Center manifold reduction for large populations of globally coupled phase oscillators.

TLDR: A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces and it is shown that there exists a finite-dimensional center manifold on a space of generalized functions.

Abstract: A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not sin θ, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

Figures

FIG. 4. Numerical results for f(h)¼ sin hþ h sin 2h. Black dots denote the order parameter calculated from Eq. (1) for N ¼ 8000, gðxÞ ¼ e x2=2= ffiffiffiffiffi 2p p using the method shown in Ref. 18. Since r is unstable when K Kc< 0, it is difficult to obtain small r. The solid lines are interpolations of black dots using quintic polynomials. The dotted lines denote the analytical results obtained by Eq. (46).

FIG. 1. Bifurcation diagrams of the order parameter for (a) f(h)¼ sin h and (b) f(h)¼ sin hþ h sin 2h. The solid lines denote stable solutions, and the dotted lines denote unstable solutions.

FIG. 3. The contour for the Laplace inversion formula.

FIG. 2. The motion of the generalized eigenvalue as K decreases. When K>Kc, it exists on the right half plane as an eigenvalue of T1. At K¼Kc, it is absorbed into the continuous spectrum and disappears from the complex plane. When K<Kc, it goes to the second Riemann sheet of the generalized resolvent and turns into a resonance pole, which is no longer an eigenvalue of T1 in the usual sense.

Full text

九州大学学術情報リポジトリ
Kyushu University Institutional Repository
Center manifold reduction for large populations
of globally coupled phase oscillators
Chiba, Hayato
Institute of Mathematics for Industry, Kyushu University
Nishikawa, Isao
Institute of Industrial Science, University of Tokyo
http://hdl.handle.net/2324/25695
出版情報：Chaos. 21 (4), pp.043103(1)-043103(10), 2011-12-12. American Institute of Physics
バージョン：
権利関係：(C) 2011 American Institute of Physics

Center manifold reduction for large populations of globally coupled
phase oscillators
Hayato Chiba
1
and Isao Nishikawa
2
1
Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
2
Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan
(Received 7 April 2011; accept ed 16 September 2011; published online 7 October 2011)
A bifurcation theory for a system of globally coupled phase oscillators is developed based on the
theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on
a space of generalized fu nctions. The dynamics on the manifold is derived for any coupling
functions. When the coupling function is sin h, a bifurcation diagram conjectured by Kuramoto is
rigorously obtained. When it is not sin h, a new type of bifurcation phenomenon is found due to the
discontinuity of the projection operator to the center subspace.
V
C
2011 American Institute of
Physics. [doi:10.1063/1.3647317]
The dynamics of systems of large populations of coupled
oscillators have been of great interest because collective
synchronization phenomena are observed in a variety of
areas. The stability of (de)synchronous states and bifur-
cations from them are the main issues to understand the
behavior of systems. Since usual stability and bifurcation
theory in dynamical systems are not applicable to such
systems when the dimensions of systems are too large,
much work has been done to understand the dynamics.
However, except for special simple cases, to investigate
bifurcations is a quite difficult problem and much
remains unclear. In this paper, a correct bifurcation
theory for such systems is proposed by means of the
theory of generalized functions, which is applicable to
large classes of coupled phase oscillators including the
Kuramoto model. To use a space of generalized functions
is suitable to study the behavior of statistical quantities
such as the order parameter. This will be demonstrated
for two cases. For the Kuramoto model, a well known
Kuramoto’s bifurcation diagram will be rigorously
obtained. For a certain system including the second har-
monic in the coup ling function, a new type of bifurcation
phenomenon will be found.
I. INTRODUCTION
Collective synchronization phenomena are observed in a
variety of areas, such as chemical reactions, engineering cir-
cuits, and biological populations.
1
In order to investigate
such phenomena, a system of globally coupled phase oscilla-
tors of the following form is often used:
dh
k
dt
¼ x
k
þ
K
N
X
N
j¼1
f ðh
j
 h
k
Þ; k ¼ 1; …; N; (1)
where h
k
(t) denotes the phase of a k-th oscillator, x
k
2 R
denotes its natural frequency drawn from some distribution
function g(x), K > 0 is the coupling strength, and
f ðhÞ¼
P
1
n¼1
f
n
e
inh
is a 2p-periodic function i ¼
ffiffiffiffiffiffiffi
1
p

.
When f(h) ¼sin h, it is referred to as the Kuramoto model.
2
In this case, it is numerically observed that if K is sufficiently
large, some or all of the oscillators tend to rotate at the same
velocity on average, which is referred to as synchroniza-
tion.
1,3
In order to evaluate whether synchronization occurs,
Kuramoto introduced the order parameter r(t)e
iw(t)
, which is
given by
rðtÞe
iwðtÞ
:¼
1
N
X
N
j¼1
e
ih
j
ðtÞ
: (2)
When a synchronous state is formed, r(t) takes a positive
value. Indeed, based on some formal calculations, Kuramoto
assumed a bifurcation diagram of r(t): Suppose N !1.If
g(x) is an even and unimodal function such that g
00
(0) = 0,
then the bifurcation diagram of r(t ) is as in Fig. 1(a).In
other words, if the coupling strength K is smaller than
K
c
:¼2=(pg(0)), then r(t) : 0 is asympto tically stable. If K
exceeds K
c
, then a stable synchronous state emerges. Near
the transition point K
c
, r is of order O((K K
c
)
1=2
). See
Ref. 3 for Kuramoto’s discussion.
In the last two decades, several studies have been per-
formed in an attempt to confirm Kuramoto’s assumption.
Daido
4
calculated steady states of Eq. (1) for any f using an
argument similar to Kuramoto’s. Although he obtained vari-
ous bifurcation diagrams, the stability of solutions was not
demonstrated. In order to investigate the stability of steady
states, Strogatz and Mirollo and coworker
5–8
performed a
linearized analysis. The linear operator T
1
, which is obtained
by linearizing the Kuramoto model, has a continuous spec-
trum on the imaginary axis. Nevertheless, they found that the
steady states can be asymptotically stable because of the ex-
istence of resonance poles on the left half plane.
8
Since the
results of Strogatz and Mirollo and coworker are based on
the linearized analysis, the effects of nonlinear terms are
neglected. However, investigating nonlinear bifurcations is
more difficult because T
1
has a continuous spectrum on the
imaginary axis, that is, a center manifold in the usual sense
is of infinite dimension. In order to avoid this difficulty,
1054-1500/2011/21(4)/043103/10/$30.00
V
C
2011 American Institute of Physics21, 043103-1
CHAOS 21, 043103 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp

Crawford and Davies
9
added noise of strength D > 0 to the
Kuramoto model. The continuous spectrum then moves to the
left side by D, and thus, the usual center manifold reduction is
applicable. However, their method is not valid when D ¼0. An
eigenfunction of T
1
associated with a center subspace diverges
as D ! 0 because an eigenvalue on the imaginary axis is em-
bedded in the continuous spectrum as D ! 0. Recently, Ott and
Antonsen
10
found a special solution of the Kuramoto model,
which allows the dimension of the system to be reduced. Their
method is applicable only for f(h) ¼sin h because their method
relies on a certain symmetry of the system.
11
Furthermore, the
reduced system is still of infinite dimension, except for the case
in which g(x) is a rational function. Pikovsky and Rosenblum
12
proposed the reduction of the system based on the construction
of constants of motion. Their method also relies on a special
form of the system. Thus, a unified bifurcation theory for
globally coupled phase oscillators is required.
In the present paper, a correct center manifold reduction
is proposed by means of the theory of generalized functions,
which is applicable for any coupling function f.Itisshown
that there exists a finite-dimensional center manifold on a
space of generalized functions, despite the fact that the contin-
uous spectrum lies on the imaginary axis. This will be demon-
strated for two cases, (I) f(h) ¼sin h and (II) f(h) ¼sin h
þh sin 2h, h 2 R, and two distribution functions g(x), (a) a
Gaussian distribution and (b) a rational function (e.g.,
Lorentzian distribution g(x) ¼1=(p(1 þx
2
))). For (I), we rig-
orously prove Kuramoto’s conjecture; that is, it is shown for
the continuous limit that when 0 < K < K
c
, the de-synchron ous
state r : 0 is locally asymptotically stable, and when K
c
< K,
a stable synchronous state r > 0 bifurcates from the trivial so-
lution with the order O((K K
c
)
1=2
). For (II), a different bifur-
cation diagram will be obtained, as was predicted by Daido.
4
The different bifurcation structure is shown to be caused by
the discontinuity of the projection to the generalized center
subspace. All omitted proofs are given in Ref. 13.
II. THE CONTINUOUS MODEL
Let us derive the continuous model (the continuum
limit) of Eq. (1) to describe the situation N !1.By
introducing the Daido’s order parameter
4
^
g
k
ðtÞ :¼
1
N
X
N
j¼1
e
ikh
j
ðtÞ
;
Equation (1) is rewritten as
dh
k
dt
¼ x
k
þ K
X
1
l¼1
f
l
^
g
l
ðtÞe
ilh
k
:
This implies that the flow of h
k
is generated by the vector
field
^
v ¼ x
k
þ K
X
1
l¼1
f
l
^
g
l
ðtÞe
ilh
k
:
Hence, we define the continuous model of Eq. (1) as the
equation of continuity
@q
t
@t
þ
@
@h
ðq
t
vÞ¼0;
v :¼ x þ K
X
1
l¼1
f
l
g
l
ðtÞe
ilh
;
8
>
>
<
>
>
:
(3)
where g
l
is defined to be
g
l
ðtÞ¼
ð
R
ð
2p
0
e
ilh
q
t
ðh; xÞgðxÞdhdx;
g(x) is a given probability density function for natural fre-
quencies, and the unknown function q
t
¼q
t
(h, x) is a proba-
bility measure on [0, 2p) parameterized by t, x 2 R .In
particular, g
1
is a continuous version of Kuramoto’s order
parameter. Roughly speaking, q
t
(h, x) denotes a probability
that an oscillator having a natural frequency x is placed at a
position h.
Setting Z
j
ðt; xÞ :¼
Ð
2p
0
e
ijh
q
t
ðh; xÞdh yields
dZ
j
dt
¼ ijxZ
j
þ ijKf
j
g
j
þ ijK
X
l6¼j
f
l
g
l
Z
jl
: (4)
The trivial solution Z
j
 0 j ¼ 61; 62; …ðÞcorresponds to
the uniform distribution q
t
: 1=2p on a circle, which
implies r : 0 (de-synchronous state). To investigate the sta-
bility of this state, we consider the linearized system. Let
L
2
(R, g(x)dx) be the weighted Lebesgue space with the
inner product
ðw; /Þ¼
ð
R
wðxÞ/ðxÞgðxÞdx:
Put P
0
(x) : 1. Then, the order parameters are written as
g
j
ðtÞ¼
ð
R
Z
j
ðt; xÞgðxÞdx ¼ðZ
j
; P
0
Þ¼ðP
0
; Z
j
Þ; (5)
and the linearized system of Eq. (4) is given by
dZ
j
dt
¼ T
j
Z
j
:¼ ijxZ
j
þ ijKf
j
ðP
0
; Z
j
Þ: (6)
Let us consider the spectra of linear operators T
j
. The spec-
trum of T
j
consists of the continuous spectrum and eigenval-
ues. The continuous spectrum is the whole imaginary axis.
Eigenvalues k of T
j
are given as roots of the equation
ð
R
1
k  ijx
gðxÞdx ¼
1
ijKf
j
: (7)
Indeed, the equation (k T
j
)v ¼0 provides
v þ ijKf
j
ðP
0
; vÞðk  ijxÞ
1
P
0
¼ 0:
FIG. 1. Bifurcation diagrams of the order parameter for (a) f(h) ¼sin h and
(b) f(h) ¼sin h þh sin 2h. The solid lines denote stable solutions, and the
dotted lines denote unstable solutions.
043103-2 H. Chiba and I. Nishikawa Chaos 21, 043103 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp

Taking the inner product with P
0
, we obtain Eq. (7).Itis
known that there exists a positive constant K
jðÞ
c
such that if
K
jðÞ
c
< K, T
j
has eigenvalues on the right half plane, so that
the j-th Fourier component of the de-synchronous state is
unstable, while if 0 < K < K
ðjÞ
c
, T
j
has no eigenvalues. In
particular, when g(x) is even and unimodal, there exists a
unique eigenvalue k ¼k
0
(K) on the right half plane for
K
jðÞ
c
< K.AsK decreases , k
0
(K) moves to the left, and at
K ¼ K
ðjÞ
c
, k
0
(K) is absorbed into the continuous spectrum on
the imaginary axis. In this manner, the eigenvalue suddenly
disappears at K ¼ K
ðjÞ
c
. For example, if f is an odd function
and if g is even and unimodal, then K
jðÞ
c
is given by
K
ðjÞ
c
¼
Imðf
j
Þ
pjf
j
j
2
gð0Þ
: (8)
In the present paper, for simplicity, we assume that K
1ðÞ
c
is
the least number among K
jðÞ
c
; K
c
:¼ inf
j
K
ðjÞ
c
¼ K
ð1Þ
c
. This
means that the first Fourier component of the coupling func-
tion f is the most dominant term (for f(h) ¼sin h þh sin 2h,
which is true if and only if h < 1). When 0 < K < K
c
, T
j
has
no eigenvalues and the spectrum consists only of the contin-
uous spectrum on the imaginary axis for any j. Thus, the dy-
namics of the lineari zed system dZ
j
=dt ¼T
j
Z
j
is quite
nontrivial. It is well known for a finite dimensional system
that the trivial solution is neutrally stable if the spectrum lies
on the imaginary axis. For infinite dimensional systems, this
is not true. Indeed, numerical simulation
8,14
suggests that the
order parameter decays exponentially to zero when
0 < K < K
c
. Strogatz and coworkers
8
found that such an
exponential decay of the order parameter is induced by a res-
onance pole, and it is rigorously proved by Chiba.
13
In Ref.
13, the spectral theory on rigged Hilbert spaces is developed
to reveal the dynamics of the linearized system.
III. A RIGGED HILBERT SPACE
A rigged Hilbert space consists of three spaces
X  L
2
R; g xðÞdxðÞX
0
: a space X of test functions, a Hil-
bert space L
2
(R, g(x)dx), and the dual space X
0
of X (a space
of continuous anti-linear functionals on X, each element of
which is referred to as a generalized function). We use Dir-
ac’s notation, where for l 2 X
0
and / 2 X, l(/) is denoted
by h/jli. For a 2 C, we have ah/ jli¼h

a/ jli¼h/ jali.
A usual function can be regarded as a generalized function
using an integral kernel. To be precise, the canonical inclu-
sion i : L
2
R; g xðÞdxðÞ!X
0
is defined as follows. For
w 2 L
2
R; g xðÞdxðÞ, we denote i(w)byjwi, which is defined
to be
iðwÞð/Þ¼h/ jwi : ¼ð/; wÞ¼
ð
R
/ðxÞwðxÞgðxÞdx: (9)
By the canonical inclusion, Eq. (4) is rewritten as an evolu-
tion equation on the dual space X
0
as
d
dt
jZ
j
i¼T

j
jZ
j
iþijK
X
l6¼j
f
l
hP
0
jZ
l
ijZ
jl
i; (10)
where T

j
is a dual operator of T
j
defined through h/jT

j
li
¼hT

j
/jli for l 2 X
0
and / 2 X,andT

j
is the adjoint
operator of T
j
. In particular, for any w 2 L
2
R; g xðÞdxðÞ,
we have h/jT

j
wi¼hT

j
/jwi¼ T

j
/; w

¼ /; T
j
w

.Thus
T

j
gives an extension of T
j
to the dual space.
Here, the strategy for the bifurcation theory of globally
coupled phase oscillators is to use the space of generalized
functions X
0
rather than a space of usual functions. The rea-
son for this is explained intuitively as follows. If we use the
space L
2
(R, g(x)dx) to investigate the dynamics, then the
behavior of q
t
itself will be obtained. However, it is neutrally
stable because of the conservation law
Ð
2p
0
q
t
ðh; xÞdh ¼ 1.
What we would like to know is the dynamics of the moments
of q
t
, in particular, the order parameter. This suggests that
we should use a different topology for the stability of q
t
.
(Note that the definition of stability depends on definition of
the topology.) For the purpose of the present study, q
t
is said
to be convergent to
^
q as t !1if and only if
ð
R
ð
2p
0
/ðxÞe
ijh
gðxÞdq
t
ðh; xÞ;
!
ð
R
ð
2p
0
/ðxÞe
ijh
gðxÞd
^
qðh; xÞ;
for any j 2 Z and / 2 X. This implies that for the Fourier
coefficients, a function Z
j
t; xðÞ2L
2
R; g xðÞdxðÞis said to
be convergent to
^
Z
j
ðxÞ as t !1if and only if
ð
R
/ðxÞZ
j
ðt; xÞgðxÞdx !
ð
R
/ðxÞ
^
Z
j
ðxÞgðxÞdx;
for any / 2 X. In other words, Z
j
(t, x) converges to
^
Z
j
ðxÞ if
and only if h/ jZ
j
i!h/ j
^
Z
j
i for any / 2 X if we regard Z
j
as a generalized function. The topology ind uced by this con-
vergence is referred to as the weak topology. By the comple-
tion of L
2
(R, g(x)dx) with respect to the weak topology, we
obtain a space of generalized functions X
0
. On the space X
0
,a
function Z
1
(t, x) converges as t !1if and only if h/jZ
1
i
converges for any / 2 X. Since the order parameter is writ-
ten as g
1
tðÞ¼hP
0
jZ
1
i, this topology is suitable for the pur-
pose of the present study. Although a solution of the
linearized system dZ
j
=dt ¼T
j
Z
j
is neutrally stable in L
2
(R,
g(x)dx)-sense because of the continuous spectrum on the
imaginary axis, we will show that it can decay to zero expo-
nentially if we use the weak topology. More generally, a
rigged Hilbert space provides strong tools for studies of
dynamics of moments of measures or probability density
functions. A suitable choice of the space X depends on a
problem, which will be explained in Sec. IV.
A rigged Hilbert space was introduced by Gelfand
15
to
generalize the theory of Schwartz distributions. See Ref. 16
for a review of Gelfand’s work and its application to quan-
tum mechanics.
IV. A SPECTRAL THEORY ON RIGGED HILBERT
SPACES
We give a summary of the spectral theory on rigged Hil-
bert spaces developed in Ref. 13. In what follows, put
f
1
¼1=(2 i) for simplicity (that is, f(h) ¼sin hþ (higher har-
monics)) and we consider the operator
043103-3 Center manifold reduction Chaos 21, 043103 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp

T
1
/ ¼ ix/ þ
K
2
ðP
0
; / ÞP
0
:
Let e
T
1
t
/ be the solution of the linearized system
dZ
1
=dt ¼T
1
Z
1
with the initial condition /(x). The linear op-
erator e
T
1
t
is called the semigroup (semiflow) generated by
T
1
. It is known that the semigroup e
T
1
t
is given by the
Laplace inversion formula
e
T
1
t
¼ lim
y!1
1
2pi
ð
xþiy
xiy
e
kt
ðk  T
1
Þ
1
dk; (11)
for t > 0, where x > 0 is chosen so that the contour is to the
right of the spectrum of T
1
(see Fig. 3(a)). The operator
(k T
1
)
1
is called the resolvent in functional analysis (in
engineering, it is called the Laplace transform). The resolv-
ent is a continuous operator on L
2
(R, g(x)dx) when k does
not lie on the spectrum of T
1
. Since T
1
has the continuous
spectrum on the imaginary axis, the inner product
(/,(k T
1
)
1
w) for w, / 2 L
2
R; g xðÞdxðÞdiverges when
k 2 iR. However, it is possible that

/; k  T
1
ðÞ
1
w

exists
on the imaginary axis and has an analytic continuation from
the right half plane to the left half plane if w and / satisfy
suitable conditions. To see this idea, it is sufficient to
consider the multiplication operator iM/ :¼ ix/; that is,
K ¼0 for T
1
. The resolvent is given by
ð/; ðk  iMÞ
1
wÞ¼
ð
R
1
k  ix
wðxÞ
/ðxÞgðxÞdx;
for any w, / 2 L
2
R; g xðÞdxðÞ. In general, the integral in the
right hand side diverges as Re(k) ! 0 because of the factor
1=(k ix). However, if w and / are holomorphic on the real
axis and the upper half plane, the integral converges as
Re(k) !þ0 and has an analytic continuation from the right
half plane to the left half plane, which is given by
ð
R
1
k  ix
wðxÞ
/ðxÞgðxÞdx
þ 2pwðikÞ
/ðikÞgðikÞ; ReðkÞ < 0:
Let X be a vector space consisting of some class of holomor-
phic functions defined near the upper half plane. The above
calculation implies that ð/; ðk  iMÞ
1
wÞ is an entire func-
tion when w, / 2 X. The second term of the above quantity
is not written as an inner product of two functions. This sug-
gests that the analytic continuation of the resolvent is no lon-
ger included in L
2
(R, g(x)dx). Since the complex number
ð/; ðk  iMÞ
1
wÞ exists for each / 2 X, we can regard
ðk  iMÞ
1
w as a functional on X; that is, an element of the
dual space X
0
. To be precise, define the generalized resolvent
A(k)ofiM to be
h/ jAðkÞw i
¼
ð/; ðk  iMÞ
1
wÞðReðkÞ > 0Þ;
ð/; ðk  iMÞ
1
wÞ
þ2pwðikÞ
/ðikÞgðikÞðReðkÞ < 0Þ:
8
>
<
>
:
Since the mapping / 7!h/jA k
ðÞ
wi is linear, jA k
ðÞ
wi is an
element of X
0
. Since jA kðÞwi is determined for each w 2 X,
A kðÞis a linear operator from X into X
0
and is analytic with
respect to k 2 C. By the definition, AðkÞw ¼ðk  iMÞ
1
w
when Re(k) > 0. Now, we have shown that the resolvent
ðk  iMÞ
1
as an operator on L
2
(R, g (x)dx) diverges on
the imaginary axis, however, if we regard it as an operator
from X into X
0
, it has an analytic continuation, which is
denoted by A(k), from the right half plane to the left half
plane.
Similarly, we can prove that the resolvent (k T
1
)
1
of
the operator T
1
does not exist as an operator on L
2
(R,
g(x)dx) when k 2 iR. If we regard it as an operator from X
into X
0
, however, it has a meromorphic continuation R
k
from
the right half plane to the left half plane. Put P/ ¼
K
2
ðP
0
; / Þ.
Then, T
1
¼ iMþP. The resolvent of T
1
is rewritten as
ðk  T
1
Þ
1
¼ðk  iMÞ
1
ðid Pðk  iMÞ
1
Þ
1
;
where id is the identity mapping. Hence, the analytic contin-
uation of (k T
1
)
1
in the generalized sense is given by
R
k
:¼ AðkÞðid P

AðkÞÞ
1
;
which is a linear operator from X into X
0
.
The generalized resolvent R
k
is a meromorp hic function
with respect to k. A pole of the generalized resolvent is
called a generalized eigenvalue. A point k is a generalized
eigenvalue if and only if the operator id P

AðkÞ is not
injective on X. For T
1
, generalized eigenvalues are given as
roots of the equation
ð
R
1
k  ix
gðxÞdx ¼
2
K
; ðReðkÞ > 0Þ; (12)
FIG. 3. The contour for the Laplace inversion formula.
FIG. 2. The motion of the generalized eigenvalue as K decreases. When
K > K
c
, it exists on the right half plane as an eigenvalue of T
1
.AtK ¼K
c
,it
is absorbed into the continuous spectrum and disappears from the complex
plane. When K < K
c
, it goes to the second Riemann sheet of the generalized
resolvent and turns into a resonance pole, which is no longer an eigenvalue
of T
1
in the usual sense.
043103-4 H. Chiba and I. Nishikawa Chaos 21, 043103 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsp

Frequently asked questions

Q1. What is the first term of Eq. (21)?

In order to apply their center manifold reduction, the initial conditions for Zj(t, x) are assumed to be included in the test functions space X so that the decomposition (21) holds (the first term of Eq. (21) is obtained as PcjZ1i and the domain of the projection is X). 

Q2. What is the meromorphic continuation of iM?

If the authors regard it as an operator from X into X0, however, it has a meromorphic continuation Rk from the right half plane to the left half plane. 

Q3. what is the synchronous state of the eigenspace?

nSince D0> 0, Kc> 0, g 00(0)< 0, the fixed point a¼ 0 (de-synchronous state) is unstable and the fixed point Eq.(32) (synchronous state) is asymptotically stable when e¼K Kc> 0. 

Q4. What is the eigenvalue of the Hilbert space?

A rigged Hilbert space consists of three spaces X L2 R; g xð Þdxð Þ X0: a space X of test functions, a Hilbert space L2(R, g(x)dx), and the dual space X0 of X (a space of continuous anti-linear functionals on X, each element of which is referred to as a generalized function). 

Q5. What is the strategy for the bifurcation theory of globallycoupled phase oscillators?

the strategy for the bifurcation theory of globallycoupled phase oscillators is to use the space of generalized functions X0 rather than a space of usual functions. 

Q6. How is the order parameter induced by a resonance pole?

numerical simulation8,14 suggests that theorder parameter decays exponentially to zero when 0<K<Kc. Strogatz and coworkers 8 found that such an exponential decay of the order parameter is induced by a resonance pole, and it is rigorously proved by Chiba.13 

Q7. What is the order parameter for h/jZ1i?

1 if and only if h/jZ1i converges for any / 2 X. Since the order parameter is written as g1 tð Þ ¼ hP0jZ1i, this topology is suitable for the purpose of the present study. 

Q8. what is the q(t) of the form ei/z1?

Since q(t) is real-valued, it turns out that it is of the form q(t)¼ 2pg(0)sin(arg(a))þO(a) andhZ1 j l0i ¼ 2pgð0Þe iargðaÞ þ OðaÞ ¼ 4 Kc e iargðaÞ þ OðaÞ:This and Eq. (A1) prove Lemma 4.1A. 

Q9. What is the eigenvalue of the coupling function f?

This means that the first Fourier component of the coupling function f is the most dominant term (for f(h)¼ sin hþ h sin 2h, which is true if and only if h< 1). 

Q10. What is the generalized eigenvalue of iM?

A point k is a generalized eigenvalue if and only if the operator id P AðkÞ is not injective on X. For T1, generalized eigenvalues are given as roots of the equationð 

Q11. What was the noise of strength of the center manifold?

A center manifold reduction for globally coupled phase oscillators was also developed by Crawford and Davies9 with noise of strength D> 0. 

Q12. What is the eigenvalue of Eq. 13?

it is easy to show that even for 0<K<Kc, k0(K) remains to exist as a root of Eq. (13) because the left hand side of Eq. (13) isan analytic continuation of that of Eq. (12).