Hysteretic Transitions in the Kuramoto Model with · PDF fileHysteretic Transitions in the Kuramoto Model with Inertia A. Torcini, S. Olmi, A. Navas, S. Boccaletti .fi.isc.cnr.it
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Hysteretic Transitions in theKuramoto Model with Inertia
A. Torcini, S. Olmi, A. Navas, S. Boccaletti
http://neuro.fi.isc.cnr.it/
Istituto dei Sistemi Complessi - CNR - Firenze, Italy
Istituto Nazionale di Fisica Nucleare - Sezione di Firenze, Italy
Centre for Biomedical Technology (UPM), Madrid, Spain
Rostock 2014 – p. 1
Pteroptix Malaccae
Usually, entrainment results in a constant phase angle equal to the difference between pacing frequency and
free-running period as it does in P. cribellata. The mechanism of attaining synchrony by Malaysian firefly Pteroptyx
malaccae is quite different. When the pacer changes, this firefly requires several cycles to reach a steady state. Once
this steady state is achieved, the phase angle difference is near zero irrespective of the pacer period. This can be
explained only by the animal adjusting the period of its oscillator to equal that of the driving oscillator. (experiments by
Hanson, 1987)
A phase model with inertia allows for adaptation of its frequency to the forcing oneB. Ermentrout, Journal of Mathematical Biology 29 , 571 (1991)
Rostock 2014 – p. 2
Plan of the Talk
Introduction of the Kuramoto model with inertia
Analogy with the damped oscillator (coexistence of stable periodic and fixed pointsolutions)
Mean field theory of the hysteretic transition (Tanaka, Lichtenberg, Oishi 1997 )
Fully coupled network of N oscillators
Existence of clusters of locked oscillators of any size between the hystereticcurves
Limits of stability of the coherent and incoherent solutions (dependence onthe size N and on the mass m)
Emergence of drifting clusters
Diluted network
Italian high voltage power grid
Rostock 2014 – p. 3
The Model
Kuramoto model with inertia
mθi + θi = Ωi +K
N
X
j
sin(θj − θi)
θi is the instantaneous phase
Ωi is the natural frequency of the i−th oscillator with Gaussian distribution
K is the coupling constant
N is the number of oscillators
By introducing the complex order parameter r(t)eiφ(t) = 1N
P
j eiθj
mθi + θi = Ωi − Kr sin(θi − φ)
r = 0 asynchronous state, r = 1 synchronized state
Rostock 2014 – p. 4
Damped Driven Pendulum
mθi + θi = Ωi − Kr sin(θi)
I = Ωi
Kr
β = 1√mKr
φ + βφ = I − sin(φ)
For sufficiently large m (small β)
For small Ωi two fixed points are present: a stable node and a saddle.
At larger frequencies Ωi > ΩP = 4π
q
Krm
a limit cycle emerges from the saddle
via a homoclinic bifurcation
Limit cycle and fixed point coexists until Ωi ≡ ΩD = Kr, where a saddle nodebifurcation leads to the disappearence of the two fixed points
For Ωi > ΩD only the oscillating solution is present
For small mass (large β), there is no more coexistence. (Levi et al. 1978)
Rostock 2014 – p. 5
Simulation Protocols
Dynamics of N oscillators
ΩM maximal natural frequency of the locked oscillators
Ω(I)P = 4
π
q
Krm
Ω(II)D = Kr
Protocol I: Increasing K
The system remains desynchronizeduntil K = K1
c (filled black circles).ΩM increases with K following ΩI
P .Ωi are grouped in small clusters(plateaus).
Protocol II: Decreasing K
The system remains synchronized untilK = K2
c (empty black circles).ΩM remains stucked to the same valuefor a large K interval than it rapidly de-creases to 0 following ΩII
D .
0
1
2
3
0 2 4 6 8 10K
0
0.5
1
ΩM
ΩM
ΩP(I)
ΩD(II)
r
K1
c
K2
c
Protocol I
Protocol II
m = 2
Rostock 2014 – p. 6
Mean Field Theory I
Tanaka et al. (TLO) in [PRL, Physica D (1997)] examined the origin of the hysteretictransition finding that
by following Protocol I and II there is a group of drifting oscillators and one oflocked oscillators which act separately
locked oscillators are characterized by < θ >= 0
drifting oscillators < θ > 6= 0
Drifting and locked oscillators are separated by a frequency:
Following Protocol I the oscillators with Ωi < ΩP are locked
Following Protocol II the oscillators with Ωi < ΩD are locked
These two groups contribute differently to the total level of synchronization in thesystem
r = rL + rD
Rostock 2014 – p. 7
Mean Field Theory II
Total level of synchronization in the system: r = rL + rD
For the locked population TLO derived the self-consistent equation
rI,IIL = Kr
Z θP,D
−θP,D
cos2 θg(Kr sin θ)dθ
where θP = sin−1(ΩP
Kr), θD = sin−1(ΩD
Kr) = π/2 .
For the drifting population the self-consistent equation is
rI,IID ≃ −mKr
Z ∞
−ΩP,D
1
(mΩ)3g(Ω)dΩ
The former equation are correct in the limit of sufficiently large masses
Rostock 2014 – p. 8
Hysteretic Behavior
Numerical Results for Fully Coupled Networks (N = 500)
The data obtained by following protocol II are quite well reproduced by the meanfield approximation rII
The mean field extimation rI by TLO does not reproduce the stepwise structure inprotocol I
Clusters of NL locked oscillators ofany size remain stable between rI
and rII
The level of synchronization of theseclusters can be theoretically obtainedby generalizing the theory of TLO toprotocols where ΩM remains constant
m = 6 0 2 4 6 8 10 12 14 16 18 20K
0
0.2
0.4
0.6
0.8
1
r
0 5 10 15 20
K0
100
200
300
400
500
NL
Rostock 2014 – p. 9
Finite Size Effects
Kc1 is the transition value from asynchronous to synchronous state
(following Protocol I)
Kc2 is the transition value from synchronous to asynchronous state
(following Protocol II)
Kc1 is strongly influenced by the size of the system
Kc2 does not depend heavily on N
0 2 4 6 8 10 12 14 16 18 20K
0
0.2
0.4
0.6
0.8
1
rN=500N=1000N=2000N=4000N=8000 N=16000
M=60 4000 8000 12000 16000N
1.5
2
2.5
3
3.5
Kc
0 4000 8000 12000 16000N1.5
2
2.5
3
3.5
Kc
m = 0.8
m = 1.0
Dashed line → KMF1 mean field value by Gupta et al (PRE 2014)
Rostock 2014 – p. 10
Finite Size Effects Kc
1
The mean field critical value has been esti-mated Gupta, Campa, Ruffo (PRE 2014) byemploying a nonlinear Fokker-Planck formu-lation for the evolution of the single oscillatordistribution ρ(θ, θ, Ω, t) for coupled oscilla-tors with inertia and noise
1
KMF1
=πg(0)
2−
m
2
Z ∞
−∞
g(Ω)dΩ
1 + m2Ω2
where g(Ω) is an unimodal distributionAcebron et al, PRE (2000)
100 1000 10000 1e+05N
1
K1M
F -K1c
100 1000 10000 1e+05N
1
m=0.8
m=1.0
Slope ~ 0.23
Slope ~ 0.22
We observe the following scaling with the system size N for fixed mass
KMF1 − Kc
1(N) ∝ N−1/5
this is true for sufficently low masses
Rostock 2014 – p. 11
Dependence On the Mass Kc
1
Kc1 increases with m up to a maximal value and than decreases at larger masses
by increasing N Kc1 increases and the position of the maximum shifts to larger
masses (finite size effects)
0 10 20 30m
2
2.5
3
3.5
4
4.5
5
5.5
K1
c
K1
MF
0 2 4 6 8
m/N1/5
0
0.2
0.4
0.6
0.8
1
ξN=16,000N=8,000N=4,000N=2,000N=1,000
The following general scaling seems to apply
ξ ≡KMF
1 − Kc1(m, N)
KMF1
= G
„
m
N1/5
«
where KMF1 ∝ 2m for m > 1
Rostock 2014 – p. 12
Dependence On the Mass Kc
2
The TLO approach fails to reproduce the critical coupling for the transition fromasynchronous to synchronous state (i.e., Kc
1), however it gives a good estimate of thereturn curve obtained with protocol II from the synchronized to the aynchronous regime
0 5 10 15 20 25 30m
1.6
1.8
2
2.2
K2
c
K2
TLO
Kc2 initially decreases with m then saturates, limited variations with the size N
KTLO2 is the minimal coupling associated to a partially synchronized state given
by TLO approach for protocol II
KTLO2 exhibits the same behaviour as Kc
2 , however it slightly understimates theasymptotic value (see the scale)
Rostock 2014 – p. 13
Drifting Clusters I
For larger masses (m=6), the synchronization transition becomes more complex, itoccurs via the emergence of clusters of drifting oscillators.
The partially synchronized state is characterized by the coexistence of
a cluster of locked oscillators with < θ >≃ 0
clusters composed by drifting oscillators with finite average velocities
The effect of these extra clusters is to induce (periodic or quasi-periodic) oscillations inthe temporal evolution of the order parameter.
-0.4 -0.2 0 0.2 0.4 0.6Ωi
-4
-2
0
2
4
<dθ
i/dt>
K=15 K=10 K=5 K=2.5
(a)
0 20 40 60 80time
0
0.2
0.4
0.6
0.8
1
r(t)
(b)
Rostock 2014 – p. 14
Drifting Clusters II
If we compare the evolution of the instantaneous velocities θi for 3 oscillators and r(t)
we observe that
the phase velocities of O2 and O3 display synchronized motion
the phase velocity of O1 oscillates irregularly around zero
the almost periodic oscillations of r(t) are driven by the periodic oscillations of O2
and O3
-0.4 -0.2 0 0.2 0.4Ωi
-2
0
2
<dθ
i/dt>
O1
O2
O3
(a)
0 10 20 30 40time
0
0.5
1
1.5
2
r(t) O1
O3O
2
(b)
Rostock 2014 – p. 15
Drifting Clusters III
The amplitude of the oscillations of r(t) and the number of oscillators in the driftingclusters NDC correlates in a linear manner
The oscillations observable in the order parameter are induced by the presence oflarge secondary clusters characterized by finite whirling velocities
At smaller masses oscillations in r(t) are present, but reduced in amplitude.These oscillations are due to finite size effects since no clusters of driftingoscillators are observed
Blue dashed line ⇒ estimatedmean field value rI by TLO
The mean field theory capturesthe average increase of the orderparameter but it does not foreseethe oscillations
0 5 10 15 20K
0
0.2
0.4
0.6
0.8
1
0 5 10 15 200
50
100
150
r min
,r m
ax
(rmax
-rmin
)*240
NDC
Rostock 2014 – p. 16
Diluted Network
Constraint 1 : the random matrix is symmetric
Constraint 2 : the in-degree is constant and equal to Nc
0 1 2 3 4 5 6K
0
0.2
0.4
0.6
0.8
1
r
Nc=N
Nc=5
Nc=10
Nc=15
Nc=25
Nc=50
Nc=125
Nc=250
Nc=500
Nc=1000
(a)
0.01 0.1 1N
c/N
0
0.5
1
1.5
2
2.5
3
Wh
N=500N=1000N=2000
0 3 6 9
K0
0.2
0.4
0.6
0.8
1
r
Wh
Diluted or fully coupled systems (whenever the coupling is properly rescaled withthe in-degree) display the same phase-diagram
For very small connectivities the transition from hysteretic becomes continuous
By increasing the system size the transition will stay hysteretic for extremely smallpercentages of connected (incoming) links
Rostock 2014 – p. 17
Diluted Network II
The TLO mean field theory still gives reasonable results (70% of broken links)
All the states between the synchronization curves obtained following Protocol I andII are reachable and stable
2 4 6 8K
0
0.2
0.4
0.6
0.8
1
r
0 2 4 6 8
K0
100
200
300
400
500
NL
These states, located in the region between the synchronization curves, arecharacterized by a frozen cluster structure, composed by a constant NL
The generalized mean-field solution r0(K, Ω0) is able to well reproduce thenumerically obtained paths connecting the synchronization curves (I) and (II)
Rostock 2014 – p. 18
Italian High Voltage Power Grid
Each node is described by the phase:
φi(t) = ωACt + θi(t)
where ωAC = 2π 50 Hz is the standardAC frequency and θi is the phase devi-ation from ωAC .Consumers and generators can bedistinguished by the sign of parameterPi:
Pi > 0 (Pi < 0)
corresponds to generated (consumed)power.
θi = α
2
4−θi + Pi + KX
ij
Ci,j sin(θj − θi)
3
5
Average connectivity < Nc >= 2.865
[ Filatrella et al., The European Physical Journal B (2008)]
Rostock 2014 – p. 19
Italian High Voltage Power Grid
We do not observe any hysteretic behavior or multistability down to K = 9
For smaller coupling an intricate behavior is observable depending on initialconditions
Generators and consumers compete in order to oscillates at different frequencies
The local architecture favours a splitting based on the proximity of the oscillators
Several small whirling clusters appear characterized by different phase velocities
The irregular oscillations in r(t) reflect quasi-periodic motions
0 10 20 30K
0
0.2
0.4
0.6
0.8
1
r
0 50 100 150 200time
0
0.1
0.2
0.3
0.4
0.5
r(t)
K=1K=4K=6K=7
(b)
Rostock 2014 – p. 20
Main Results
We have studied the synchronization transition for a globally coupled Kuramoto modelwith inertia for different system sizes and inertia values.
The transition is hysteretic for sufficiently large masses
Clusters of locked oscillators of any size coexist within the hysteretic region
A generalization of TLO theory is capable to reproduce all the possiblesynchronization/desynchronization hysteretic loops
The presence of clusters composed by drifting oscillators induces oscillatorybehaviour in the order parameter
The properties of the hysteretic transition have been examined also for random dilutednetwork.
The main properties of the transition are not affected by the dilution
The transition appears to become continuous only when the number of links pernode becomes of the order of few units
S. Olmi, A. Navas, S. Boccaletti, A. Torcini, submitted to PRE
Rostock 2014 – p. 21
Extension of the Mean Field Theory
In principle one could fix the discriminating frequency to some arbitrary value Ω0 andsolve self-consistently
r = rL + rD
rI,IIL = Kr
Z θ0
−θ0
cos2 θg(Kr sin θ)dθ rI,IID ≃ −mKr
Z ∞
−Ω0
1
(mΩ)3g(Ω)dΩ
This amounts to obtain a solution r0 = r0(K, Ω0) by solving
Z θ0
−θ0
cos2 θg(Kr0 sin θ)dθ − m
Z ∞
−Ω0
1
(mΩ)3g(Ω)dΩ =
1
K
with θ0 = sin−1(Ω0/Kr0). The solution exists if Ω0 < ΩD = Kr0.
⇒ A portion of the (K, r) plane delimited by the curve rII(K) is filled with the curvesr0(K) obtained for different Ω0 values.
Rostock 2014 – p. 22
Hysteretic Behavior
Fully Coupled Networks
A step-wise structure emerges at larger masses do to the break down of theindependence of the whirling oscillators
The number of locked oscillators NL follows the same step-wise structure
NL remains constant until it reaches the descending curve
0 2 4 6 8 10K
0
0.2
0.4
0.6
0.8
1
r
1 2 3 4 5
K0
100
200
300
400
500
NL
(a)
KF
G
0 2 4 6 8 10 12 14 16 18 20K
0
0.2
0.4
0.6
0.8
1
r
0 5 10 15 20
K0
100
200
300
400
500
NL
(b)
KF
G
Rostock 2014 – p. 23
Italian High Voltage Power Grid
By following Protocol II
the system stays in one cluster up to K = 7
at K = 6 wide oscillations emerge in r(t) due to the locked clusters that havebeen splitted in two (is this also the origin for the emergent multistability?)
By lowering further K several whirling small clusters appear and r becomesirregular
0 50 100
-1
0
1
2
3
0 50 100
-1
0
1
2
3
0 50 100
-1
0
1
2
3
0 50 100
-1
0
1
2
3
0 50 100
-0.2
-0.1
0
0.1
0 50 100-0.5
0
0.5
1
1.5
2
2.5
3
0 50 1000
1
2
3
K=2 K=3
K=4
K=6
K=1
K=7
Oscillator Index
<dθ
i/dt>
Rostock 2014 – p. 24
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