169 .
Prog. Theor. Phys. Vol. 83, No.2, February 1990, Progress Letters
Phase Turbulence and Mutual Entrainment in a Coupled Oscillator System
Hidetsugu SAKAGUCHI
Department of Physics, College of General Education, Kyushu University
Fukuoka 810
(Received August 9, 1989)
A coupled oscillator system is considered which has a pacemaker region. Phase waves are sent
out from the pacemaker, but under a certain condition the phase waves become chaotic. The mean
wave number and phase fluctuations of the turbulent phase waves are determined self-consistently by
the condition of the mutual entrainment.
Phase waves are observed in several systems such as the BelousovcZhabotinsky
and a nematic liquid crysta1. 2) In such systems there are pacemakers in the
center of the wave patterns and they send out the phase waves. We study such phase
waves and their instability with a one-dimensional oscillator chain: 3 )-5)
r~actionl)
(1)
where Wj is a complex variable of the j-th oscillator and it is rewritten with its
amplitude and phase as Wj=Rjexp(ir,Dj), and Wj is the oscillator's natural frequency.
We consider a case in which the w/s are locally larger than the outer region. The
third term on the r. h. s. of Eq. (1) indicates diffusion-type coupling .. Indeed Eq. (1)
becomes the Ginzburg-Landau equation:
(2)
by the continuum approximation.
We carried out some numerical simulations of Eq. (1). Figure 1 shows a result
of a numerical simulation when cl=-O.6, C2=1.0 and Wj=wl=1.015 for 400~j~600
and Wj=W2=1.0 for the other j. The figure shows a phase distribution r,Dj. The
phase distribution is approximately expressed as
for
1~j<400,
wt
400~j~600
-Qj+wt
600<j~1000
,
.
(3)
It means a phase wave with the wave number Q is sent out from the central pacemaker region where the natural frequency is larger than the outer region. When Wj
=const=wo, a solution of uniform osc~llation Wj=exp(i(wo-c2)t) exists, but the
uniform solution becomes unstable and 'a spatially nonuniform and chaotic solution
appears, if the two parameters Cl and C2 satisfy the condition CIC2< -1. If CIC2 is a
little smaller than -1, the amplitude R j is nearly 1 and long-wavelength phase
fluctuations dominate. The phase turbulence is well described by the KuramotoSivashinsky equation: 3 ),4),6)
Progress Letters
170
e
e
Vol. 83, No.2
e
e
d,
~
<0
We
Wo
(/)0
(/)0
<.;
IN
a..
<d,
Ir-
a..
20. 00
4 O. 00
SPACE
60. 00
BO. 00
100. 00
xlO'
Fig. 1. Phase distribution for C, = -0.6, C2=1.0,
wI=1.015 and W2=1.0.
20. 00
40.00
SPACE
60.00
BO. 00
100.00
xlO'
Fig. 2. Phase distribution for C, = -1.4, C2= 1.0, w,
=1.01 and W2=1.0.
(4)
Even if a pacemaker region exists, the phase distribution is expected to become
chaotic, when CIC2 is smaller than -1. We carried out a numerical simulation for Cl
= -1.4 and C2=1.0. The initial condition is nearly constant, i.e., R j =l and ¢j~O.
Figure 2 shows a snap-shot phase distribution when Wj= Clh = 1.01 for 400;;:;;j;;:;; 600 and
Wj=W2=1.0 for the other j. The phase distribution has fluctuations but is similar to
Fig. 1 on the average. Namely, the pacemaker region sends out the phase waves with
nearly constant wave number toward both sides. Chaotic phase fluctuations are
added to the regular phase waves. Closely looking, the chaotic fluctuations are
different between the flat region and gradient regions of the phase distribution. The
fluctuations seem to be larger but more regular in the phase gradient regions. To see
the features more clearly we show the time development of the fluctuations in Fig. 3.
In Fig. 3 local minimum points of the phase distribution are plotted for each time. In
the flat pacemaker region 400;;:;;j;;:;;600 the local minimum points are created or
collapse to disappear. This is characteristic to the Kuramoto-Sivashinsky turbulence described by Eq. (4). In the outer regions, the K-S turbulence is seen in the early
time when the mean phase gradient is still zero. When the phase waves sent out from
the pacemaker region arrive at each point, the mean phase gradient become nonzero.
Where the mean phase gradient is nonzero, the local minimum points are rather stable
and drift toward the boundary with a nearly constant velocity.
To understand the relation between the phase gradient and chaotic fluctuations,
we derive an equation for the phase distribution ¢(x, t) from the Ginzburg-Landau
equation (2) with w(x)=wo=const. A solution to Eq. (2) is
(5)
Wo(x, t)=Aexp{i(wt- Qx)} ,
where A=;j1-Q2 and W=WO-C2+(C2-Cl)Q2. This solution indieates the phase
wave solution with the wave number Q. Equation (2) is rewritten as
W=(l- Q2+ iWo-iclQ2) W-(1 + iC2) IWI 2W+(1 + iCl)(dx 2W-2iQdx W)
,
(6)
where W= We iQX • The uniformly rotating solution to Eq. (6) is Wo=Ae'xp{i¢o(x, t)},
where ¢o(x, t)=wt. The third term including spatial derivatives can be considered
as a perturbation to the uniformly rotating solution if the spatial variation of W is
slow and the wave number Q is small. Then we can get an equation of ¢(x, t) by
}Jro~ress
February 1990
]Setters
171
expanding it systematically in powers of ax and Q following Refs. 3) and 4). The
solution can be assumed as
W= Wo(¢(x, t»+ u(¢(x, t»,
(7)
¢=w+£2(¢(x, t»,
where £2(¢) is a functional of ¢(x, t) and u(¢) must satisfy the solvability condition:
(8)
Z(¢)u(¢)=O,
1
( cos(¢),
where Z(¢)=-( - C2, 1)
. (,f.,)
w
-SIn 'P ,
sin(¢»)
cos(¢) ,
Le., u( ¢) must be orthogonal to the left null eigenvector of the linear operator. After
a tedious calculation w.e can obtain
(9)
where f20=wo- C2+(C2- C1)Q2, £21 = -2Q(C2- C1), £22 (1)=1 + C1C2, £22(Z)=cz- C1, £23(1)
=2QC1(1 + d), £24 (1)= - d(1 + czZ)/2 and £24 (2)= -2C1(1 + d). The coefficients satisfy
£2z(Z)=1/2 aZf20/aQz=-1/2 a£2daQ and £24(2)=-a£23(1)/aQ. Namely, Eq. (9) can be
obtained directly from a modified K-S equation:
(10)
by putting cf;= ¢- Qx. When C1CZ< -1, £2Z(1) becomes negative and the phase
instability occurs. The second term of Eq. (9) is a drift term, i.e., a disturbance drifts
with a velocity of -£21=2Q(cZ-C1). This corresponds to the drift motion of the local
minimum points seen in Fig. 3. When Q is not zero, the fifth term, i.e., the dispersion
term appears. By neglecting the last term in Eq. (9) and introducing u=¢x, the
following equation is obtained
(11)
This equation was studied by Toh and Kawahara and periodic trains of soliton-like
pulses are found numerically.7) The dispersion term stabilizes the pulse trains. The
pulse trains correspond to the rather regular fluctuations of the phase distribution in
Fig. 3.
In the above argument the mean wave number Q was assumed and an equation
for slow fluctuations of the phase wave was derived. How is the wave number· Q
determined? We consider the case that all the oscillators are mutually entrained.
Then each oscillator has the same frequency w. For C1CZ> -1 there are no phase
fluctuations as seen in Fig. 1. Since the pacemaker region is rather large in our
system, ax¢, axz¢ ... are all small enough and the frequency w is approximately given
by Eq. (9) as
172
Progress Letters
VoL 83, No.2
o
o
o
'"
o
o
o
N
D~
a
X
7
<D
Do
X 0
o
o
6
~
o
6
g
20. 00
40. 00
SPACE
60. 00
80. 00
/'
ct. 00
100.00
0.04
"10'
Fig. 3. Time development of local minimum
points in the phase distribution for Cl = -1.4, C2
=1.0, (01=1.01 and (02=1.0.
for
0.08
,<) 0"
Fig. 4. Wave number of the phase wave for Cl
= -0.6 and C2=1.0. The points show numer·
ically obtained values and the dashed line
shows the theoretical values.
400~j~600
for the other j .
(12)
From this condition the wave number is given by Q2~(aJl-aJ2)/(C2-CI). Figure 4
compares this result with the numerically obtained wave number for CI = -0.6 and
C2=1.0. For CIC2< -1 the phase fluctuations cannot be ignored. If we integrate
Eq. (9) with respect to x over a certain long interval [Xl, X2] neglecting the last term,
and divide it by (X2-XI),
<¢>=~+ Q2(2)«oxcp)2> + [QI cp+ Q2(1) oxCP + Q3(I)Ox2cp+Q4(I)Ox3cp]i~/(X2- Xl),
(13)
where < > indicates an average over the interval and [jU~= I(xz)- l(xI). If the
interval is long enough, the last term of Eq. (13) can be neglected. Then the mean
frequency aJ is approximately given by
(14)
where - indicates an average over the interval and a certain long time. In the case
of Fig. 2, Eq. (14) is rewritten as
aJ= aJI- cz+(cz- CI)(OXCP)IZ
aJz- C2+(CZ- CI){ QZ+ (oxcp)ZZ}
for
400~j~600,
for the other j ,
(15)
where (oxcp)I.zare respectively the mean square phase fluctuations for the pacemaker
region and outer region. The features of the phase fluctuations are changed by the
mean phase gradient Q through the dispersion term in Eq. (9). On the other hand, the
mean phase gradient Q is determined by Eq. (15), i.e., the condition of mutual
entrainment as
173
~ro~ress ~etters
February 1990
w
lD
/
/
,
o
/
N
o
;(
/
/
<0
o
/
o
/
/
o
o
+---~--r--'---.--,
0 0 . 00
0.04
0.08
~dO-lW,-lJ2
(a)
o
o
+=~.-
0 0 . 00
__.--,__-.__-.
O. 04
O. 08
xl 0- 1 Wl-UI2
(b)
Fig. 5. (a) Mean frequency w for Cl = -1.4 and C2= 1.0. The points indicate numerically obtained
values, the solid curve shows the curve (14) for 400;;;;;j;;;;;600 and the dashed line shows the curve for
700;;;;;j;;;;;900.
(b) Square mean wave number Q2 for Cl = -1.4 and C2=1.0. The points indicate numerically
obtained values, the dashed line shows a simple law Q2=(Wl- (2)/(C2- Cl) and the solid line shows
the curve (16) for 700;;;;;j;;;;; 900.
(16)
Nameiy, if the phase fluctuations are different in the two regions, the mean phase
gradient deviates from the simple law QZ=(ah - wz)/(cz- Cl) which is realized for CICZ
> - 1. .It can be said that the mean phase gradient and the mean square of phase
fluctuations are self-consistently determined. Figure 5(a) shows the mean frequency
W obtained in two ways. The six points show the mean frequency obtained directly
by numerical simulations and the solid line and the dashed line show respectively the
mean frequency obtained through Eq. (15) using numerically obtained (oxc/J)2 for the
pacemaker region and outer region. The rather good agreement shows that the
above approximate argument is valid. Figure 5(b) shows the square mean phase
gradient QZ. The five points show numerically obtained values, and the dashed line
shows the simple law QZ=(WI-WZ)/(CZ-Cl) which would be satisfied if the phase
fluctuations are statistically uniform, i.e., (OXc/J)lZ=(OXc/J)zz. The solid line shows the
square mean wave number numerically obtained through Eq. (16). It is seen that the
mean phase gradient or the mean wave number of the phase wave is effectively
decreased since the phase fluctuations are larger in the outer region.
In summary we studied a one-dimensional oscillator chain with a pacemaker
region. Phase waves are sent out from the pacemaker region but the regular phase
waves may be unstable and weak chaotic fluctuations appear under a certain condition. A phase equation with a dispersion term was derived to understand the phase
turbulence of the phase waves and mutual entrainment in the phase turbulence. It
was shown that the mean wave number and the fluctuations of the phase wave are
mutually coupled and are self-consistently determined. There are many other turbulent systems in which the mean flow and the fluctuations about the mean flow are
mutually coupled. In our systerri-the coupling is relatively easily seen through the
phase equation (9) and the condition of mutual entrainment (15).
174
Progress Letters
1)
2)
3)
4)
5)
6)
7)
Vo!. 83, No.2
A. N. Zaikin and A. M. Zhabotinsky, Nature 225 (1970), 535.
S. Nasuno, M. Sano and Y. Sawada, J. Phys. Soc. Jpn. 58 (1989), 875.
Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, Heidelberg, New
York, Tokyo, 1984).
Y. Kuramoto, Pi-og. Theor. Phys. Supp!. No. 64 (1978), 346.
H. Sakaguchi, Prog. Theor. Phys. 80 (1988), 743.
G. 1. Sivashinsky, Acta Astronautica 6 (1979), 569.
S. Toh and T. Kawahara, J. Phys. Soc. Jpn. 54 (1985), 1257.