# Phase-shift inversion in oscillator systems with periodically switching couplings.

**ABSTRACT** A system's response to external periodic changes can provide crucial information about its dynamical properties. We investigate the synchronization transition, an archetypical example of a dynamic phase transition, in the framework of such a temporal response. The Kuramoto model under periodically switching interactions has the same type of phase transition as the original mean-field model. Furthermore, we see that the signature of the synchronization transition appears in the relative delay of the order parameter with respect to the phase of oscillating interactions as well. Specifically, the phase shift becomes significantly larger as the system gets closer to the phase transition, so that the order parameter at the minimum interaction density can even be larger than that at the maximum interaction density, counterintuitively. We argue that this phase-shift inversion is caused by the diverging relaxation time, in a similar way to the resonance near the critical point in the kinetic Ising model. Our result, based on exhaustive simulations on globally coupled systems as well as scale-free networks, shows that an oscillator system's phase transition can be manifested in the temporal response to the topological dynamics of the underlying connection structure.

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arXiv:1111.3734v1 [cond-mat.stat-mech] 16 Nov 2011

Phase-shift inversion in oscillator systems with periodically switching couplings

Sang Hoon Lee,1, ∗Sungmin Lee,1,2Seung-Woo Son,3and Petter Holme1,4

1IceLab, Department of Physics, Ume˚ a University, 901 87 Ume˚ a, Sweden

2Department of Biotechnology, Norwegian University of Science and Technolgy, N-7491 Trondheim, Norway

3Complexity Science Group, University of Calgary, Calgary T2N 1N4, Canada

4Department of Energy Science, Sungkyunkwan University, Suwon 440–746, Korea

(Dated: November 17, 2011)

A system’s response to external periodic changes can provide crucial information about its dynam-

ical properties. We investigate the synchronization transition, an archetypical example of a dynamic

phase transition, in the framework of such a temporal response. The Kuramoto model under period-

ically switching interactions has the same type of phase transition as the original mean-field model.

Furthermore, we see that the signature of the synchronization transition appears in the relative delay

of the order parameter with respect to the phase of oscillating interactions as well. Specifically, the

phase shift becomes significantly larger as the system gets closer to the phase transition so that the

order parameter at the minimum interaction density can even be larger than that at the maximum

interaction density, counterintuitively. We argue that this phase-shift inversion is caused by the

diverging relaxation time, in a similar way to the resonance near the critical point in the kinetic

Ising model. Our result shows that an oscillator system’s phase transition can be manifested in the

temporal response to the topological dynamics of the underlying connection structure.

PACS numbers: 05.45.-a, 05.45.Tp, 05.45.Xt, 64.60.an

I. INTRODUCTION

A collection of interacting coupled oscillators is one of

the most intensively studied systems showing a dynamic

phase transition called a synchronization transition [1–3].

Such systems have been used as models to describe vari-

ous phenomena such as the pacemakers of the heart, the

collection of amoeba, large-scale ecosystems [4–9], and

fiber-optic networks of optoelectronic oscillators [10]. As

the coupling strength increases, such systems undergo

a phase transition from the desynchronized (disordered)

state to the synchronized (ordered) state characterized

by critical phenomena. The most well-known theoreti-

cal formalism is the Kuramoto model [1] where the os-

cillators interact with each other by a sinusoidal cou-

pling strength with respect to the phase differences of

the two oscillators. In a system of globally coupled Ku-

ramoto oscillators, the exact value of critical coupling

strength KMF

c

and the transition nature depending on

the concavity of the natural frequency distribution are

exactly shown in the thermodynamic limit [3]. Recently,

the effects of finite-size systems with various dimensions

including the heterogeneity in the degree distributions

(“scale-free” networks) [11] or the possibility of nega-

tive coupling (“contrarian” oscillators) [12] are studied

as well.

Besides the steady-state behavior after the initial tran-

sient behavior, the dynamical aspect of the oscillators

has also been studied recently [13, 14]. Especially, the

effects of temporally varying interaction structures [15]

themselves are worthwhile to investigate since we can

∗Electronic address: sanghoon.lee@physics.umu.se

systematically analyze the response of systems to such

structural changes of interactions [16, 17]. For instance,

the mobile oscillator are considered in Refs. [18–20] as

an example of temporally switching interactions. In this

paper, we take the Kuramoto model with periodically

switching interactions and analyze the temporal response

in the collective phase of oscillators to such periodicity

in interactions. Such a periodically switching interac-

tions may have implications to biological or social com-

munications among compartments under natural circa-

dian rhythms, for instance [21].

In terms of the temporally averaged order parame-

ter and its amplitude of oscillation, we verify that the

model undergoes the same synchronization transition as

the mean-field (MF) Kuramoto model [3] with the finite-

size scaling (FSS) analysis. Furthermore, with numerical

simulations and analogy to the kinetic Ising model [22],

we show that the signature of the synchronization tran-

sition is revealed in the relative phase shift of the order

parameter with respect to the density of interactions, as

a resonance-like phase-shift inversion. The result is also

consistent with the diverging relaxation time at the crit-

ical coupling strength [13, 14]. Therefore, we claim that

the system’s temporal response to the temporally chang-

ing interactions can be used as an indicator of a phase

transition.

II.MODEL

We consider the Kuramoto-type oscillator dynamics [3]

composed of N oscillators as

dφi

dt

= ωi+2K

N

N

?

j=1

Tij(t)sin(φi− φj), (1)

Page 2

2

where {φi} is the set of oscillators’ phases, {ωi} is the set

of natural frequencies of oscillators given by the Gaus-

sian distribution with average 0 and unit variance, and

the interaction Tijmediated by the edge between the os-

cillators i and j is subject to the periodic function as

Tij(t) =

?1 if sin(Ωt + θij) > 0

0 otherwise,

(2)

where Ω is the frequency of switching common to all the

edges and {θij} is the set of edges’ intrinsic phases, which

is randomly assigned from the uniform distribution of the

interval [0,π). By choosing such an interval, the edge

densities change periodically with the exact triangle wave

subject to the frequency Ω, from fully connected edges to

isolated oscillators without interactions. Note that the

factor 2 in Eq. (1) is to directly compare the coupling

strength K of our model with ?Tij? = 1/2 to the K value

in the globally coupled Kuramoto model, where ?x? refers

to the temporal average of x(t) over a period.

In addition to the original control parameter of cou-

pling strength K in the Kuramoto model, the edge fre-

quency Ω driving the system is a crucial parameter as

well. The response of oscillators to the toggling interac-

tions is measured by the temporal behavior of conven-

tional phase order parameter ∆(t) where

∆(t)exp[iψ(t)] =

1

N

N

?

i=1

exp[iφi(t)]. (3)

The order parameter’s dynamics depends on the periodic

change of interactions with the density

ρ(t) =

2?

N(N − 1)

i<jTij(t)

. (4)

We provide the java applet simulating this model sys-

tem, for readers to observe the behavior of a system of

relatively small sizes [23]. From now on, we present our

numerical simulation results and their implications.

III. RESULTS

A typical temporal behavior of oscillators is illustrated

in Fig. 1, in case of Ω = 1.00. Note that the oscillating

behavior of the order parameter ∆(t) driven by switching

edges (with the same frequency) is shown even in case of

quite low values of K, for which the original MF Ku-

ramoto model corresponds to the disordered phase. An-

other interesting aspect is the relative phase shift of ∆(t)

with respect to ρ(t). In other words, the maximum or

minimum of ∆(t) occurs after some time when the maxi-

mum of minimum of ρ(t) is reached, as shown in Fig. 1(a).

Moreover, as K is increased, the delay for maximum

values and the delay for minimum values become more

asymmetric, as clearly shown in Fig. 1(b). In spite of such

temporal oscillations of ∆(t), the temporally averaged

value ?∆? (averaged after the transient time ≃ 20) shows

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

∆(t), ρ(t)

t

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4

ρ(t > 20)

0.5 0.6 0.7 0.8 0.9 1

∆(t > 20)

(b)

K=0.20

1.00

1.60

2.50

6.00

FIG. 1: A typical behavior of the order parameter ∆(t) and

the interaction density ρ(t). The system size N = 800, the

edge frequency Ω = 1.00, and the time series of ∆(t) de-

pending on K are shown as smooth curves in (a) and ρ(t)

(independent of K) as black triangle waves in (a). The ∆–ρ

diagram for t > 20 after the transient behavior is shown in

(b). All the curves are results averaged over 50 samples.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-10 -8-6-4

(K - Kc)N1/ –ν

-2 0 2 4 6

〈∆〉 Nβ/ –ν

N = 100

200

400

800

FIG. 2: FSS scaling collapse of the temporally averaged order

parameter ?∆? for various system sizes with the critical ex-

ponents of the original MF Kuramoto model (Kc = 1.62(1),

β = 1/2, and ¯ ν = 5/2). The edge frequency is Ω = 1.00.

exactly the same universality class of synchronization

transition as the MF Kuramoto model with β = 1/2, and

¯ ν = 5/2 [3, 11], as described in the FSS scaling collapse

with ?∆? = N−β/¯ νf?(K − Kc(Ω))N1/¯ ν;Ω?(the scaling

function f itself, as well as Kc, depends on Ω) shown

in Fig. 2. We suggest the slightly larger value of Kc(Ω)

than that of the MF model KMF

from additional quenched randomness caused by {θij}

c

= 2?2/π ≃ 1.596 stem

Page 3

3

(a)

0.5 1 1.5 2

1

2

3

4

5

6

7

8

K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

(b)

0.5 1 1.5 2

SΩ

1

2

3

4

5

6

7

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

AΩ

(c)

0.5 1 1.5 2

Ω

1

2

3

4

5

6

7

8

K

0

0.1

0.2

0.3

0.4

0.5

0.6

S

(d)

0.5 1 1.5 2

Ω

1

2

3

4

5

6

7

8

0

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 3: The difference A between maximum and minimum

values of ∆(t) oscillation and the area S enclosed by the loop

in ∆–ρ the diagram are shown in the K–Ω plane. The system

size N = 800, and the color-coded values are A (a), AΩ (b),

S (c), and SΩ (d). The time series ∆(t) for t > 20 after the

transient behavior is used, and all the results are averaged

over 50 samples.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4

K

5 6 7 8

AΩ

(a)

Ω=0.40

0.60

0.80

1.00

1.20

1.30

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4

K

5 6 7 8

SΩ

(b)

1.40

1.50

1.60

1.70

1.80

1.90

2.00

FIG. 4: AΩ (a) and SΩ (b) as functions of K, for various Ω

values, for the system size N = 800. The time series ∆(t)

for t > 20 after the transient behavior is used, and all the

results are from averaged over 50 samples. The black vertical

lines correspond to the MF critical coupling strength KMF

2?2/π.

c

=

for edges, similarly to the lower Tc(the disordered phase

extended) for the kinetic Ising model under the external

oscillating field [22]. Since we have confirmed the MF

transition for ?∆?, from now on we focus the temporal

behavior of ∆(t), especially with respect to the oscillating

interaction strength ρ(t).

To characterize the oscillating behavior more quantita-

tively, we systematically measure the difference between

maximum and minimum values of ∆(t) oscillation de-

noted as A (roughly twice the amplitude of ∆(t)) and

the enclosed area in the ∆–ρ diagram denoted as S, for

a wide range of Ω and K values. Note that the data

shown here are from the system size N = 800, but we

have checked that there exists no significant finite-size

effects based on the simulations results with smaller and

larger systems. As shown in Fig. 3, the scaling behavior

of A and S are quite similar, and notably both A and S

seem to be inversely proportional to Ω for Ω ? 1.20. This

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

4

5

6

7

8

K

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

!("max) - !("min)

FIG. 5: Height difference between ∆(ρmax) and ∆(ρmin) for

N = 800. The time series ∆(t) for t > 20 after the tran-

sient behavior is used, and all the results are averaged over

50 samples.

scaling behavior makes the AΩ and SΩ in Figs. 3(b) and

(d) depend only on K for Ω ? 1.20, and is also clearly

observable from the fact that the curves for Ω ? 1.20

are collapsed in Fig. 4. Since Ω represents the angular

velocity of the fluctuation of edges (hence that of ∆(t)),

AΩ corresponds to the “linear velocity” of ∆(t), which is

shown to be conserved for a given value of K. More im-

portantly, the synchronization transition is clearly shown

in the steep change of A and S near the critical coupling

strength KMF

c

for the original MF Kuramoto model, as

shown in Figs. 3 and 4. Therefore, we conclude that the

signature of the MF synchronization transition of Ku-

ramoto model is resurfaced in terms of A and S.

Besides the amplitude of ∆ and enclosed area in the

∆–ρ diagram, a notable property of ∆–ρ diagram in

Fig. 1(b) is successive change in the shapes of enclosed

regions, as K is increased. First, the desynchronization

process of ∆(t) after the onset of decreasing phase of ρ(t)

is slower than the synchronization process of ∆(t) after

the onset of increasing phase of ρ(t) in general. This

asymmetry is natural, since the response of the order pa-

rameter to such linear functions is integrated form, i.e.,

quadratic functions depending on the initial values.

We show that the synchronization transition is also

observed in the phase shift of ∆(t) with respect to ρ(t)

as well.To systematically analyze such a phase shift

caused by the temporal delay of ∆(t), we check the rel-

ative height difference of ∆ at the right and left ends

in Fig. 1(b), corresponding to ∆(ρmax) and ∆(ρmin) re-

spectively, also affects the shape of enclosed regions. In

Fig. 5, we plot such height differences in the K–Ω space,

and the height difference occurs mainly in small Ω and

large K regimes. Interestingly, near K ≃ 1.6 in rela-

tively large Ω regimes, the difference becomes negative,

which corresponds to ∆(ρmax) < ∆(ρmin). We call this

phenomenon phase-shift inversion. K ≃ 1.6 is close to

the MF critical coupling constant KMF

suggest that it is related to critical slowing down near

the critical point in a phase transition, as shown in the

diverging relaxation time τ at KMF

c

again, and we

c

[13, 14], given by

Page 4

4

1 =

like phase-shift inversion phenomenon occurs similarly

to the frequency matching condition around the phase

transition in the MF kinetic Ising model [22]. In other

words, the MF synchronization transition in the Ku-

ramoto model is also reflected in the temporal response

to the periodically switching interactions.

Finally, as an alternative model, we consider the

Kuramoto model under the interactions whose overall

strength oscillates as ρ(t) in Eq. (4), but the edges Tij

are drawn randomly. In other words, only the density

of edges is periodic and the interactions are completely

random. Interestingly, we have found that the results

are similar to our original model with Eq. (2) in case of

large K > Kc, but show significant difference for small

K < Kc. The fact may reflect the intrinsic difference of

the ordered and disordered phases of the synchronization

transition, but we do not have clear explanation for the

difference yet.

?π/8K exp[1/(2τ2)]erfc[1/(√2τ)]. This resonance-

IV.SUMMARY AND DISCUSSIONS

We have studied the Kuramoto model under period-

ically switching interactions, checked that it shows the

same MF synchronization transition as the original Ku-

ramoto model with the FSS analysis, and found that the

phase transition is observed in terms of dynamical prop-

erties such as the amplitude of oscillation of the order

parameter and its relative phase shift with respect to

the overall strength of interactions. Especially, the lat-

ter causes the phase-shift inversion phenomenon that the

significantly large phase shift near the phase transition

let the order parameter at the minimum interaction den-

sity to be larger than that at the maximum interaction

density. Such observations strongly suggest our model is

related to the MF kinetic Ising model with the frequency

matching condition around the phase transition [22] and

would be worth investigating further. Since periodicity

plays important roles in many parts of the nature, es-

pecially for systems under natural circadian rhythms or

external stimuli, our results suggest that the temporal

response to such periodicity can be an important cue to

characterize the system. In our simple model, the phase

transition is already known as MF type so that we can

confirm the result, but it could be more useful if a simi-

lar method is applied to more complicated or previously

unknown systems.

We remark that the current work deals with the pe-

riodic interactions based on the globally connected sub-

strate structure, but the effects of other types of sub-

strate structures or more complex forms of temporally

varying structures would be worthwhile to investigate in

the future. More nontrivial temporal responses may arise

for those cases, due to the complications caused by the

substrate interaction itself.

Acknowledgments

We greatly appreciate Beom Jun Kim, Hang-Hyun Jo,

Michael Gastner, Thilo Gross, and Sergey Dorogovtsev

for their valuable comments. This work was supported

by the Swedish Research Council (SHL and PH), the

Wenner-Gren Foundation (SL), and the WCU program

through NRF Korea funded by MEST R31-2008-10029

(PH).

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