Uni e si a Poli `ecnica de Ca alunya
Facul a de Ma em`a iques i Es ad´ıs ica
Mas e in Ad anced Ma hema ics and Ma hema ical Enginee ing
Mas e ’s hesis
Synch oniza ion in Ne wo ks o Coupled
Oscilla o s using a phase-ampli ude
amewo k
Laia Poma Palla `es
Supe ised by Gemma Hugue Casades and Ma ina Vegu´e Llo en e
Janua y, 2025
I would like o exp ess my g a i ude o my ad iso s, Gemma Hugue and Ma ina Vegu´e, o gi ing
me he oppo uni y o wo k wi h hem, and o hei ime, suppo , and encou agemen h oughou his
p ojec . Thei guidance and eedback ha e been essen ial in shaping he di ec ion o his wo k.
I also wan o gi e special hanks o my iends and amily o hei suppo .
Abs ac
Ne wo ks o coupled nonlinea oscilla o s can exhibi a wide ange o eme gen beha io s depending on he
connec i i y ma ix. In his p ojec , we in es iga e he dynamics o modula and homogeneous di ec ed
ne wo ks o oscilla o s desc ibed by he phase-ampli ude equa ions. We employ a ecen ly de eloped dimen-
sion educ ion echnique (Vegu´e e al., 2023) o analyze a ious synch oniza ion egimes, such as comple e
synch oniza ion, clus e synch oniza ion, and desynch oniza ion. Ou goal is o adap his echnique o a
ne wo k o oscilla o s go e ned by he phase-ampli ude equa ions and o assess he accu acy o he educed
model h ough nume ical simula ions.
Keywo ds
Complex ne wo ks, dynamical sys ems, dimensionali y educ ion, synch oniza ion egimes, phase-ampli ude
equa ions, oscilla o s
1
Con en s
1 In oduc ion 3
2 S a emen o he p oblem 4
2.1 Phase-ampli ude educ ion .................................. 4
2.1.1 Singleoscilla o .................................... 4
2.1.2 Ne wo k o coupled oscilla o s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Applica ion o a pa icula oscilla o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Dimensionali y educ ion 11
3.1 T ans o ma ion o he complex ci cle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Spec al educ ion in he complex ci cle . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Resul s 17
4.1 Analysis o a 2-dimesional ne wo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Phase Synch oniza ion Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Simula ions .......................................... 19
4.3.1 Pa ame e s o he model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.2 Simula ions o equilib ium dynamics in he educed model . . . . . . . . . . . . . 21
4.3.3 Simula ions o he comple e ne wo k . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Explo a ions wi h an al e na i e model 28
5.1 Al e na i e coupling unc ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Simula ions .......................................... 28
5.2.1 Simula ions wi h homogeneous coupling . . . . . . . . . . . . . . . . . . . . . . . 29
5.2.2 Simula ions wi h he connec i i y ma ix . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusions 33
A Ma hema ical de i a ions o he a e aging me hod 35
B Py hon code used o he simula ions 37
2
1. In oduc ion
The s udy o he dynamics o ne wo ks o coupled oscilla o s is a cen al opic in a ious ields o science
and enginee ing, anging om neu oscience o physics and social dynamics. Synch oniza ion egimes a e
undamen al o unde s and he collec i e beha io o complex sys ems and ha e p ac ical applica ions in
ields such as neu al synch oniza ion, powe g id s abili y, and biological pa e n o ma ion. Howe e , he
high-dimensional na u e o hese ne wo ks is a big challenge when pe o ming analy ical and compu a ional
s udies. Cons uc ing a educed ep esen a ion o a la ge complex sys em emains an open p oblem,
highligh ing he alue o de eloping educ ion echniques ha simpli y ne wo k dynamics while p ese ing
hei essen ial ea u es.
In his p ojec , we ocus on applying he spec al educ ion in oduced in [9] o a ne wo k o coupled
oscilla o s ep esen ed in phase-ampli ude coo dina es. The phase-ampli ude educ ion le e ages he in-
he en p ope ies o limi cycles o desc ibe he dynamics o each oscilla o using only wo a iables: he
phase and he ampli ude de ia ion along he leas con ac ing di ec ion. On he o he hand, he spec al
educ ion is a dimensionali y- educ ion echnique o dynamical sys ems on di ec ed ne wo ks o ganized in
clus e s, whe e he nodes wi hin he same clus e ha e simila ac i i y.
We apply hese me hodologies o a canonical model o an And ono -Hop bi u ca ion, as in oduced in
[8], o explo e whe he spec al educ ion can accu a ely app oxima e he dynamics o ne wo ks composed
o such oscilla o s. Ou goal is o de e mine whe he he educed sys em can ai h ully cap u e he synch o-
niza ion egimes o he ull ne wo k. This includes examining he in luence o ne wo k pa ame e s, such as
he s eng h o gene al coupling ϵand he phase o se ψ, on he synch oniza ion p ope ies. Addi ionally,
we also begin an in es iga ion in o an al e na i e model wi h di e en coupling unc ions, as p esen ed in
[6], se ing he s age o u u e esea ch ela ed o his wo k.
The s uc u e o he p ojec is he ollowing. In Sec ion 2 we in oduce he heo e ical ounda ions
o oscilla o s and phase-ampli ude educ ion, including concep s such as isoch ons and isos ables, as well
as a gene al o m o his pa ame e iza ion. We also p esen he equa ions in Ca esian coo dina es o he
oscilla o p esen ed in [1], and de i e i s exp ession in he gene al o m o he phase-ampli ude educ ion.
In Sec ion 3, we apply he dimension- educ ion echnique ha we discussed p e iously, o he gene al o m
o he phase-ampli ude pa ame e iza ion. In Sec ion 4, we s udy analy ical p ope ies o a wo-dimensional
ne wo k o he p esen ed oscilla o s, in oduce a di ec ed homogeneous ne wo k di ided in o wo g oups,
and p esen some nume ical simula ions o bo h he comple e ne wo k and he educed wo-dimensional
sys em. Finally, in Sec ion 5, we p esen some open wo k using an al e na i e model wi h he same
oscilla o s bu di e en coupling unc ions.
All he plo s shown in his manusc ip , wi hou e e ence, ha e been de eloped by he au ho using
Py hon. The code used o hese igu es and simula ions can be ound in he Appendix B.
3
Synch oniza ion in ne wo ks o couple oscilla o s
2. S a emen o he p oblem
2.1 Phase-ampli ude educ ion
Unde s anding he dynamics o oscilla o y sys ems o en equi es modelling complex, high-dimensional
sys ems using simpli ied app oaches. In his sec ion, we in oduce phase-ampli ude a iables, a amewo k
which allows ep esen ing he beha io and dynamics o an oscilla o x∈Rdusing only wo a iables ha
desc ibe he oscilla o y phase and de ia ions om he limi cycle along he slowes con ac ing di ec ion.
The ollowing explana ion o phase-ampli ude a iables is based on [1] and [7].
2.1.1 Single oscilla o
We begin by conside ing a single oscilla o ha is desc ibed by he ollowing sys em o di e en ial equa ions
˙x=X(x), (1)
whe e X:Rd→Rdis an analy ic ec o ield. The low is deno ed by ϕ (x), and he sys em has a
T-pe iodic a ac ing hype bolic limi cycle Γ.
Ou goal is o ind a local analy ic di eomo phism
K:T× B ⊂ T×Rd−1→Rd
(θ,σ)7→ K(θ,σ) = x(2)
in such a way ha he dynamics o he ec o ield Xin hese new a iables is gi en by
˙
θ=1
T=ω, ˙σ= Λσ, wi h Λ =
λ1
...
λd−1
, (3)
whe e λ1, ... , λd−1a e he cha ac e is ic exponen s o he pe iodic o bi . No ice ha he a iable θ o a es
a cons an speed ωand he a iables σicon ac a a e λi. I is impo an o no ice ha he cha ac e is ic
exponen s λia e nega i e because we a e assuming ha he limi cycle Γ is hype bolic a ac ing. Mo eo e ,
we will suppose ha hese exponen s a e eal and dis inc , ha is, λi∈Rand λd−1<... < λ1<0.
Using hese new a iables, we can see ha he e olu ion o he low ϕ (x) is desc ibed by
ϕ (K(θ,σ)) = K(θ+ω ,eΛ σ). (4)
I is in e es ing o no e ha he limi cycle Γ co esponds o he poin s whe e σequals ze o, i.e.,
Γ := nx∈Rd|x=K(θ, 0) o θ∈To. (5)
The e o e, he unc ion γ(θ) = K(θ, 0) p o ides a pa ame e iza ion o he limi cycle Γ in e ms o he
phase a iable θ.
As we a e conside ing a hype bolic limi cycle, o any poin pin a neighbo hood o he limi cycle Ω
con ained in i s basin o a ac ion, he e exis s a poin q∈Γ such ha
lim
→∞|ϕ (p)−ϕ (q)|= 0. (6)
4
We say ha pand qha e he same asymp o ic phase, and we de ine he isoch on Iθas he se o all poin s
wi h he same asymp o ic phase θ, ha is
Iθ=np∈Ω|lim
→∞|ϕ (p)−ϕ (γ(θ))|= 0o. (7)
The di eomo phism Ksa is ies he ollowing p ope y
∀x∈ Iθx=K(θ,σ) o some σ∈Rd−1. (8)
Mo eo e , we can de ine a scala unc ion Θ ha assigns o any poin xin Ω, i s asymp o ic phase θ, ha
is,
Θ: Ω ⊂Rd→T
x7→ Θ(x) = θ.(9)
Wi h all his, he le el cu es o Θ co espond o he isoch ons Iθo he oscilla o
Iθ={x∈Ω|Θ(x) = θ}. (10)
Analogously, we can also de ine d−1 scala unc ions Σi(i= 1, ..., d−1) ha map any poin x∈Ω
o he ampli ude a iable σi
Σi: Ω ⊂Rd→R
x7→ Σi(x) = σi.(11)
The le el cu es o hese unc ions a e called isos ables
Ai
σi={x∈Ω|Σi(x) = σi}. (12)
To ge a be e idea o hese concep s, in Figu e 1, we show an example o some isoch ons and isos ables
o he And ono -Hop oscilla o p esen ed in [8], which we will s udy in mo e de ail in he nex sec ion, o
di e en alues o i s pa ame e s.
No ice ha he ec o unc ion (Θ(x), Σ1(x), ... , Σd−1(x)) a e he in e se o K, sa is ying he ollowing
p ope y:
K(Θ(x), Σ1(x), ... , Σd−1(x)) = x. (13)
So, i we de i a i e on bo h sides:
Id×d=DK(θ,σ)
∇Θ(x)
∇Σ1(x)
.
.
.
∇Σd−1(x)
(14)
and he e o e,
DK(θ,σ)−1=
∇Θ(x)
∇Σ1(x)
.
.
.
∇Σd−1(x)
. (15)
5
Synch oniza ion in ne wo ks o couple oscilla o s
Now o equa ions (21b) and (21c).
˙σk=λσk+
2
X
m=1
˜
G(σ)
m(zk,zk,σk,hm
k),
hm
k=
N
X
j=1
wkj ˜
Hm(zk,zk,σk,zj,zj,σj).
Joining e e y hing, we ge he phase-ampli ude pa ame e iza ion o he oscilla o s on he complex uni
ci cle
˙zk=i2πzkω+
2
X
m=1
i2πzk˜
G(z)
m(zk,zk,σk,hm
k), (43a)
˙σk=λσk+
2
X
m=1
˜
G(σ)
m(zk,zk,σk,hm
k), (43b)
hm
k=
N
X
j=1
wkj ˜
Hm(zk,zk,σk,zj,zj,σj). (43c)
3.2 Spec al educ ion in he complex ci cle
Now ha we ha e de ined ou sys em in he complex uni ci cle (43), we can p oceed o apply he spec al
educ ion. To do his, i s we de ine he ollowing mac oscopic obse ables
Zν=X
k∈Gν
aνkzk, Σν=X
k∈Gν
aνkσk,Hm
ν=X
k∈Gν
aνkhm
kν= 1 ... n, (44)
whe e aν=aνkN
k=1 is he educ ion ec o o he g oup Gν, and sa is ies he ollowing condi ions
N
X
k=1
aνk= 1, aνk≥0∀k,aνk= 0 i k/∈Gν. (45)
Using equa ions o ou sys em (43), we de i e he exac dynamics o hese mac oscopic obse ables
˙
Zν=X
k∈Gν
aνk˙zk=i2πω X
k∈Gν
aνkzk+i2π
2
X
m=1 X
k∈Gν
aνkzk˜
G(z)
m(zk,zk,σk,hm
k)
˙
Zν=i2πωZ+i2π
2
X
m=1 X
k∈Gν
aνkzk˜
G(z)
m(zk,zk,σk,hm
k), (46)
˙
Σν=X
k∈Gν
aνk˙σi=λX
k∈Gν
aνkσk+
2
X
m=1 X
k∈Gν
aνk˜
G(σ)
m(zk,zk,σk,hm
k)
12
˙
Σν=λΣν+
2
X
m=1 X
k∈Gν
aνk˜
G(σ)
m(zk,zk,σk,hm
k), (47)
Hm
ν=X
k∈Gν
aνkhm
k=X
k∈Gν
aνk
N
X
j=1
wkj ˜
Hm(zk,zk,σk,zj,zj,σj)
Hm
ν=
n
X
ρ=1 X
k∈GνX
j∈Gρ
aνkwkj ˜
Hm(zk,zk,σk,zj,zj,σj). (48)
Ou objec i e is o exp ess his as an ODEs sys em in e ms o he mac oscopic a iables. Howe e ,
his canno be achie ed in gene al. Ins ead, we app oxima e he dynamics by using he i s -o de Taylo
se ies o he unc ions zk˜
G(z)
m(zk,zk,σk,hm
k), ˜
G(σ)
m(zk,zk,σk,hm
k) and ˜
Hm(zk,zk,σk,zj,zj,σj) a ound he
obse ables in o de o ge a closed- o m sys em. I is impo an o no e, ha his app oxima ion only
makes sense when he ac i i ies o he nodes wi hin he same g oup a e close o each o he , since in ha
case, he nodes will also be close o hei co esponding obse ables.
Le ’s s a wi h he unc ions zk˜
G(z)
m(zk,zk,σk,hm
k)
zk˜
G(z)
m(zk,zk,σk,hm
k)≈ Zν˜
G(z)
m(Zν,Zν, Σν,Hm
ν)+(Zν˜
G(z)
m,1(Zν,Zν, Σν,Hm
ν) + ˜
G(z)
m(Zν,Zν, Σν,Hm
ν))(zk− Zν)
+Zν˜
G(z)
m,2(Zν,Zν, Σν,Hm
ν)(zk− Zν) + Zν˜
G(z)
m,3(Zν,Zν, Σν,Hm
ν)(σk−Σν)
+Zν˜
G(z)
m,4(Zν,Zν, Σν,Hm
ν)(hm
k− Hm
ν),
whe e ˜
G(z)
m,l(Zν,Zν, Σν,Hm
ν) e e s o he de i a i e o ˜
G(z)
m(Zν,Zν, Σν,Hm
ν) wi h espec o he l- h
a iable. Analogously,
˜
G(σ)
m(zk,zk,σk,hm
k)≈˜
G(σ)
m(Zν,Zν, Σν,Hm
ν) + ˜
G(σ)
m,1(Zν,Zν, Σν,Hm
ν)(zk− Zν)
+˜
G(σ)
m,2(Zν,Zν, Σν,Hm
ν)(zk− Zν) + ˜
G(σ)
m,3(Zν,Zν, Σν,Hm
ν)(σk−Σν)
+˜
G(σ)
m,4(Zν,Zν, Σν,Hm
ν)(hm
k− Hm
ν),
whe e ˜
G(σ)
m,l(Zν,Zν, Σν,Hm
ν) e e s o he de i a i e o ˜
G(σ)
m(Zν,Zν, Σν,Hm
ν) wi h espec o he l- h
a iable.
˜
Hm(zk,zk,σk,zj,zj,σj)≈˜
Hm(Zν,Zν, Σν,Zρ,Zρ, Σρ) + ˜
Hm,1(Zν,Zν, Σν,Zρ,Zρ, Σρ)(zk− Zν)
+˜
Hm,2(Zν,Zν, Σν,Zρ,Zρ, Σρ)(zk− Zν) + ˜
Hm,3(Zν,Zν, Σν,Zρ,Zρ, Σρ)(σk−Σν)
+˜
Hm,4(Zν,Zν, Σν,Zρ,Zρ, Σρ)(zj− Zρ) + ˜
Hm,5(Zν,Zν, Σν,Zρ,Zρ, Σρ)(zj− Zρ)
+˜
Hm,6(Zν,Zν, Σν,Zρ,Zρ, Σρ)(σj−Σρ)
whe e ˜
Hm,l(Zν,Zν, Σν,Zρ,Zρ, Σρ) e e s o he de i a i e o ˜
Hm(Zν,Zν, Σν,Zρ,Zρ, Σρ) wi h espec o
he l- h a iable.
13
Synch oniza ion in ne wo ks o couple oscilla o s
Plugging his in o equa ions (46), (47) and (48), espec i ely,
˙
Zν≈i2πωZν+i2π
2
X
m=1
Zν˜
G(z)
m(Zν,Zν, Σν,Hm
ν), (49)
˙
Σν≈λΣν+
2
X
m=1
˜
G(σ)
m(Zν,Zν, Σν,Hm
ν), (50)
Hm
ν≈
n
X
ρ=1 "Wνρ ˜
Hm(Zν,Zν, Σν,Zρ,Zρ, Σρ) + ˜
Hm,1(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj zk− WνρZν
+˜
Hm,2(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj zk− WνρZν
+˜
Hm,3(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj σk− WνρΣν
+˜
Hm,4(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj zj− WνρZρ
+˜
Hm,5(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj zj− WνρZρ
+˜
Hm,6(Zν,Zν, Σν,Zρ,Zρ, Σρ)
X
k∈Gν,j∈Gρ
aνkwkj σj− WνρΣρ
#,
(51)
whe e
Wνρ := X
k∈Gν,j∈Gρ
aνkwkj . (52)
As we wan hese equa ions o depend solely on he mac oscopic a iables, we impose he ollowing
X
k∈Gν,j∈Gρ
aνkwkj zk=µνρZν,(53a)
X
k∈Gν,j∈Gρ
aνkwkj zk=µνρZν, (53b)
X
k∈Gν,j∈Gρ
aνkwkj σk=µνρΣν, (53c)
14
X
k∈Gν,j∈Gρ
aνkwkj zj=λνρZρ, (54a)
X
k∈Gν,j∈Gρ
aνkwkj zj=λνρZρ, (54b)
X
k∈Gν,j∈Gρ
aνkwkj σj=λνρΣρ. (54c)
These condi ions a e in ui i e, as hey show ha he weigh ed sum o e ms in ol ing he mic oscopic
ac i i ies o k o all k∈Gνis p opo ional o he mac oscopic obse able o his g oup.
Now, ins ead o (53) and (54), as he mic oscopic a iables a e a bi a y, we can simply w i e, espec-
i ely
X
j∈Gρ
wkj aνk=µνρaνk∀k∈Gν, (55)
X
k∈Gν
wkj aνk=λνρaρj∀j∈Gρ. (56)
O , in ma ix o m
Kνρˆ
aν=µνρˆ
aν, (57)
WT
νρˆ
aν=λνρˆ
aρ. (58)
He e ˆ
aνis a ec o which con ains he componen s o he ec o aν ha co espond o elemen s o
Gν.Kνρ is he diagonal ma ix o size mν×mνwhe e (Kνρ)k,k=Pj∈Gρwikj, and Wνρ is a ma ix o size
mν×mρgi en by Wνρ = (wikjl)k,l, whe e Gν={i1, ... , imν}and Gρ={j1, ... , jmρ}. F om now on, we will
e e o equa ions (57) and (58) as compa ibili y equa ions. A comp ehensi e and o mal ma hema ical
explana ion o how o sol e hem app oxima ely can be ound in [9].
An in e es ing obse a ion is ha i he unc ions ˜
Hm(zk,zk,σk,zj,zj,σj) do no depend on zk,zk
and σk(o , equi alen ly, he unc ions Hm(θk,σk,θj,σj) do no depend on θkand σk), he compa ibili y
equa ion (57) is no necessa y, and we only need o sol e (58). This is he case o he oscilla o (32)
p esen ed in Sec ion 2.2.
When 57 and 58 a e ul illed o all ν,ρ∈ {1, ... , n}, we ge he app oxima e dynamics
˙
Zν≈i2πωZν+i2π
2
X
m=1
Zν˜
G(z)
m(Zν,Zν, Σν,Hm
ν), (59a)
˙
Σν≈λΣν+
2
X
m=1
˜
G(σ)
m(Zν,Zν, Σν,Hm
ν), (59b)
Hm
ν≈
n
X
ρ=1
Wνρ ˜
Hm(Zν,Zν, Σν,Zρ,Zρ, Σρ). (59c)
I is impo an o no e ha he unknowns in equa ions (57) and (58) a e he educ ion ec o s ˆaν,
as well as he pa ame e s µνρ and λµρ. The e o e, he aim is o de e mine he bes way o weigh he
15
Synch oniza ion in ne wo ks o couple oscilla o s
ac i i ies wi hin each g oup, whe e ”bes ” e e s o ensu ing ha he dynamics o he obse ables emain
as close as possible o he dynamics o he closed sys em (59). Addi ionally, he equa ions ypically do no
ha e an exac solu ion; he me hod p oposed in he a icle p o ides an app oxima e solu ion whene e he
connec ion ma ices a e posi i e.
Finally, we obse e ha he mic oscopic dynamics de ined in (43) sha e he same o m as he mac o-
scopic dynamics in (59). This indica es ha he phases associa ed wi h he mac oscopic complex obse -
ables mus ollow dynamics analogous o hose o he mic oscopic phases θi. The e o e, we can de ine
mac oscopic obse ables Θνassocia ed wi h Zν, which ela e o hem in he same way as he mic oscopic
a iables θi ela e o zi. Using hese obse ables, we ge he app oxima e dynamics
˙
Θν≈ω+
2
X
m=1
G(θ)
m(Θν, Σν,Hm
ν), (60a)
˙
Σν≈λΣν+
2
X
m=1
G(σ)
m(Θν, Σν,Hm
ν), (60b)
Hm
ν≈
n
X
ρ=1
WνρHm(Θν, Σν, Θρ, Σρ). (60c)
16
4. Resul s
We now aim o apply he dimensionali y educ ion echnique o he model in oduced in Sec ion 2.2. In
his sec ion, we will examine his me hod in he con ex o a ne wo k comp ising No hese oscilla o s (32),
pa i ioned in o wo dis inc g oups. To achie e his, we i s conduc an analy ical s udy o he e olu ion
o a wo-dimensional sys em using a e aging echniques. Following his, we pe o m a se ies o nume ical
simula ions o compa e he synch oniza ion s a es o he exac and educed sys ems, ocusing on how hese
s a es a y wi h di e en pa ame e alues.
4.1 Analysis o a 2-dimesional ne wo k
Le us conside wo weakly coupled oscilla o s o he o m (32). We de ine φias he phase de ia ion o
he oscilla o idue o he in e acions om he o he oscilla o s:
θi=ω +φi. (61)
By combining (32a) and (61), we can de i e an exp ession o ˙φi
˙φi=ϵ⟨∇Θ(ω +φi,σi), X
j
wij F(K(ω +φj,σj))⟩. (62)
Fo simplici y, abusing o no a ion, om now on we will conside F(K(ω +φj,σj)) = F(ω +φj,σj).
Since we a e in e es ed in he synch oniza ion o hese wo oscilla o s, we de ine a new a iable, χ:=
φ2−φ1, which ep esen s he phase di e ence be ween hem. The dynamics o his a iable χwill be
he ocus o ou s udy, as unde s anding i s beha io is equi alen o analyzing he s abili y o he a ious
synch oniza ion s a es be ween he wo oscilla o s. To do his, we will employ he me hod o a e aging [4],
which ans o ms he sys em by means o a nea -iden i y change o a iable in o he o m
˙φi=ϵ
2
X
j=1
wij
T
2
X
k=1 ZT
0
∇Θ(k)(ω +φi,σi)F(k)(ω +φj,σj)d . (63)
De ining τ:= ω +φiand no ing ha ω +φj=τ−φi+φj=τ+(φj−φi), we ew i e he exp ession
as
˙φi=ϵ
2
X
j=1
wij
2
X
k=1 Z1
0
∇Θ(k)(τ,σi)F(k)(τ+ (φj−φi), σj)dτ. (64)
We simpli y his o
˙φi=ϵ
2
X
j=1
wij H(φj−φi,σi,σj), (65)
whe e
H(χ,σi,σj) =
2
X
k=1 Z1
0
∇Θ(k)(τ,σi)F(k)(τ+χ,σj)dτ. (66)
Using his, we exp ess he de i a i e o χas:
˙χ= ˙φ2−˙φ1, (67)
17
Synch oniza ion in ne wo ks o couple oscilla o s
˙χ=ϵw22H(0, σ2,σ2)−ϵw11H(0, σ1,σ1) + ϵw21H(−χ,σ2,σ1)−ϵw12H(χ,σ1,σ2). (68)
As s a ed ea lie , ou goal is o ind he equilib ium poin s o his a iable. To achie e his, we subs i u e
∇Θ and Fin o H(χ,σi,σj) o ob ain a mo e explici o m.
Z1
0
∇Θ(1)(τ,σi)F(1)(τ+χ,σj)dτ=
Z1
0
B(σi)(νsin(ΦK(τ,σi)) + γcos(ΦK(τ,σi)))(K(τ+χ,σj)xcos ψ−K(τ+χ,σj)ysin ψ)dτ,
Z1
0
∇Θ(2)(τ,σi)F(2)(τ+χ,σj)dτ=
Z1
0
B(σi)(γsin(ΦK(τ,σi)) −νcos(ΦK(τ,σi)))(K(τ+χ,σj)ycos ψ+K(τ+χ,σj)xsin ψ)dτ.
Pe o ming some de i a ions (p o ided in Appendix A), we a i e a he ollowing simpli ied exp ession
H(χ,σi,σj) = 1
2πν
Rk(σj)
Rk(σi)(νsin D − γcos D) (69)
whe e
D= 2πχ +ψ+γln (1 −2ασi)
2ν−γln (1 −2ασj)
2ν, (70)
No ice ha , when σiequals σj, he unc ion H(χ,σi,σj) becomes independen o hese pa ame e s.
So we can de ine he cons an :
H0:= H(0, σ,σ) = 1
2πν (νsin(ψ)−γcos(ψ)). (71)
The e o e, he a iable χ eaches equilib ium when
(w22 −w11)H0+w21H(−χ,σ2,σ1)−w12H(χ,σ1,σ2) = 0. (72)
I is impo an o no e ha his equilib ium does no depend on he alue o ϵ.
Using his, we will be able o p edic he equilib ium poin s o ou educed sys em analy ically. This
will be explo ed in de ail in Sec ion 4.3.2.
4.2 Phase Synch oniza ion Index
When we pe o m he simula ions, we aim o compa e he e olu ion o he exac sys em wi h he educed
sys em. In o de o do so, we will use he so-called Synch oniza ion Index (SI), which is a quan i a i e
measu e o he synch oniza ion be ween oscilla o s. I is de ined as [5]
=1
n
n
X
k=1
ei2πθk
. (73)
. F om now on we will conside he case in which n= 2
=|ei2πθ1+ei2πθ2|
2. (74)
18
θ1
θ2
0
3
2π
π
π
2
Figu e 2: G aphical ep esen a ion o he Synch oniza ion Index (SI). The ec o s ei2πθ1and ei2πθ2co -
espond o he phases o he wo oscilla o s on he uni ci cle, and hei ec o sum =ei2πθ1+ei2πθ2
de e mines he alue o , calcula ed as =1
2| |.
Figu e 2 p o ides a g aphical ep esen a ion o his measu e, whe e he oscilla o s’ phases θ1and θ2a e
ep esen ed as uni ec o s, and he SI is compu ed using he modulus o hei sum. I can be seen ha
he SI anges be ween 0 and 1, whe e a alue o 0 indica es ha he wo oscilla o s a e in an i-phase, and
a alue o 1 ha hey a e in phase.
I is wo h no ing ha his index depends only on he di e ence o phase, i.e. θ1−θ2, a he han he
wo phase alues. Le , be he sum ei2πθ1+ei2πθ2
=ei2πθ1+ei2πθ2= cos (2πθ1) + isin (2πθ1) + cos (2πθ2) + isin (2πθ2)
= (cos (2πθ1) + cos(2πθ2)) + i(sin (2πθ1) + sin (2πθ2))
| |2= (cos (2πθ1)+cos (2πθ2))2+(sin (2πθ1)+sin 2πθ2)2= 2+2(cos (2πθ1) cos (2πθ2)+sin (2πθ1) sin (2πθ2))
| |2= 2(1 + cos (2πθ1−2πθ2)).
And using (74), we ge he alue o he SI in e ms o he phase di e ence
=1
2| |= 1 + cos (2π(θ1−θ2))
2. (75)
4.3 Simula ions
In his sec ion, we p esen a se ies o simula ions o a ne wo k o 100 nodes, compa ing he synch oniza ion
dynamics obse ed in he exac model wi h hose in he educed model. Bu be o e di ing in o he simula-
ions, we i s discuss he chosen alues o he oscilla o pa ame e s and he s uc u e o he connec i i y
ma ix.
19
Synch oniza ion in ne wo ks o couple oscilla o s
4.3.1 Pa ame e s o he model
Fi s , we ocus on he pa ame e s chosen o he oscilla o s. Going back o Sec ion 2.2, we ha e o se
he alues o ou pa ame e s: α,ν,wand γ. To simpli y hings, we will choose hem in such a way ha
R= 1 and ω= 1. Fo his, we se ν=−αand w= 1 −γ. Fu he mo e, ollowing he pape [1], we will
se γ=αa. In his way, he only ee pa ame e s a e αand a, which we will change along he simula ions.
These wo pa ame e s play an impo an ole in he dynamics o he oscilla o s. The pa ame e α
con ols he s eng h o he con ac ion o he limi cycle: o small alues o α he con ac ion is weak,
while o la ge alues i becomes s onge . On he o he hand, he pa ame e acon ols he ans e sali y
o he isoch ons o he limi cycle. This can be seen in Figu es 3 and 4, espec i ely. Figu e 3 shows ha
an oscilla o wi h a la ge alue o α eaches as e he limi cycle (σ= 0), and Figu e 4 shows how he
cu a u e o he isoch ons inc eases wi h he alue o he pa ame e a.
Figu e 3: E olu ion o he ampli ude a iable σo wo oscilla o s o he o m (32) wi h a= 1 and wo
di e en alues o α:α= 0.1 (blue), and α= 2 (o ange).
Figu e 4: Isoch ons o he s udied oscilla o (32) o α= 1 and h ee di e en alues o a:a= 0, a= 2,
and a= 5, shown om le o igh .
Ano he pa ame e ha has o be se is ψ, which encodes phase in o ma ion abou he connec ions in
he ne wo k. In he analysis o he 2-dimensional ne wo k ha we pe o med in Sec ion 4.1, we ha e seen
ha his pa ame e a ec s he poin s o equilib ium o χ=φ2−φ1. We will s udy his in mo e de ail in
he ollowing sec ion.
In addi ion o he oscilla o pa ame e s, we mus also speci y he de ails o he ne wo k s uc u e. We
will wo k wi h a ne wo k con aining 100 nodes, di ided e enly in o wo g oups o 50 nodes each. The
20
Figu e 5: The le squa e ep esen s he connec i i y ma ix, and he igh one he in/ou -deg ees o he
nodes in he ne wo k.
connec i i y be ween nodes is desc ibed by a homogeneous connec i i y ma ix, as well as he pa ame e ϵ
which con ols he s eng h o he gene al coupling. The ma ix is conside ed homogeneous in he sense
ha he in-deg ees and ou -deg ees o nodes wi hin he same g oup exhibi li le a iabili y. We can see a
ep esen a ion o his ma ix in Figu e 5. The e a e ou dis inc sec ions: he op-le sec ion ep esen s
he connec ions wi hin g oup 1, he op- igh sec ion ep esen s he connec ions om nodes in g oup 1 o
nodes in g oup 2, he bo om-le sec ion co esponds o he connec ions om nodes in g oup 2 o nodes
in g oup 1, and he bo om- igh sec ion ep esen s he connec ions wi hin g oup 2. Each elemen wij o
he connec i i y ma ix is equal o 1/Ni he e is a connec ion om node j o node i, o 0 o he wise.
Howe e , in p ac ice, o cases whe e he e is no connec ion, we se wij o a e y small alue o ensu e ha
he connec i i y ma ix is s ic ly posi i e, as his is equi ed o sol e he compa ibili y equa ions (57) and
(58). Fo mo e de ails on cons uc ing homogeneous ne wo ks, see he sec ion Homogeneous Ne wo ks in
[9].
Fo he simula ions in ol ing he educed model, whe e he ne wo k is simpli ied o wo dimensions, we
u ilize he g ouped weigh s as de ined in equa ion (52).
4.3.2 Simula ions o equilib ium dynamics in he educed model
In his sec ion, we wan o s udy he equilib ium o χin he educed model o ou pa ame e s. This
analysis is pe o med unde he condi ion σ1=σ2. Since H(χ,σi,σj) does no depend on hese pa ame e s
when his condi ion holds, we will w i e i simply as H(χ). Fu he mo e, we de ine
G(χ) := w21H(−χ)−w12H(χ). (76)
As p e iously men ioned, he equilib ium poin s o χdepend on he alue o ψ. Figu e 6 illus a es he
in e sec ion o G(χ) and −H0(w22 −w11) o di e en alues o ψ, ep esen ing hese equilib ium poin s.
We a e also in e es ed in he s abili y o hese equilib ium poin s, which we s udy by analyzing he sign
o he de i a i e o G(χ) + H0(w22 −w11) wi h espec o χ:
(G(χ) + H0(w22 −w11))′=G′(χ) = −w21H′(−χ)−w12H′(χ), (77)
21
Synch oniza ion in ne wo ks o couple oscilla o s
5. Explo a ions wi h an al e na i e model
As we discussed in he p e ious sec ion, he s eng h o he gene al coupling ϵdoes no appea o in luence
he synch oniza ion egimes o he model we s udied (26). Mo i a ed by his obse a ion, we aim o explo e
a model whe e his pa ame e plays a signi ican ole. In his subsec ion, we in oduce a di e en coupling
unc ion [6] o he canonical model o he And ono -Hop oscilla o [8] and ou line i s pa ame e iza ion
in phase-ampli ude a iables, which will o m he ounda ion o subsequen simula ions and analysis.
5.1 Al e na i e coupling unc ions
The canonical model o he And ono -Hop oscilla o [8] wi h he coupling unc ions p esen ed in Nicks
e al. [6], has he ollowing o m,
˙xi=xi−(xi−c2yi) 2+ϵ
N
X
j=1
wij (xi−xj−c1(yj−yi)), (79a)
˙yi=yi−(yi+c2xi) 2+ϵ
N
X
j=1
wij (yi−yj+c1(xj−xi)), (79b)
whe e =qx2
i+y2
i.
We can see ha he pa o he equa ions co esponding o he dynamics o a single oscilla o is a
pa icula case o he model we s udied ea lie in (26). Speci ically, his occu s when α= 1, ν=−1,
w= 0, and γ=−c2,
xi−(xi−c2yi) 2=xi(1 − 2)−yi(−c2 2), (80a)
yi−(yi+c2xi) 2=yi(1 − 2) + xi(−c2 2). (80b)
Thus, we can use he same isomo phism K(θ,σ) de ined in (23) o ob ain he phase-ampli ude pa ame-
e iza ion o his model, which will ha e oscilla ions o adius R= 1, equency ω=−c2and Floque
exponen λ=−2.
Howe e , he coupling dynamics be ween he nodes di e om hose o he p e ious model. Conse-
quen ly, when exp essing his sys em in he gene al o m shown in (21), he only changes pe ain o he
de ini ions o he unc ions Hk, which a e now gi en by [6]
H1(θi,σi,θj,σj) = xi−xj−c1(yj−yi), (81)
H2(θi,σi,θj,σj) = yi−yj+c1(xj−xi), (82)
whe e (xi,yi) = K(θi,σi) and (xj,yj) = K(θj,σj).
5.2 Simula ions
Ou aim is o ind a scena io in which he s abili y o he synch ony s a e depends on he gene al s eng h
o he coupling, ϵ. To his end, we begin by conduc ing simula ions whe e he connec i i y ma ix is de ined
28
wi h wij = 1/N, ep oducing he scena io s udied in Nicks e al. [6], in which hey show he s abili y o he
synch ony s a e o di e en alues o c1,c2and ϵ. A e ep oducing hese esul s, we ex end he analysis
by using ou ne wo k’s connec i i y ma ix o explo e whe he he e exis s a se o pa ame e s c1and c2
o which ϵplays a signi ican ole in egula ing he s abili y o he synch ony s a e.
I is impo an o no e ha hese explo a ions emain un inished and open. The esul s p esen ed he e
ep esen p og ess owa ds unde s anding his phenomenon, bu u he in es iga ions a e needed o ully
cha ac e ize he in e play be ween he coupling s eng h and s abili y in his al e na e model.
5.2.1 Simula ions wi h homogeneous coupling
In his sec ion, ou objec i e is o ep oduce he s udy conduc ed in [6] o a pa icula se o he pa ame e s
c1and c2in o de o ind a case in which ϵplays a ole on he s abili y o synch ony. Figu e 17, p esen s
he indings o his pape abou his. The egion below he blue cu e co esponds o he alues o he
pa ame e s o which he synch ony s a e is uns able, and he egion abo e he pa ame e s o which i is
s able.
Figu e 17: Ma ginal s abili y cu es o synch ony o di e en pa ame e alues. Ex ac ed om [6].
Fo ou expe imen s, we se c1=−1.4 and c2= 1.1, and we explo ed he e olu ion o he nodes o
ϵ= 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, wi h σand θuni o mly dis ibu ed in = 0s.
In his case, as he coupling weigh s we e de ined as wij =1
N o i,j= 1, ... , N, i makes no sense
o conside he pa i ion be ween wo g oups, and he e o e we can’ compu e he synch oniza ion index
be ween hem. Ins ead, we p esen he esul s by plo ing he alues o θiand σia = 0s, = 500s and
= 1000s, and also by showing he ime e olu ion o θiin he ime window 0= 950s and = 1000s.
Fo he simula ions in which ϵ≤0.5 he nodes did no each he synch oniza ion s a e, as exempli ied
in Figu e 18 o ϵ= 0.5. In con as , o ϵ= 0.6 and ϵ= 0.7, all nodes ended up synch onized wi h one
ano he , his is shown in Figu e 19 o ϵ= 0.6.
This obse a ion is close o, bu does no exac ly ma ch, he esul s p esen ed in [6], whe e he
synch oniza ion s a e o his se o pa ame e s is obse ed o ϵ≳0.4, as seen in Figu e 17.
5.2.2 Simula ions wi h he connec i i y ma ix
Now, we wan o ind a simila scena io bu using he ne wo k ha we ha e been s udying un il now,
i.e., se ing he weigh s wij acco ding o he connec i i y ma ix shown in Figu e 5, and conside ing he
pa i ions in o wo g oups: G1={1, ... , 50}and G2={51, ... , 100}.
To explo e his, we pe o med simula ions o se e al se s o pa ame e s c1and c2, a ying ϵ o in es-
iga e he ole ha he gene al coupling plays in he synch ony o he sys em. In Figu e 20, we p esen an
29
Synch oniza ion in ne wo ks o couple oscilla o s
Figu e 18: Time e olu ion o he sys em (79) wi h homogeneous weigh s wij =1
N. Pa ame e s used o
he simula ion: c1=−1.4, c2= 1.1, and ϵ= 0.5. In = 0s, he nodes a e dis ibu ed uni o mly. Top:
Values o σand θa = 0s, = 500s and = 1000s. Bo om: E olu ion o he θ a iables in he ime
window 0= 950s and = 1000s.
example showing he synch oniza ion index o he exac sys em o di e en alues o ϵ. We obse e ha
o ϵ≲0.6, he phase di e ence be ween he wo g oups does no appea o each an equilib ium. The
ac ha he in e al shown on he plo is no jus [0, 1] o hese cases may jus be because he dynamics
a e oo slow and/o he ime window being oo small. Fo , ϵ≳0.6 he wo g oups each an equilib ium a
he synch ony s a e.
In Figu e 21, we ake a close look a a case in which he phase di e ence be ween he wo g oups does
no s abilize. Simila o wha was obse ed in he o he model when ψ= 0, he nodes wi hin he same
g oup p esen di e en phase alues. This sugges s ha he educed sys em may no be able o cap u e
he dynamics co ec ly in hese cases.
Howe e , his pa o he wo k emains open, and o ully cha ac e ize he ela ionship be ween he
coupling s eng h ϵand he s abili y o synch ony in his con ex , as well as o e alua e he pe o mance
o he educed model, a deepe analysis is necessa y. This should include addi ional simula ions and also
an analy ical in es iga ion o he educed sys em.
30
Figu e 19: Time e olu ion o he sys em (79) wi h homogeneous weigh s wij =1
N. Pa ame e s used o
he simula ion: c1=−1.4, c2= 1.1, and ϵ= 0.6. In = 0s, he nodes a e dis ibu ed uni o mly. Top:
Values o σand θa = 0s, = 500s and = 1000s. Bo om: E olu ion o he θ a iables in he ime
window 0= 950s and = 1000s.
Figu e 20: Synch oniza ion index o di e en alues o ϵ o he exac dynamics o he sys em (79) in he
ime window 0= 950s and = 1000s. Pa ame e s used o he simula ion: c1=−0.8 and c2= 1.1.
31
Synch oniza ion in ne wo ks o couple oscilla o s
Figu e 21: Time e olu ion o he sys em o he mic oscopic a iables. The wo colo s indica e wo g oups.
Pa ame e s used o he simula ion: c1=−0.8, c2= 1.1 and ϵ= 0.1. In = 0, he nodes wi hin he same
g oup ha e he same alue.
32
6. Conclusions
In his p ojec , we ha e s udied wo main concep s: he phase-ampli ude educ ion o oscilla o s, and he
applica ion o he spec al educ ion p esen ed in Vegu´e e al. [9] o a ne wo k o oscilla o s desc ibed by
his pa ame e iza ion. This in ol ed using wo mac oscopic obse ables o each g oup in he ne wo k, one
ep esen ing he phase and he o he ep esen ing he ampli ude. Ou objec i e was o de e mine whe he
he educed sys em could accu a ely cap u e he synch oniza ion egimes o he o iginal ne wo k.
Fi s , we s udied a ne wo k o oscilla o s o he model p esen ed in [8], which co esponds o a canonical
model o an And ono -Hop bi u ca ion. Analyzing he esul s ob ained wi h he nume ical simula ions,
we obse ed ha he educed sys em is only able o cap u e he dynamics o he exac sys em when he
mic oscopic a iables wi hin he same g oup exhibi simila ac i i ies. This is in ac a basic condi ion in
o de o he spec al educ ion o wo k e ec i ely, as when he ac i i ies wi hin a g oup a e no simila , i
does no make sense o app oxima e all hese a iables wi h one mac oscopic obse able. Addi ionally, we
ound ha he s eng h o he gene al coupling in he ne wo k, ϵ, does no a ec his synch oniza ion. We
ha e s udied his beha io bo h analy ically, o he educed 2-dimensional sys em, and h ough simula ions
o he comple e sys em. Ne e heless, we ha e seen ha he pa ame e ψ∈[0, 2π), which encodes phase
in o ma ion abou he connec ions be ween he nodes, does play a key ole in his. Pa icula ly, we ha e
seen ha o alues o ψapp oxima ely be ween 4 and 6, he nodes wi hin he same g oup end o exhibi
simila alues o θand σ, and so he educed sys em is able o app oxima e co ec ly he dynamics o he
comple e ne wo k.
Following his, and wi h he aim o inding a scena io in which he s eng h o he couplings in he
ne wo k in luences he s abili y o he synch oniza ion egimes, we s a ed explo ing a ne wo k wi h he same
oscilla o s bu a di e en dynamical coupling. We s a ed by eplica ing pa o he s udy conduc ed in he
pape whe e he model is p esen ed [6], using homogeneous coupling weigh s wij . This in ol ed pe o ming
simula ions o a ious se s o pa ame e s and compa ing ou indings wi h espec o he s abili y o he
synch ony s a e wi h he ones shown in he pape . Subsequen ly, we ied o ex end his analysis o ou
ne wo k, employing he connec i i y ma ix p esen ed in p e ious simula ions.
Howe e , his pa o he wo k emains incomple e and equi es u he explo a ion. Fo ins ance, an
analy ical s udy simila o he one conduc ed in Sec ion 4.1 o a 2-dimensional ne wo k wi h his model
would be an in e es ing pa h o u u e esea ch.
Addi ionally, se e al o he u u e di ec ions could be pu sued. One po en ial ex ension o he p ojec
could be o epea his s udy wi h he e ogeneous connec i i y ma ices, as his ype o ne wo k is explo ed
in he o iginal pape o he spec al educ ion [9]. Mo eo e , we could s udy he e iciency o he educed
sys em on ne wo ks wi h mo e han wo clus e s, analyzing he s abili y o he synch ony and splay s a es.
33
Synch oniza ion in ne wo ks o couple oscilla o s
Re e ences
[1] O iol Cas ej´on, An oni Guillamon, and Gemma Hugue . Phase-Ampli ude Response Func ions o
T ansien -S a e S imuli. J. Ma h. Neu osci., 3(1):1–26, Decembe 2013.
[2] F.C. Hoppens ead and E.M. Izhike ich. Weakly Connec ed Neu al Ne wo ks. Applied Ma hema ical
Sciences. Sp inge New Yo k, 2012.
[3] G. Hugue . Dynamics o coupled oscilla o s using phase ampli ude a iables: E icien algo i hms and
igo ous esul s. Pe sonal communica ion, 2024.
[4] Eugene M. Izhike ich. Dynamical Sys ems in Neu oscience: The Geome y o Exci abili y and Bu s ing.
The MIT P ess, 2007.
[5] Yoshiki Ku amo o. Chemical Oscilla ions, Wa es, and Tu bulence. Sp inge , Be lin, Ge many, 1984.
[6] R. Nicks, R. Allen, and S. Coombes. Insigh s in o oscilla o ne wo k dynamics using a phase-isos able
amewo k. Chaos, 34(1), Janua y 2024.
[7] Albe o P´e ez-Ce e a, Te e M-Sea a, and Gemma Hugue . Global phase-ampli ude desc ip ion o
oscilla o y dynamics ia he pa ame e iza ion me hod. Chaos: An In e disciplina y Jou nal o Nonlinea
Science, 30(8):083117, 08 2020.
[8] S e en H. S oga z. Nonlinea Dynamics and Chaos: Wi h Applica ions o Physics, Biology, Chemis y
and Enginee ing. Wes iew P ess, 2000.
[9] Ma ina Vegu´e, Vincen Thibeaul , Pa ick Des osie s, and An oine Alla d. Dimension educ ion o
dynamics on modula and he e ogeneous di ec ed ne wo ks. PNAS Nexus, 2(5):pgad150, 05 2023.
34
A. Ma hema ical de i a ions o he a e aging me hod
In Sec ion 4.1, we employed he a e aging me hod o s udy he synch oniza ion dynamics o a 2-dimensional
ne wo k. He e, we p esen in mo e de ail he ma hema ical de i a ions pe o med o ob ain Equa ion (69).
Recall ha we de ined
H(χ,σi,σj) =
2
X
k=1 Z1
0
∇Θ(k)(τ,σi)F(k)(τ+χ,σj)dτ. (83)
We will s a by de eloping he i s e m o he summa ion by subs i u ing he unc ions ∇Θ(1)(τ,σi) and
F(1)(τ+χ,σj)
Z1
0
∇Θ(1)(τ,σi)F(1)(τ+χ,σj)dτ=
Z1
0
B(σi)(νsin(ΦK(τ,σi)) + γcos(ΦK(τ,σi)))(K(τ+χ,σj)xcos ψ−K(τ+χ,σj)ysin ψ)dτ=
B(σi)Rk(σj)Z1
0
(νsin(ΦK(τ,σi)) + γcos(ΦK(τ,σi)))(cos(ΦK(τ+χ,σj)) cos ψ−sin(ΦK(τ+χ,σj)) sin ψ)dτ.
Fo simplici y, we de ine A:= ΦK(τ,σi) and B:= ΦK(τ+χ,σj) (This is an abuse o no a ion, i is
impo an o no e ha bo h Aand Ba e dependen on τ). Subs i u ing hem in he i s e m we ha e
B(σi)Rk(σj)Z1
0
(νsin A+γcos A)(cos Bcos ψ−sin Bsin ψ)dτ=
B(σi)Rk(σj)Z1
0
(νcos ψsin Acos B+γcos ψcos Acos B−νsin ψsin Asin B−γsin ψcos Asin B)dτ=
B(σi)Rk(σj)
2Z1
0
cos ψ(ν(sin(A+B) + sin(A−B)) + γ(cos(A+B) + cos(A−B)))
−sin ψ(ν(cos(A−B)−cos(A+B)) + γ(sin(A+B)−sin(A−B))) dτ
B(σi)Rk(σj)
2Z1
0
sin(A+B)(νcos ψ−γsin ψ) + sin(A−B)(νcos ψ+γsin ψ)
+ cos(A+B)(γcos ψ+νsin ψ) + cos(A−B)(γcos ψ−νsin ψ)dτ,
whe e
A+B= 2π(2τ+χ)−γln(1 −2ασi) + ln(1 −2ασj)
2ν,
A−B= 2π(−χ) + γln(1 −2ασi)−ln(1 −2ασj)
2ν.
Analogously, o he second e m o he summa ion
Z1
0
∇Θ(2)(τ,σi)F(2)(τ+χ,σj)dτ=
Z1
0
B(σi)(γsin(ΦK(τ,σi)) −νcos(ΦK(τ,σi)))(K(τ+χ,σj)ycos ψ+K(τ+χ,σj)xsin ψ)dτ=
35
Synch oniza ion in ne wo ks o couple oscilla o s
B(σi)Rk(σj)
2Z1
0
sin(A+B)(γsin ψ−νcos ψ) + sin(A−B)(γsin ψ+νcos ψ)
cos(A+B)(−γcos ψ−νsin ψ) + cos(A−B)(γcos ψ−νsin ψ)dτ.
Subs i u ing bo h e ms on equa ion (83)
H(χ,σi,σj) = −Rk(σj)
2πνRk(σi)Z1
0
sin(A−B)(γsin ψ+νcos ψ) + cos(A−B)(γcos ψ−νsin ψ)dτ.
No ing ha A−Bdoes no depend on τ
H(χ,σi,σj) = −Rk(σj)
2πνRk(σi)(sin(A−B)(γsin ψ+νcos ψ) + cos(A−B)(γcos ψ−νsin ψ))
Rk(σj)
2πνRk(σi)(sin(B−A)(γsin ψ+νcos ψ)−cos(B−A)(γcos ψ−νsin ψ))
Rk(σj)
2πνRk(σi)γ
2(−2 cos(B−A+ψ)) + ν
2(2 sin(B−A+ψ))
Rk(σj)
2πνRk(σi)(−γcos(B−A+ψ) + νsin(B−A+ψ)) .
Finally, de ining D:= B−A+ψ,
H(χ,σi,σj) = 1
2πν
Rk(σj)
Rk(σi)(νsin D − γcos D), (84)
whe e
D= 2πχ +ψ+γln (1 −2ασi)
2ν−γln (1 −2ασj)
2ν. (85)
36
B. Py hon code used o he simula ions
He e we p esen he Py hon code used o pe o m all he simula ions in his p ojec , along wi h some
commen s.
Impo ed py hon lib a ies
1impo numpy as np
2impo ma h
3impo cma h
4 om scipy.in eg a e impo sol e_i p
5 om scipy.op imize impo sol e, oo _scala
6impo ma plo lib.pyplo as pl
De ini ion o he unc ions o he model
He e we de ine he unc ions o he i s model we s udied as p esen ed in Sec ion 2.2.
Func ions o he di eomo phism K(θ,σ). They co espond o equa ions (23), (24) and (25).
1de di eo_K( he a, sigma, pa ams):
2# pa ams: pa ame e s o he model (alpha, gamma, nu, psi)
3alpha, gamma, nu, psi = pa ams
4R_k = ma h.sq (alpha/(nu*(2*alpha*sigma - 1)))
5Phi_k = 2*ma h.pi* he a - (gamma*ma h.log(1-2*alpha*sigma))/(2*nu)
6 e u n (R_k*ma h.cos(Phi_k), R_k*ma h.sin(Phi_k))
7
8de g adien _ he a_1( he a, sigma, pa ams):
9alpha, gamma, nu, psi = pa ams
10 R_k = ma h.sq (alpha/(nu*(2*alpha*sigma - 1)))
11 Phi_k = 2*ma h.pi* he a - (gamma*ma h.log(1-2*alpha*sigma))/(2*nu)
12 B = -(2*ma h.pi*nu*R_k)**(-1)
13 e u n B*(nu*ma h.sin(Phi_k)+gamma*ma h.cos(Phi_k))
14
15 de g adien _ he a_2( he a, sigma, pa ams):
16 alpha, gamma, nu, psi = pa ams
17 R_k = ma h.sq (alpha/(nu*(2*alpha*sigma - 1)))
18 Phi_k = 2*ma h.pi* he a - (gamma*ma h.log(1-2*alpha*sigma))/(2*nu)
19 B = -(2*ma h.pi*nu*R_k)**(-1)
20 e u n B*(gamma*ma h.sin(Phi_k)-nu*ma h.cos(Phi_k))
21
22 de g adien _sigma_1( he a, sigma, pa ams):
23 alpha, gamma, nu, psi = pa ams
24 R_k = ma h.sq (alpha/(nu*(2*alpha*sigma - 1)))
25 Phi_k = 2*ma h.pi* he a - (gamma*ma h.log(1-2*alpha*sigma))/(2*nu)
26 A = (alpha * R_k)/(nu*(2*alpha*sigma-1))
27 C = -(nu*A)**(-1)
28 e u n C*ma h.cos(Phi_k)
29
30 de g adien _sigma_2( he a, sigma, pa ams):
31 alpha, gamma, nu, psi = pa ams
32 R_k = ma h.sq (alpha/(nu*(2*alpha*sigma - 1)))
37
Synch oniza ion in ne wo ks o couple oscilla o s
8 o g oup_idx, g oup_slice in enume a e(g oup_slices):
9sigma_g oup = (sigma_ alues[:, g oup_slice])
10 size_g oup = sigma_g oup.shape[1]
11 o iin ange(size_g oup):
12 pl .plo ( imes amps, sigma_g oup[:, i],colo =colo s[g oup_idx])
13
14 # Fo ma ing he plo
15 pl . i le(’Sigmas Time by G oups’)
16 pl .xlabel(’Time’)
17 pl .ylabel(’sigma’)
18 pl .axhline(0, colo =’black’, linewid h=0.8, lines yle=’--’)# Line a y=0 o e e ence
19 pl .ylim(np.min(sigma_ alues - 0.2),np.max(sigma_ alues + 0.2))
20 pl .g id()
21 pl .show()
Func ion o plo he alues o θand σa h ee poin s in ime. These plo s show θas he phase a iable
and σas he adial a iable (wi h an o se so ha we can ep esen nega i e alues o sigma).
1de plo _ he a_sigma( he a_ alues, sigma_ alues, g oup_slices, ime_plo s, imes eps,
sigma_in e al):
2T,n = he a_ alues.shape
3 he a_scaled = he a_ alues*2*ma h.pi
4sigma_min, sigma_max = sigma_in e al
5 ig, axs = pl .subplo s(1, 3, subplo _kw=dic (p ojec ion=’pola ’), igsize=(15, 5))
6colo s = pl .cm. ab10( ange(len(g oup_slices)))
7o se = 0.2 - sigma_min
8
9sigma_ex = max(-sigma_min,sigma_max)
10 adii_labels = [-sigma_ex , 0, sigma_ex ]
11 adii_ icks_all = np.linspace(-sigma_ex , sigma_ex , num=5) + o se
12
13 o i, _idx in enume a e( ime_plo s):
14 ax = axs[i]
15 ax.se _ i le( "Time = { imes eps[ _idx]}s")
16
17 o g oup_idx, g oup_slice in enume a e(g oup_slices):
18 he as = he a_scaled[ _idx, g oup_slice]
19 sigmas = sigma_ alues[ _idx, g oup_slice]
20 ax.sca e ( he as, sigmas + o se , label= ’G oup {g oup_idx + 1}’, colo =colo s[
g oup_idx])
21
22 ax.se _ylim(0, np.max(sigma_ alues+o se )+0.05) # Ensu e ou e poin s a e no c opped
23 ax.se _y icks( adii_ icks_all)
24 ax.se _y icklabels([])
25 o ick in adii_labels:
26 ax. ex (
27 0.39,
28 ick + o se + 0.01,
29 "{ ick:.2 }",
30 ha=’le ’, a=’bo om’, on size=10, colo ="black"
31 )
32 ax.se _x icklabels([])
33 ax.spines[’pola ’].se _ isible(False)
44
34 pl . igh _layou ( ec =[0, 0.03, 1, 0.95])
35 pl .show()
This unc ion does he same as he p e ious one bu adding he exac and educed mac oscopic obse ables.
1de comple e_plo _ he a_sigma( alues, g oup_slices, ime_plo s, imes eps, sigma_in e al):
2 he a_nodes, sigma_nodes, he a_exac _g oup, sigma_exac _g oup , he a_ educed,
sigma_ educed = alues
3m,n = he a_nodes.shape
4 he a_nodes = he a_nodes*2*ma h.pi
5 he a_exac _g oup = he a_exac _g oup*2*ma h.pi
6 he a_ educed = he a_ educed*2*ma h.pi
7sigma_min, sigma_max = sigma_in e al
8 ig, axs = pl .subplo s(1, 3, subplo _kw=dic (p ojec ion=’pola ’), igsize=(15, 5))
9colo s = pl .cm. ab10( ange(len(g oup_slices)))
10 o se = 0.2 - sigma_min
11
12 sigma_ex = max(-sigma_min,sigma_max)
13 adii_labels = [-sigma_ex , 0, sigma_ex ]
14 adii_ icks_all = np.linspace(-sigma_ex , sigma_ex , num=5) + o se
15
16
17 o i, _idx in enume a e( ime_plo s):
18 ax = axs[i]
19 ax.se _ i le( "Time = { imes eps[ _idx]}s")
20
21 o g oup_idx, g oup_slice in enume a e(g oup_slices):
22 he as = he a_nodes[ _idx, g oup_slice]
23 sigmas = sigma_nodes[ _idx, g oup_slice]
24 ax.sca e ( he as, sigmas + o se , label= ’G oup {g oup_idx + 1}’, colo =colo s[
g oup_idx], ma ke =’+’)
25
26 exac _g ouped_ he a = he a_exac _g oup[ _idx][g oup_idx]
27 exac _g ouped_sigma = sigma_exac _g oup[ _idx][g oup_idx]
28 educed_ he a = he a_ educed[ _idx][g oup_idx]
29 educed_sigma = sigma_ educed[ _idx][g oup_idx]
30
31
32 ax.sca e (
33 exac _g ouped_ he a, exac _g ouped_sigma + o se ,
34 label= ’Exac ed g ouped (G oup {g oup_idx + 1})’,
35 colo =colo s[g oup_idx], ma ke =’o’, s=100
36 )
37 ax.sca e (
38 educed_ he a, educed_sigma + o se ,
39 label= ’Reduced (G oup {g oup_idx + 1})’,
40 colo =colo s[g oup_idx], ma ke =’s’, s=100
41 )
42
43 ax.se _ylim(0, np.max(sigma_nodes+o se )+0.05) # Ensu e ou e poin s a e no c opped
44 ax.se _y icks( adii_ icks_all)
45 ax.se _y icklabels([])
46 o ick in adii_labels:
45
Synch oniza ion in ne wo ks o couple oscilla o s
47 ax. ex (
48 0.39,
49 ick + o se + 0.01,
50 "{ ick:.2 }",
51 ha=’le ’, a=’bo om’, on size=10, colo ="black"
52 )
53 ax.se _x icklabels([])
54 ax.spines[’pola ’].se _ isible(False)
55 pl . igh _layou ( ec =[0, 0.03, 1, 0.95])
56 pl .show()
This unc ion has a simila ou pu as he wo p e ious ones, bu i jus plo s he alues o θ
1de plo _ he a( he a_ alues,g oup_slices, imes eps):
2m,n = he a_ alues.shape
3 he a_scaled = he a_ alues*2*ma h.pi
4 ig, axs = pl .subplo s(1, 3, subplo _kw=dic (p ojec ion=’pola ’), igsize=(15, 5))
5colo s = pl .cm. ab10( ange(len(g oup_slices)))
6
7
8 o iin ange(m):
9ax = axs[i]
10 ax.se _ i le( "Time = { imes eps[i]}s")
11
12 o g oup_idx, g oup_slice in enume a e(g oup_slices):
13 he as = he a_scaled[i, g oup_slice]
14 ax.sca e ( he as, np.ones(len( he as)), label= ’G oup {g oup_idx + 1}’, colo =
colo s[g oup_idx])
15
16 # Fo ma
17 ax.se _y icklabels([])
18 ax.se _x icklabels([])
19 ax.spines[’pola ’].se _ isible(False)
20 pl . igh _layou ( ec =[0, 0.03, 1, 0.95])
21 pl .show()
Plo ing he o bi s
Code use o plo he isoch ons and isos ables o a single oscilla o .
1 he a_ ixed = [0, 0.3, 0.7]
2sigma_min, sigma_max = ge _sigma_in e al(model_pa ams,5)
3sigma_ ange = np.linspace(sigma_min, 0.35, 400)
4
5sigma_ ixed = [0, -1, 0.3]
6 he a_ ange = np.linspace(0, 1, 400, endpoin =False)
7
8
9colo s = pl .cm. ab10( ange(6))
10 pl . igu e( igsize=(8, 8))
11
12 #plo isoch ones
46
13 o i, he a in enume a e( he a_ ixed):
14 x_ als = []
15 y_ als = []
16 o sigma in sigma_ ange:
17 x, y = di eo_K( he a, sigma, model_pa ams)
18 x_ als.append(x)
19 y_ als.append(y)
20 pl .plo (x_ als, y_ als, label= ’Isoch on o $ he a$= { he a}’, colo =colo s[i])
21 pl .sca e (x_ als, y_ als, colo =colo s[i], s=10)
22
23 #plo isos ables
24 o i, sigma in enume a e(sigma_ ixed):
25 x_ als = []
26 y_ als = []
27 o he a in he a_ ange:
28 x, y = di eo_K( he a, sigma, model_pa ams)
29 x_ als.append(x)
30 y_ als.append(y)
31 pl .plo (x_ als, y_ als, label= ’Isos able o $ sigma$= {sigma}’, colo =colo s[i+3],
lines yle=’--’)
32
33 pl .xlabel(’x’)
34 pl .ylabel(’y’)
35 pl . i le( ’O bi s wi h Fixed $ he a$’)
36 pl .legend()
37 pl .g id(T ue)
38 pl .axis(’equal’)
39 pl .show()
A e aging analysis
De ini ion o he unc ions H and G o he a e aging analysis. They co espond o equa ions (69) and (76),
espec i ely. equilib ium a ge G makes e e ence o −H0(w22 −w11) as when G(χ0) equals his alue, i
means ha χ0is an equilib ium poin .
1de a e age_H(chi, sigma_i, sigma_j, model_pa ams):
2alpha, gamma, nu, psi = model_pa ams
3aux_i = (gamma*ma h.log(1-2*alpha*sigma_i))/(2*nu)
4aux_j = (gamma*ma h.log(1-2*alpha*sigma_j))/(2*nu)
5D = 2*ma h.pi*chi + psi + aux_i - aux_j
6R_k_i = ma h.sq (alpha/(nu*(2*alpha*sigma_i - 1)))
7R_k_j = ma h.sq (alpha/(nu*(2*alpha*sigma_j - 1)))
8 e u n (1/(2*ma h.pi*nu))*(R_k_j/R_k_i)*(nu*ma h.sin(D) - gamma*ma h.cos(D))
9
10 de a e age_G(chi, sigma_i, sigma_j, weigh s, model_pa ams):
11 e u n weigh s[1,0]*a e age_H(-chi, sigma_j, sigma_i, model_pa ams) - weigh s[0,1]*
a e age_H(chi, sigma_i, sigma_j, model_pa ams)
12
13 de equilib ium_ a ge _G(sigma_i, sigma_j, weigh s, model_pa ams):
14 e u n a e age_H(0, sigma_j, sigma_j, model_pa ams)*weigh s[1,1] - a e age_H(0, sigma_i,
sigma_i, model_pa ams)*weigh s[0,0]
47
Synch oniza ion in ne wo ks o couple oscilla o s
15
16 de equilib ium_G(chi, sigma_i, sigma_j, weigh s, model_pa ams):
17 chi = chi[0] i isins ance(chi, (np.nda ay, lis )) else chi
18 e u n a e age_G(chi, sigma_i, sigma_j, weigh s, model_pa ams) + equilib ium_ a ge _G(
sigma_i, sigma_j, weigh s, model_pa ams)
De ini ion o he de i a i es o H and G, used o assess he s abili y o he equilib ium poin s.
1de d_a e age_H(chi, sigma_i, sigma_j, model_pa ams):
2alpha, gamma, nu, psi = model_pa ams
3aux_i = (gamma*ma h.log(1-2*alpha*sigma_i))/(2*nu)
4aux_j = (gamma*ma h.log(1-2*alpha*sigma_j))/(2*nu)
5D = 2*ma h.pi*chi + psi + aux_i - aux_j
6R_k_i = ma h.sq (alpha/(nu*(2*alpha*sigma_i - 1)))
7R_k_j = ma h.sq (alpha/(nu*(2*alpha*sigma_j - 1)))
8 e u n (1/nu)*(R_k_j/R_k_i)*(nu*ma h.cos(D) + gamma*ma h.sin(D))
9
10 de d_a e age_G(chi, sigma_i, sigma_j, weigh s, model_pa ams):
11 e u n -weigh s[1,0]*d_a e age_H(-chi, sigma_j, sigma_i, model_pa ams) - weigh s[0,1]*
d_a e age_H(chi, sigma_i, sigma_j, model_pa ams)
12
13
14 de s able( oo , sigma_i, sigma_j, weigh s, model_pa ams):
15 e u n d_a e age_G( oo , sigma_i, sigma_j, weigh s, model_pa ams) < 0
Code o make he plo in Figu e 6
1psi_ alues = np.linspace(0, 2*ma h.pi,3, endpoin =False)
2
3chi_ alues= np.linspace(0, 1, 500)
4G_ alues_i = []
5eq_i = []
6
7 o psi_i in psi_ alues:
8model_pa ams_i = alpha, gamma, nu, psi_i
9G_ alues = np.a ay([a e age_G(chi, sigma_i, sigma_j, g ouped_weigh s, model_pa ams_i) o
chi in chi_ alues])
10 G_ alues_i.append(G_ alues)
11 equilib ium = equilib ium_ a ge _G(sigma_i, sigma_j,g ouped_weigh s, model_pa ams_i)
12 cons an _ alues = np. ull(500, equilib ium)
13 eq_i.append(cons an _ alues)
14
15 ig, axes = pl .subplo s(1, 3, igsize=(18, 6), sha ex=T ue, sha ey=T ue)
16
17
18 colo s = pl .cm. ab10( ange(2))
19 o i, psi in enume a e(psi_ alues):
20 axes[i].plo (chi_ alues, G_ alues_i[i], label= ’G($ chi$)’, colo =colo s[0])
21 axes[i].plo (chi_ alues, eq_i[i], label= ’$-H_0 (w_{22} - w_{11})$’, colo =colo s[1],
lines yle=’--’)
22 axes[i].se _ i le( ’$ psi = {psi:.2 }$’)
23 axes[i].se _xlabel( ’$ chi$’)
24 i i == 0:
48
25 axes[i].se _ylabel(’Func ion Value’)
26 axes[i].g id(T ue)
27 axes[i].legend()
28
29
30 pl . igh _layou ( ec =[0, 0, 1, 0.95])
31 pl .show()
Code o compu e he equilib ium poin s o χ o di e en ψ alues.
1psi_ alues = np.a ange(0, 2*ma h.pi,0.1)
2chi_ alues = np.a ay([0, 0.5])
3 oo s_sigma = []
4sigma = 0.2
5sigma_i = sigma
6sigma_j = sigma
7min_chi = 1
8signi ican _psi= -1
9 o psi_i in psi_ alues:
10 oo s = []
11 o chi in chi_ alues:
12 model_pa ams_i = alpha, gamma, nu, psi_i
13 oo = sol e(equilib ium_G, chi, a gs=(sigma_i, sigma_j, g ouped_weigh s,
model_pa ams_i))
14 oo = oo [0]%1
15 i no any(np.isclose( oo , exis ing_ oo ) o exis ing_ oo in oo s):
16 oo s.append( oo )
17 i oo < min_chi and s able( oo , sigma_i, sigma_j, g ouped_weigh s, model_pa ams_i
):
18 min_chi = oo
19 signi ican _psi = psi_i
20 oo s_sigma.append( oo s)
21
22 p in (signi ican _psi)
Code o plo hese poin s based on hei s abili y, as shown in Figu e 7
1pl . igu e( igsize=(8, 6))
2
3 o i, oo s in enume a e( oo s_sigma):
4model_pa ams_i = alpha, gamma, nu, psi_ alues[i]
5s able_ oo s = [ oo o oo in oo s i s able( oo , sigma_i, sigma_j, g ouped_weigh s
, model_pa ams_i)]
6uns able_ oo s = [ oo o oo in oo s i no s able( oo , sigma_i, sigma_j,
g ouped_weigh s, model_pa ams_i)]
7
8# Plo s able oo s (blue) and uns able oo s ( ed) o each K alue
9pl .sca e ([psi_ alues[i]] * len(s able_ oo s), s able_ oo s, colo =’blue’)
10 pl .sca e ([psi_ alues[i]] * len(uns able_ oo s), uns able_ oo s, colo =’ ed’)
11
12 pl .xlabel(’psi’)
13 pl .ylabel(’Roo s (y- alues)’)
14 pl . i le(’Plo o Equilib ium Poin s Based on S abili y’)
49
Synch oniza ion in ne wo ks o couple oscilla o s
15
16
17 pl .sca e ([], [], colo =’blue’, label=’S able Equilib ium’)
18 pl .sca e ([], [], colo =’ ed’, label=’Uns able Equilib ium’)
19
20 pl .legend()
21 pl .g id(T ue)
22
23 pl .show()
Synch oniza ion index
Func ions o compu e he synch oniza ion index in a gi en ime window.
1de compu e_ ( he a1, he a2):
2exp1 = cma h.exp(2*ma h.pi*1j* he a1)
3exp2 = cma h.exp(2*ma h.pi*1j* he a2)
4 e u n abs(exp1+exp2)/2
5
6de compu e_ _ ime_window( he as, ime_window):
7 _ ec o = np.a ay([compu e_ ( he as[ ime, 0], he as[ ime, 1]) o ime in ime_window])
8 e u n np.min( _ ec o ), np.max( _ ec o )
Code o pe o m he simula ions o di e en alues o ϵ(K in he code).
1R_del a = -1
2sigma_min, sigma_max = ge _sigma_in e al(model_pa ams,R_del a)
3
4omega = a/(2*ma h.pi) # equency
5lambda_ = -2*alpha # con ac ion alue
6
7# micoscopic a iables a = 0
8 he a_nodes[0,:] = he a_nodes_0
9sigma_nodes[0,:] = sigma_nodes_0
10 x_nodes_ini ial = np.conca ena e([ he a_nodes[0,:], sigma_nodes[0,:]])
11
12 # mac oscopic a iables a = 0
13 ( he a_g oups[0,:], sigma_g oups[0,:]) = ge _ educed_obse ables( educ ion_ ec o s,
he a_nodes[0,:], sigma_nodes[0,:])
14 x_g oups_ini ial = np.conca ena e([ he a_g oups[0,:], sigma_g oups[0,:]])
15
16 K_ alues = np.a ange(0.05,0.95,0.1)
17
18 _min = T_ -50
19 _max = T_
20 ime_window = np.whe e(( imes eps >= _min) & ( imes eps <= _max))[0]
21
22 _min_exac = np.emp y(len(K_ alues))
23 _max_exac = np.emp y(len(K_ alues))
24 _min_ educed = np.emp y(len(K_ alues))
25 _max_ educed = np.emp y(len(K_ alues))
26
50
27 o j,Kin enume a e(K_ alues):
28 p in ("Compu ing o K = ", K)
29 y:
30 # Sol e o exac sys em
31 sol = sol e_i p(de i a i es, (0, T_ ), x_nodes_ini ial, _e al = imes eps, a gs = (
omega, lambda_, weigh s*K, model_pa ams,N,R_del a),a ol=1e-8, ol=1e-8)
32 he a_nodes = sol.y[:N, :].T
33 sigma_nodes = sol.y[N:, :].T
34 exac _ he a_g oups = np.emp y((T,2))
35 exac _sigma_g oups = np.emp y((T,2))
36 o iin ange(T):
37 (exac _ he a_g oups[i,:], exac _sigma_g oups[i,:]) = ge _ educed_obse ables(
educ ion_ ec o s, he a_nodes[i,:], sigma_nodes[i,:])
38 _min_exac [j], _max_exac [j] = compu e_ _ ime_window(exac _ he a_g oups,
ime_window)
39 p in (" exac : ", _min_exac [j], _max_exac [j])
40 excep Excep ion as e:
41 p in ( "Exac sys em ailed o K={K}: {e}")
42 _min_exac [j] = np.nan
43 _max_exac [j] = np.nan
44
45 y:
46 # Sol e o educed sys em
47 sol = sol e_i p(de i a i es, (0, T_ ), x_g oups_ini ial, _e al = imes eps, a gs =
(omega, lambda_, g ouped_weigh s*K, model_pa ams,n, R_del a),a ol=1e-8, ol=1e
-8)
48 he a_g oups = sol.y[:n, :].T
49 sigma_g oups = sol.y[n:, :].T
50 _min_ educed[j], _max_ educed[j] = compu e_ _ ime_window( he a_g oups, ime_window
)
51 p in (" educed: ", _min_ educed[j], _max_ educed[j])
52 excep Excep ion as e:
53 p in ( "Reduced sys em ailed o K={K}: {e}")
54 _min_ educed[j] = np.nan
55 _max_ educed[j] = np.nan
Code o plo he esul s o he p e ious simula ions.
1pl . igu e( igsize=(8, 6))
2
3
4pl .sca e (K_ alues, _min_ educed, colo =’g een’, ma ke =’o’, label=’ min educed’)
5pl .sca e (K_ alues, _max_ educed, colo =’g een’, ma ke =’o’, label=’ max educed’)
6pl .sca e (K_ alues, _min_exac , colo =’blue’, ma ke =’o’, label=’ min exac ’)
7pl .sca e (K_ alues, _max_exac , colo =’blue’, ma ke =’o’, label=’ max exac ’)
8
9 o iin ange(len(K_ alues)):
10 pl .plo ([K_ alues[i], K_ alues[i]], [ _min_exac [i], _max_exac [i]], colo =’blue’, lw
=1, lines yle=’--’)
11 pl .plo ([K_ alues[i], K_ alues[i]], [ _min_ educed[i], _max_ educed[i]], colo =’g een’
, lw=1, lines yle=’--’)
12
13 pl .xlabel( ’$ epsilon$’)
51
Synch oniza ion in ne wo ks o couple oscilla o s
14 pl .ylabel(’ ’)
15 pl . i le(’’)
16 pl .legend()
17 pl .g id(T ue)
18 pl .ylim(-0.1, 1.1)
19
20 pl .show()
52
