenceofanontrivialsolutionfortheaforementionedlinearsystemsisguaranteedbytheFredholm theorem[23].Asweshalldiscussinamoment,thesolvabilitycondition(seeAppendixA)isdirectlysatisfiedfor

ν

=1and

ν

=2_,whileitmustbeexplicitlyimposedfor

ν

₌3.

Letusstart byfocusingon

ν

=1.Thecorrespondingrighthandsideofsystem(6)isB(_j1)=0.Itiseasytoseethatthe analyticalsolutionis

u (_j1)

(

t,

τ

)

=Wj

(

τ

)

U 0eiω0t+c.c. (7)

whereU0 therighteigenvectorofLcorresponding totheeigenvaluei

ω

0.The complexvariableWj(

τ

)istheamplitudeof

theperturbation.Itsdynamicsissupposed tobe slow,i.e.ruledbytherescaled time

τ

.As weshallsee,by imposingthe solvabilityconditionfor

ν

₌3willreturnanequationforthetimeevolutionofW_j(

τ

).

Movingto

ν

=2,wefindB(_j2)=M0u(j1)u(

1)

j andwetrytocastthesolutionofthelinearsystemintheform

u (_j2)=W2

jV 2e2iω0t+

γ

u (_j1)+

|

Wj

|

2V 0.

Inserting this ansatz into (6) and grouping together terms that do not depend on t, one finds V0=−2L−10 M0U0U¯0

where the bar stands for the complex conjugate. In the same way, collecting terms proportional to e2iω₀t _{yields V}

2=

(

2i

ω

0I2− L0

)

−1M0U0U0.

The third constant term appearing in Eq. (6) is explicitly given by B(_j3)₌



L1−_dd_τI2



u(_j1) ₊D_kN₌₁



_jku(_k1)₊ 2M0u(j1)u( 2) j +N0u(j1)u( 1) j u( 1)

j .Byimposingthesolvabilitycondition,oneeventuallyobtainsthefollowingCGLequationfor

Wj(

τ

) d d

τ

Wj

(

τ

)

=

σ

Wj− g

|

Wj

|

2_W j+d N  k=1



jkWk (8) where

σ

=

σ

Re+i

σ

Im=

(

U∗0

)

†L1U0, g=gRe+igIm= −

(

U∗0

)

†



2M0V2U¯0 +2M0V0U0+3N0U0U0U¯0

and d=dRe+idIm=

(

U∗₀

)

†_DU

0 arecomplexnumbers.

Usingthefollowingscaletransformation

τ

σ

Re →

τ

dRe

σ

Re



i j→



i j

gRe

σ

Re Wj→Wj,

introducingthreenewcoefficients c0=

σ

Im

σ

Re c1= dIm dRe c2= gIm gRe

andmakingthefurthertransformationWj→Wjexp(ico

τ

),wefindthefollowingreducedCGLE

d d

τ

Wj=Wj−

(

1+ ic2

)

|

Wj

|

2_W j+

(

1+ic1

)

 k



jkWk (9)

forthecomplexamplitudeWjofthej-thoscillator.

Eq.(9)displaysauniformlysynchronizedsolutiongivenby

Wj

(

τ

)

≡ WLC=e−ic2τ (10)

for j=1,...,N.Inthisrespectitprovidesalocalapproximationforthedynamicsoftheoriginalreaction-diffusionsystem neartheHopfbifurcation,namelywhenself-organizedoscillationsrise.ThedynamicsoftheensemblemadeofNoscillators issensitivetothecouplingimposed,thislatterbeingencodedintheLaplacianoperator.Mutualinterferencesamong oscil-lators,asmediatedbytheweakdiffusivecouplinghereatplay,candisruptthecoherencyofthesynchronousstate,making thesystemtoevolvetowardsa differentattractor.Thestabilityofthesynchronizeddynamicscanbe assessedviaalinear stabilityanalysis whichis carriedout forCGLE, andwhoseconclusion can be readily translatedto theoriginal reaction-diffusion framework. Havingobtaineda universal description ofthe dynamics ofthe systemin termsof anormal mode representationallowsonetoobtaingeneralconditionsoftheonsetoftheinstability,whichprescindthespecificcontextof analysis.

Toinvestigatethe linearstabilityofsolution(10),we briefly review themethod describedin[20,24]andpoint tothe specificaspectsthatrelatetothedirectednatureofthecouplings,asdiscussedin[25].Noticethatinthislatterpaper,the CGLEisassumedasthereferencemodel,whilehereitisobtainedasalocalapproximationofagenericreaction-diffusion systemdefinedonadirectednetwork,underweakdiffusivecouplings.

Tocarryoutthelinearstabilitycalculation,wesubstitutethefollowingamplitudeandphaseperturbation[20,24]

Wj

(

τ

)

=WLC

(

τ

)(

1+

ρ

j

(

τ

))

eiθj(τ )

intoEq.(9)andwelinearizearound

ρ

_j=0and

θ

_j=0obtaining



˙

ρ

j ˙

θ

j



=



−2 0 −2c2 0



ρ

j

θ

j



+ N  k=1



jk



1 −c1 c1 1



ρ

k

θ

k



. (11)

Using theLaplacianeigenvaluesandeigenvectorsthat, bydefinition,satisfy



(α) ₌

_

(α)

_

(α) _for

α

_=1,...,N,the

inho-mogeneousperturbations

ρ

jand

θ

jcanbeexpandedas



ρ

j

θ

j



= N  α=1



ρ

(α)

θ

(α)



eλ(α)τ



(α)_j (12)

where

ρ

(α)_and

_θ

(α)_{denotetheexpansioncoefficients, while}

_λ

(α) _{controlstheexponentialgrowthofthe}

_α

_thmode.Ifthe

real part of

λ

(α) ₍

_λ

(α)

Re ) is negative, perturbations shrink to zero,otherwise they grow exponentially. Plugging expansion

(12)intoEq.(11)yieldsthefollowingcondition

det



J _α−

λ

(α)_I2



_{=0 (13)} with J _α=



−2+



(α) _−c1

_

(α) −2c2+c1



(α)



(α)



.

F. Di Patti et al. / Commun Nonlinear Sci Numer Simulat 56 (2018) 447–456 451

Constrain(13)returnsanequation for

λ

(α) _{asa functionof}

_

(α) _{knownasthedispersion relation.Fora symmetric}

undi-rectednetwork,astheoneexploredbyNakao[20,24],theeigenvalues



(α) _{oftheLaplacianoperatorsarerealandthis}

ob-servationtranslatesinsimpleconditionfortheinstabilitytodevelop.Astraightforwardcalculation,allowsonetoconclude thatthe perturbationcan grow,andso destroythe synchronyofthesystem, provided 1+c1c2<0.When theunderlying

graphofcouplingisinsteaddirected,thespectrumoftheLaplacianoperatoriscomplexandtheinstabilitycansetinalso when1+c1c2>0.Forthisreason,thegeneralizedclassofinstabilitythatisencounteredwhenthesystemisconfinedona

asymmetricsupportisreferredtoastotopologydriven.Theconditionswhichresultin

λ

(α)_Re >0whenthenetworkof cou-plingsisdirected,or,equivalently,when



(α) _iscomplex(

_

(α)₌

_

(α)

Re +i



(α)Im),havebeenworkedoutin[25],buildingon

therecipeoutlinedin[21].Theconclusionof[25] areherebrieflyreviewedforthesakeofcompleteness.Indeed,

λ

_Re(α)>0 if S2

(

(α)Re

)

S1

(

(α)Re

)





Im(α)

2 (14) where S2

(

αRe

)

=C2,4

(

(α)Re

)

4− C2,3

(

(α)Re

)

3+C2,2

(

Re(α)

)

2− C2,1



Re(α) S1

(

αRe

)

=C1,2

(

αRe

)

2− C1,1



Re(α)+C1,0 with C2,4=1+c21 C2,3=4+2c1c2+2c21 C2,2=5+4c1c2+c21 C2,1=2+2c1c2 C1,2=c41+c21 C1,1=2c31c2+2c21 C1,0=c21

(

1+c22

)

.

Whencondition(14)ismetforsomeeigenvalue



(α)_{,thelimitcycleloosesits stabilityandthesystemevolvestowardsa}

differentattractors,whichisnonhomogeneousinspace.

Thenext sectionisaimedatchallengingthevalidityofEq.(9),asanapproximateequationforthenonlinearevolution ofan injectednon homogeneousperturbation. Toanticipateourfindings, wewill showthat theCGLE allows tocorrectly delineatethe region of parameters that yields stableandsynchronized self-sustained oscillators,and so accurately mark the transitionto the pattern forming domain. In both cases,i.e. when uniform oscillations or complexpatterns in den-sityare displayed,we obtain asatisfyingdegree ofcorrespondencebetweentheoriginal reaction-diffusionmodelandits correspondingCGLE.ToperformthenumericaltestswewillmakeuseoftheBrusselatormodel,asareferencecasestudy.

3. Numericalvalidationofthetheory:theBrusselatormodel

Tochallengethe analysisperformedinthe previoussection, we hereconsideraspecific reaction-diffusionsystem, the Brusselatormodel,whichallowsforself-sustainedoscillations.Thegoverningequationsread:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

˙

φ

j=A−

(

B+1

)

φ

j+

φ

2j

ψ

j+Dφ N  k=1



jk

φ

k ˙

ψ

j=B

φ

j−

φ

2j

ψ

j+Dψ N  k=1



jk

ψ

k (15)

where A, B stand for non negative constants parameters. This system admits a homogeneous steady state given by

(

φ

∗_,

_ψ

∗

₎

₌

₍

_A,B/A

₎

_{.ByactingontheparameterB,whilekeepingA fixed,onecan induceaHopfbifurcation,whichopens}

uptheroutetoself-sustainedoscillations.ThebifurcationoccursforBc=1+A2,asexplainedintheannexedAppendixB.

Abovethebifurcationpoint,theBrusselatordynamicscanbeapproximatedbyaCGLE.Followingthederivationreportedin theAppendixB,whichbuildsontheseminalpaperbyKuramoto[18],oneendsupwiththecompactexpressionforc1and

c2: c1=−A D_φ− D_ψ D_φ+D_ψ c2= 4− 7A2_+4A4 3A

(

2+A2

)

. (16)

The CGLE is therefore completely characterized andit can be readily employed to asses the stability of the uniform oscillatorystate,namelythespatiallyextendedconfigurationthatisobtainedasthesynchronousreplicaofNself-sustained

Fig. 1. Panel (a): eigenvalues of the Laplacian for a balanced Newman–Watts [26] network generated with p = 0 . 27 and N = 100 nodes. The algorithm to built the network is detailed in [22] . The shaded area marks the instability region for the Brusselator model, as obtained under the CGL normal form representation. Here, A = 1 . 55 , B c = 1 + A 2 , D φ= 0 . 6 and D ψ = 4 . 9 . The perturbative parameter is = 0 . 07 . For the case at hand, A c = 1 . 48 . Panel (b): the

real part of the dispersion relation (magenta triangles) for the same choice of network and parameters as in panel (a). The black line originates from the continuous theory. Panel (c): pattern relative to species φobtained by numerical integration of system (15) . Panel (d): pattern relative to species φ

calculated using the CGL approximation.

oscillators.FixthevaluesofD_φ andD_ψ.Then,thestabilityofthesynchronousstateisultimatelycontrolled byA,thesole parameterthatonecanfreelytune.WewilldenotewithActhevalueofA(ifitexists),abovewhichstabilityiseventually

lost.Moreover,wewillsettheparametersc1andc2 soastoenforcestabilityonasymmetricsupport,namely1+c1c2<0.

TheinstabilitythatmaterializesforA>Acishenceindirectlyreflectingtheasymmetryoftheimposedcouplings.

TheresultsoftheanalysisisdepictedinFig.1(a)and(b).Theshaded greyareaintheplane (



(α)_Re ,



(α)Im)delimitsthe

regionofinstability,asitfollowsinequality(14).SymbolsrepresenttheeigenvaluesoftheLaplacianmatrixinthecomplex plane andfollowthespecificchoice ofthenetwork operated.Asexplained inthecaptionofFig.1,we hereworkwitha balancedNewman-Wattsgraph,generatedbyfollowingtherecipediscussedin[22].FortheselectedvalueofA(largerthan thecriticalvalueAc=1.48),twoeigenvaluesoftheLaplacianmatrixprotrudeinsidetheshadedregion,thustriggeringthe

instability.Thisisindeedatopologydriveninstability,asitcannotrealizeifthesamechoiceofparametersisoperatedforan homologoussystemhostedonasymmetricsupport.Toarguealongthisline,considerasymmetricnetworkofcouplings:the spectrumoftheLaplacianmatrixisnowreal(



(α)_Im =0)andshouldhencefallontherealaxisofFig.1(a).Thegreyregion wheretheinstabilitymaterializes,intersectstherealaxisonlyintheorigin,wherethetrivialzeroeigenvalue(corresponding totheuniform,fullysynchronized,limit cyclestate)sits.ForthechosenvaluesofA andB,hencec1 andc2,itistherefore

necessarytoactivateanimaginarycomponentof



(α)_Im,toinvadetheregionofthecomplexplanedeputedtotheinstability. Tostate it differently,forthesame choiceofparameters, theinstability sets inonlyifa suitable degreeof asymmetryis enforcedinthenetworksofconnections,whileitisboundtofadeaway otherwise.ThisscenarioisconfirmedbyFig.1(b) wherethe (magenta online)triangles representthereal partofthe dispersionrelation,

λ

Re, asafunction of−



(α)Re . Also

inthiscase, the dispersionrelation ispunctuallypositive incorrespondenceoftwo specificentries −



(α)Re .The solid line

curve representsthehomologousdispersionrelationwhenthediffusive couplingisinstead declinedonasymmetric(and continuous)support.

PatternsemergingfromtheaforementionedinstabilityaredisplayedinFig.1.Panel(c)showspatternsrelativetospecies

φ

obtainedvia a numericalintegration of system(15),while panel(d) originates fromthe CGLE (8). The degreeof corre-spondenceissatisfactoryandtestifiesaposterioriontheadequacyoftheproposedapproximation.

F. Di Patti et al. / Commun Nonlinear Sci Numer Simulat 56 (2018) 447–456 453

Fig. 2. Panel (a): eigenvalues of the Laplacian for a balanced Newman–Watts [26] network generated with p = 0 . 27 and N = 100 nodes. The algorithm to built the network is detailed in [22] . The shaded area marks the instability region for the Brusselator model with A = 1 . 35 < A c , D φ= 0 . 6 and D ψ = 4 . 9 . The perturbative parameter is = 0 . 07 . Panel (b): the real part of the dispersion relation (magenta triangles) for the same choice of network and parameters as in panel (a). The black line originates from the continuous theory. Panel (c): pattern relative to species φobtained by numerical integration of system (15) . Panel (d): pattern relative to species φcalculated using the CGL equation.

Fig.2refers toA<Ac:thesynchronizedlimit cyclesolutionispredictedtobestable,followingtheCGLapproximation.

Directintegrationoftheoriginalreactiondiffusionsystemconfirmsthisconclusion.Thedegreeofcorrespondencebetween originalandapproximatedschemesis,alsointhiscase,satisfying.

4. Conclusions

Synchronizationis a fascinatingfield of investigation that hasapplications ranging frombiology to computer science, justto namefew examples.Whenthisphenomenon occurson complexnetworks,thestabilityofthesynchronizedstates isstronglyaffectedbythetopologyofthegraph.Whenthenetworkisdirected,external nonhomogeneousperturbations candestabilize synchronousoscillations alsoifthe parametersare setto valuesthat fallin theregion ofstabilityforthe dynamicsformulatedonasymmetricsupport.

Startingfromthesepremises,wehereconsideredareactiondiffusionsystemdefinedonadirected(andbalanced)graph. Thesystemismadeto evolveinthevicinityofasupercritical Hopfbifurcationandweakdiffusive couplingsareassumed toholdbetweenadjacentnodes.BygeneralizingtheworkofNakao[20]tothesettingwhereasymmetryinthecouplings isenforced,andadapting themultiple-scales approachdiscussed inKuramoto [18]tothecaseofa discrete,network like support,wederivedanormalformequationtoapproximatethelocalevolutionoftheunderlyingreaction-diffusionsystem. ThisisaComplexGinzburg-Landauequation(CGLE)whosecoefficientsclearlydependontheparametersofthemodel,but alsoonthetopologicalcharacteristicsofthehostingnetwork.TheCGLE canbeeffectivelyemployedtoprobethestability oftheperiodicuniformsolutionandworkout theconditionsfortheonsetofthesymmetrybreakinginstabilitythat even-tuallydestroythesynchronizedregime. NumericaltestsperformedfortheBrusselatormodel,confirmtheadequacyofthe proposed approximation.Patterns obtained underthe CGLE schemeresemblecloselythose displayedupon integration of theoriginalreaction-diffusionsystem.

Acknowledgments

ThisworkhasbeensupportedbytheprogramPRIN2012fundedbytheItalianMinisterodell’Istruzione,dellaUniversità edellaRicerca(MIUR).D.F.acknowledgesfinancialsupportfromH2020-MSCA-ITN-2015projectCOSMOS642563.Thework

ofT.C.presentsresearch resultsoftheBelgianNetworkDYSCO(DynamicalSystems,Control,andOptimization),fundedby theInteruniversityAttractionPolesProgramme,initiatedbytheBelgianState,SciencePolicyOffice.

AppendixA. Thesolvabilitycondition

WeconsideralinearoperatorA,andtwocomplexvectorsu(t)andb(t)ofthesamelength.AccordingtotheFredholm theorem,thelinearsystemAu

(

t

)

=b

(

t

)

issolvableif



v

(

t

)

,b

(

t

)



=0forallvectorsv(t)solutionofA∗v

(

t

)

=0,whereA∗is theadjointoperatorsatisfying



A∗y,x



=



y,Ax



∀

x,y.Theangularbracketsdenotethescalarproductthatwe heredefine as



v

(

t

)

,b

(

t

)



=2π/ω0

0 v†

(

t

)

b

(

t

)

dt,thesymbol†standingfortheconjugatetranspose.

With referenceto Eq.(6), thefirst requirement ofthe Fredholm theoremconsists in findingv(t) such that

(

∂

/

∂

tI2− L0

)

∗v

(

t

)

=0.Bypartial integration,andrecalling thatL0 isa realmatrix,we findthat

(

∂

/

∂

tI2− L0

)

∗=−

(

∂

/

∂

tI2N+L0

)

T.

Asa consequence,the systemtobe solvedis−

(

∂

/

∂

tI2+L0

)

Tv

(

t

)

=0.Inanalogy withEq.(7),wesearchv(t)intheform v

(

t

)

=U∗₀eiω0t _{forsomevectorU}∗

0.Substitutingthisansatz intotheprevious equation,wefindL0TU∗0=−i

ω

0U∗0.Tosimplify

calculation,itisconvenienttonormalizeU∗₀sothat

(

U∗₀

)

†_U 0=1.

Having defined U∗_0, one can explicitly write down the solvability condition



U∗₀eiω₀t_,B(ν)

j

(

t,

τ

)



=0. Since B(ν)j

(

t,

τ

)

turns out to be periodic functions of period 2

π

/

ω

0, it is appropriate to express them in the form B(ν)j

(

t,

τ

)

=

+∞

l=−∞



B(ν)_j

(

τ

)



l

eilω₀t_{. If we multiply this series by}

₍

_U∗

0eiω0t

)

† we again obtain periodic functions that, when

inte-grated over the period 2

π

/

ω

0 give zero. The only exception holds for l=1 which gives



U∗0eiω0t,



B(ν)_j

(

τ

)



1 eiω0t



= 2π/ω0 0

(

U∗0

)

†



B(ν)_j

(

τ

)



1dt.Theintegranddoesnotdependontimetandthereforetheintegraliszeroonlyiftheintegrand

itselfisidenticallyequaltozero.Forthisreasonthesolvabilityconditionreducesto

(

U∗₀

)

†



_B(ν)

j

(

τ

)



1=

0

∀

j.

AppendixB.DetailsofthecoefficientsoftheGLfortheBrusselatormodel

InthissectionweexplicitlyderivethecoefficientsoftheCGLequationfortheBrusselatormodel. TheJacobianmatrixassociatetosystem(15)evaluatedattheequilibriumpointreads

L 0=



B− 1 A2 −B −A2



.

Settingthe systemat thebifurcation point means requiringL0 to havea pairof imaginaryeigenvalues, which translates

intotrL0=0.ThisrelationimpliesaconstrainonthecriticalvalueoftheparameterBwhichmustbechosenasBc=1+A2.

Withthisassumption,L0 canberewrittenas L 0=



A2 _A2

−

(

1+A2

₎

_−A2



anditspairofimaginaryeigenvaluesare

λ

0=±i

ω

0=±iA.TherighteigenvectorsofL0andLT0correspondingto,respectively,

eigenvaluesiAand−iAaregivenby

U 0=



1 −1+ i A



U ∗₀=1₂



1+iA iA



.

Toproceedwiththemultiscaleanalysis,we perturbBc asB=Bc+



2Bc andweimmediatelyfindtheentriesofmatrix

L1 L 1=

(

1+A2

)



1 0 −1 0



andthevalueof

σ

whichturnsouttobe

(

1+A

2

₎

F. Di Patti et al. / Commun Nonlinear Sci Numer Simulat 56 (2018) 447–456 455

Theoperators_M0andN0aregivenby

(

M0xy

)

1=−

(

M0xy

)

2 =1+A2 A x1y1+A

(

x1y2+x2y1

)

(

N0xyz

)

1=−

(

N0xyz

)

2 =1 3

(

x1y1z2+x2y1z1+x1y2z1

)

where(· )idefinestheithcomponentofatwo-dimensionalvector.Thesepreliminaryquantitiesallowustocalculate

M0U 0U 0=

⎛

⎜

⎝

1 A− A + 2i −1 A+A− 2i

⎞

⎟

⎠

M0U 0U ¯0=

⎛

⎜

⎝

1 A− A −1 A+A

⎞

⎟

⎠

(B.1) and N0U 0U 0U ¯0=

⎛

⎜

⎜

⎝

−1+ i 3A 1− i 3A

⎞

⎟

⎟

⎠

.

FromEq.(B.1)wecanmakeexplicitV0andV2 V 0= 2

(

A2_{− 1}

₎

A3



0 1



V 2=− 1 3

⎛

⎜

⎜

⎝

−4 A+2



₁ A2− 1



i −1 A





₁ A2 − 5



+2



2 A− A



i



⎞

⎟

⎟

⎠

andthus M0U ¯0V 2=

⎛

⎜

⎜

⎜

⎝

−1+ 1 A2+i 2 3



A2_{− A}4_{− 1} A3



+1− 1 A2− i 2 3



A2_{− A}4_{− 1} A3



⎞

⎟

⎟

⎟

⎠

M0U 0V 0=

⎛

⎜

⎜

⎝

2

(

A2_{− 1}

₎

A2 −2

(

A2− 1

)

A2

⎞

⎟

⎟

⎠

.

NoticethatourequationsforU0andV2 areslightlydifferentfromthosereportedin[18].Despitethisminordifference,we

endupwithsamevaluesofdandg.Finally,wecanwritedownthetwolastcoefficientsoftheCGLequation: d= 1 2[Du+Dv− iA

(

Du− Dv

)

] g= 1₂



A2₊₂ A2 +i



4A4_{+4− 7A}2 3A3



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