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)

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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

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(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

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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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