collective dynamics of coupled modulated oscillators with random pinning

Physica D 56 (1992) 23-35 North-Holland Plllm

Collective dynamics of coupled modulated oscillators with random pinning

Bo C h r i s t i a n s e n , P r e b e n A l s t r ~ m a n d M o g e n s T. L e v i n s e n

Physics Laboratory, University of Copenhagen, H.C. Orsted Institute, Universitetsparken 5, DK-2100 Copenhagen ~, Denmark

Received 30 August 1991 Revised manuscript received 25 November 1991 Accepted 26 November 1991 Communicated by A.V. Holden

We present a study of a large pool of coupled oscillators in the presence of a modulated external field. Random distributed pinning phases introduce a disordering element. We find that phase locking of the oscillator community to the harmonics of the frequency of the applied field always is associated with a complete loss of coherence between the oscillators. The phase-lock regions form islands in parameter space, the size of which decreases for increasing coupling strength to vanish completely at a critical value. By stability considerations the shape of the phase-locked islands is reduced to a two points boundary problem of a first order non-autonomous differential equation and an approximation is found for high frequencies. The structure of the coherent states is discussed and a first order approximation is found in the limit of strong coherence.

1. Introduction

Recent ly collective p h e n o m e n a in large pools of coupled oscillators have a t t racted much atten-

tion [1-5]. One of the interesting features of such

populat ions is the possibility of spontaneous syn- chronization, which in nature has been observed

in a wide range o f physical, chemical, biological and medical systems. Examples are charge den-

sity waves [6], oscillating chemical reactions [7], fireflies flashing in unison [8, 9], the human circa-

dian rhythm [10] and an audience applauding the pr ima ballerina. The analysis of these large sys-

tems with many degrees of f reedom has involved elements f rom both statistical mechanics and non-l inear dynamics.

We report a study of coupled non-l inear oscillators with r andom pinning in the presence of an al ternating external field. Denot ing the phase o f

the p t h oscillator by xp we have

~p = s in (ap - X p ) + E + A sin( tot)

K N + - ~ ~ , s i n ( x j - - X p ) ( p = l , . . . , N ) ,

y=l

(1)

where N is the number of oscillators, ap is the

pinning phase of the p t h oscillator, E is the dc part of the external field, A and to are respectively the ampli tude and f requency of the ac par t of the external field, and K is the coupling

strength. For convenience the coupling is chosen

to be of infinite range, and the coupling term is normalized with the factor 1/N. The choice of

coupling may seem unrealistic but is justified by its relative simplicity. It is of ten found that an

infinite range model captures the essentials of

0167-2789/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved

24 B. Christiansen et aL / Coupled oscillators with random pinning

more refined models. However, a few physical systems well described by infinite range couplings do exist. As an example we mention an array of resistively shunted Josephson junctions [11, 12].

We have without loss of generality written eq. (1) in normalized form with the coefficient of s i n ( u p - x p ) set to unity. Moreover, the pinning phases ap are distributed randomly in the interval [0,2zr] with a constant probability distribution. In the limit of large N, fluctuations can be ignored and the homogeneous distribution can be modelled by Up = (2ar/N)p.

The terms on the right hand side of eq. (1) have each a specific influence on the time development of xp. The pinning term tends to align every individual oscillator Xp to its constant pinning phase up, the coupling term tends to align all the phases to a mutual phase and the alternating field induces a common rotation and oscilla- tion. The competition between these tendencies gives rise to a rich and complicated behavior of the full system.

The rest of the paper is organized as follows. In section 2 we give a short review of the limiting cases of pure dc field (A = 0) and no coupling (K - - 0). In section 3 we present numerical calculations of the structure and size of the phase- locked regions. These results are partly explained by linear perturbation theory. In section 4 we consider the limit of high frequencies, where an approximation can be found for the size of the phase-locked regions. Section 5 is devoted to the study of the complication structure of the coherent state in the quasiperiodic regions. In section 6 this state is studied by perturbation methods for strong coherence. The paper is closed by the summary and discussion in section 7.

2. The dc limit and the uncoupled oscillator

The system has previously been studied in detail under two different simplifying assumptions. When the oscillators are uncoupled the system reduces to the celebrated resistively shunted

Josephson junction equation. This model corresponds to a damped driven pendulum without inertia and a huge amount of literature exists on this and related models [13]. Recently Strogatz et al. considered the system under the assumption of the external field being purely dc. In this section we give a short account of the relevant features of the two cases.

Strogatz et al. [14] have studied the system in detail by numerical and analytical means in a constant external field (A = 0). Using mean-field methods they proved in the low field region E < 1 that for weak coupling the oscillators evolve to a non-coherent and pinned stationary state,

xp = ap + arcsin E. (2)

While this state always is a formal solution to eq. (1), it loses stability at a depinning threshold E-r(K). The threshold ET(K) was found analyti- cally:

E T ( K ) = ~ - - 1 2 ~K , K < 2,

= 0 , K > 2 . (3)

However, when E exceeds a switching threshold Ee(K) less than ET(K) a stable coherent state occurs. This state is moving in the sense that the oscillators have a non-zero average velocity. In

2.0

K ~ moving

K c / 1.0 "'" ~ At''/ ET

pinned E ~"~'"-... \

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

E

Fig. 1. The depinning threshold E T and the switching threshold Ep as function of the coupling strength K for the non-modulated system. (From ref. [14].)

B. Christiansen et al. / Coupled oscillators with random pinning 25

o8 l v

,

0.0 017 i 110 0.4 Ep E T

r 1.0[

0.5

0.0 0.4

it

i 1'.0. Ep 0'.7 ET

E

Fig. 2. The collective frequency 0 (here denoted v) and the order parameter r as function of E for K = 1. (From ref. [14].)

the hysteretic region E p ( K ) < E < ET(K) where two stable states exist, the actual state depends on the initial conditions. Fig. 1 shows the two

thresholds ET( K) and Ep( K). In the high field region E > 1 no pinned state

exists and the system always evolve to a coherent moving state. Defining the mean-field parameters r and 0 by rexp(iO)=~=lexp(ixy)/N the no- tion coherence can be given a quantitative mean- ing. The phase 0 is an average of the phases xp, and r is an order parameter . For r = 0 the oscillators are totally incoherent, while non-vanishing r implies the presence of coherence. The order parameter grows with increasing coherence and r = 1 describes the absolute coherent state Xp = xy (p , j = 1,..., N). Fig. 2 shows the order parameter r and the collective frequency/~ as function of E. Note the finite jump in both r and 0 at the switching threshold Ep(K) .

Without the coupling term ( K = 0), eq. (1) reduces to the resistively shunted Josephson junction equation (RSJ). This has as mentioned above been thoroughly studied, and its behavior is well known [13]. The competit ion between the intrinsic frequency and the frequency of the applied field induces harmonic steps in the E-R characteristics, where R is the rotation number ~ / 2 ~

1.20

1.oo - ~

080 ET

060

0./.(3

0.2C

I

I I i I i

o

10.00

: '.,.

r t , i l , 2000 3000 L0.O0 5000 A

1.20

1.00

0.80

0.60

O.t,O

0.20

I i l I

I I i I i I 10,00 20.00 30,00 A

i I '

b

~0.00 58.00

1.201 ,

I00~

0801 EpX~

0"20f i

1000 i I I I 20.00 :30.00 A

I

c

~o,oo 5O.00

Fig. 3. The zeroth (no rotation) harmonic step shown in (E, A) space for (a) the Josephson junction equation i.e. K=O, (b) K = 0 . 4 and (c) K=0.8. In (b) and (c) where hysteresis is present both E T and E e are drawn.

2 6 B. Christiansen et al. / Coupled oscillators with random pinning

7.00

6 , 5 0 1

6.00

I i

I I 10.00

/ /

. /

I t I I I i 20.00 3 0 , 0 0 40 .00 50 .00

A

(here and elsewhere an overline denotes the time average). For not too small frequencies the size of the steps AE as a function of A resembles

Bessel functions and vanish at some isolated val-

ues of A. For small frequencies the Bessel func-

tion behavior is distorted, although it still gives the points where A E vanish. Figs. 3a and 4a

show for to = 2"rr the zeroth step (i.e. the non-

rotating region with R = 0) and the first harmonic step respectively.

3. The phase-locked islands

7.00

6.50

E

6.00

I i i i r I r I

I I I i I L I 10.00 20 .00 30 .00 t ,0.00

A

b

1 50.00

7.00

6.50

600

i I I I l

~'-'~E T

X...._.~ ~

l

J I t 1 L I i I i 10.00 20.00 30 .00 4 0 . 0 0 5 0 . 0 0

A

Fig. 4. The first harmonic step shown in (E, A) space for (a) the Josephson junction equation i.e. K = 0, (b) K = 0.4 and (c) K = 0.8. In (b) and (c) where hysteresis is present both ET and Ep are drawn.

We now return to the full system. Figs. 5 and 6

show, for A = 4 and A = 10 respectively, numerically calculated E - R characteristics for a series

of increasing values of the coupling strength K.

The rotation number is now defined by the collective frequency R = 0 / 2 rr although we find that all oscillators have the same mean velocity Jp, as

would be expected by symmetry reasons. In the figures the zeroth and the first harmonic steps are seen. We observe that all step sizes decrease with

increasing K. The decrease is very slow for small K getting increasingly faster until the steps seem to vanish completely at some critical value of K. To illustrate this point further, figs. 3 and 4 show

for K = 0, K = 0.4, and K = 0.8 the threshold E T in the (E, A) space, where the phase-locked state

ceases to exist, for the zeroth and the first harmonic step, respectively. The effect of the coupling-is now apparent. While for K = 0 (RSJ) the

steps vanish only for isolated values of A a whole interval exists for K > 0 where the steps are miss- ing. The phase-locked regions lie as islands in the (E, A) space, centered around the maxima of the RSJ steps. For increasing coupling strength

the islands shrink to disappear for a finite value of K.

For A = 0 Strogatz et al. argue that r and 0 in the limit N ~ oo are independent of time. This is in general not the case for A ~ 0, where both r and 0 are found to oscillate. Fig. 7 shows, for the


7.50 ' I ' t ' I ' / I / ' / I ' / /,,- j'// /" ,,"'//,G'

5.00 K=0/ 0.2/ 0.~/ 0.6/" 0.s/ to / /i // // // // //// //"

/////'/ ( -- / .. ../ ....... /

l I I I I

0.00 0.50 1.00 1.50 2.00 2.50 R

Fig. 5. E-R curves for a series of increasing coupling strengths: K = 0,0.2,..., 1.0. The zeroth and the first harmonic step is seen. The amplitude and frequency of the alternating field is A = 4 and to = 2"n'.

7.50

5.00

2.50 ! , , , , , ,/,/,/,/,/,/

0.00 0.50 1.00 1.50 2.00 R

Fig. 6. As fig. 5 but-now with A = 10.

2.50

same parameters as in fig. 5, the time average of the order parameter r as function of E. We observe that ~ = 0 on the steps and ~ > 0 elsewhere, reaching a maximum close to unity approximately halfway between two steps. This means that when the oscillators a r e phase locked to the external drive they are mutual incoherent,

while they are coherent when they are not phase

locked. By plotting x u as function of the pinning phase

ap we find that in the incoherent phase-locked state the oscillators are always linearly related,

Xp - a p = xy - a / . (4)

28 B. Christiansen et aL / Coupled oscillators with random pinning

1.20

1.00

0.80

0.60

0.40

0.20

_ ' I ; I v I ' ' t

_ I

I I I ~ L i 1.50 3.00 4.50 6.00 7.50

E

Fig. 7. The order pa ramete r r as function of E for A = 4, K = 0.4 and to = 2at. On the phase locked steps the oscillators are complete incoherent and r = 0.

The same relation holds for the pinned state given by eq. (2) in the absence of an alternating field. In the present case, however, the Xp'S are time dependent. The incoherent state is effectively decoupled as the sum T.~v=t sin(Xp-Xy) is identically zero. The time development of the system is then determined by the first order non- autonomous RSJ equation

2 = - s i n ( x ) + E + A sin(tot) ( 5 )

and xp = x + ap. Without any ac field the stabilizing effect of the

pinning wins over the destabilizing effect of the coupling for E < Er, making the pinned incoherent state x i = ot i -t- arcsin E stable. For E > E r other stabilizing effects are needed to stabilize the incoherent state. In the presence of an ac field one such effect comes from the disposition of an uncoupled oscillator phase locked on the harmonic steps to choose the phase so that x i = a i for t = 2 ~ r n / t o , where n is an integer. This addi- tional effect favors the incoherent state by increasing its stability when phase locked.

We now use linear perturbation theory to study the stability of the incoherent state in detail.

Substituting xp = x + ap + ~'p, where ~'p is an in- finitesimal perturbation, in eq. (1), we get to first order

~, = - c o s ( ~ . - x , ) ~-,

+ N E (~j - ~,)cos (1 - p ) , (6) j = l

o r

~,= -cos(x)G + ~ E ~jcos ( j - p ) j = l

(7)

(here we have used E~=t c o s ( 2 ~ r / N ) ( j - p ) - - 0 ) .

In vector notation this can be written

~=A~ ' , (8)

where

K A = - c o s x E + ~ A , (9)

B. Christiansen et al. /Coupled oscillators with random pinning 29

¢: is the unit N × N matrix, and ity cri terion is K < K+, where

A i j = c o s ( ~ - ( j - i ) ) ( i , j = l . . . . . N ) . (10)

We note that this is a l inear system of first o rder

differential equat ions with a time dependen t co-

efficient cos x. The per turbat ion will grow if the

time average of at least one eigenvalue of A is

positive, i.e., the stability cri terion is

m

m a x A i < 0, (11) i = l . . . N

where A i are the eigenvalues of A. These eigen-

values can be written A i = - c o s x + ( K / N ) A i,

where /~i a r e the eigenvalues of A. In the highly symmetrical case under consider-

at ion we find, that the only eigenvalues of A are

A = 0 and A = N / 2 #~. These eigenvalues are N - 2 and 2 times degenera te , respectively. W h e n N

is odd the eigenvectors can be chosen as w o, w k,

and v k ( k = 1 . . . . . ( N - 1)/2) , where

w~ = (cos ( a l k ) , cos( a 2 k ) . . . . . cos(o tNk) ) , (12)

and

v k = ( s i n ( a l k ) , s i n ( a 2 k ) . . . . , s i n ( a N k ) ) . (13)

W h e n N is even we can choose the eigenvectors

w 0, wN/2, w k, and v k (k = 1 . . . . . ( N - 2) /2) . The eigenvalue N / 2 corresponds to k = 1. The stabil-

#IThe matrix A is a circulant matrix, i.e. each row is a circular shift of the preceding row and the first row is a circular shift of the last row. An N × N circulant matrix A has the eigenvalues

N - 1 2~r r .exp(i km) m ~ 0

and the corresponding eigenvectors are

K-r = 2 cos x. (14)

At K = Kar the incoherent state loses stability to the long wavelength per turbat ions (in a space),

while per turbat ions of a shorter wavelength stay

stable until eq. (5) itself loses stability. We em-

phasize that the number of oscillators do not

enter explicitly in the eq. (14) and only enters the calculation through the assumption of a l inear

distribution of pinning phases.

The edges of the n th harmonic phase-locked

step for fixed A and K can now be de termined by the following procedure . For an initially guess

of E we find the stable #2 solution of the two

points boundary problem given by eq. (5) and the condit ions x ( t = 0) = x ° and x ( t = 2xr/to) = x ° +

2am. For this solution cos x is calculated and the

guess on E is adjusted (e.g. by Newton ' s method)

to approach K = 2cos x. This procedure is con- t inued until K = 2 cos x to the required accuracy.

The curves in fig. 3 and fig. 4 are found in this

way. In the case without modula t ion (A = 0), we

have x = arcsin E for the pinned steady state,

and eq. (14) simplifies to eq. (3). F rom eq. (14) we can now unders tand the

mechanism responsible for the destruct ion of the

phase-locked islands. Fig. 8 shows the typical circular shape of cos x as function of E for fixed A. The two values of E, E 1 and E~, where

cos x = 0, is the lower and upper edge of the RSJ

step. For finite K only the interval ~ u [ EK, E K ] given by the intersection of K + ( E ) and the line K supports the stable incoherent state. For K

greater than the maximum value K c of 2 cos x, no phase locking is possible. In the generic case the

maximum is of second order and the widths of the phase-locked intervals d isappear square-root like ( ~ ~ - K ) at the critical coupling strength

g c • In the two intervals [E0 I, Ek ] and [Eta, E~] a

delayed decay to the coheren t state can be ob-

.2at . ( w k ) j = e x p ( l - - ~ - ( j - 1)k). #2Both a stable x s and an unstable x u solution exist. They

are related by Xs(t) = w -xu(Ir/ to - t).

30 B. Christiansen et al. / Coupled oscillators with random pinning

1 . 5 - -

1.2 KC

KT 0.8

0.4

T T ' [ T - - 1

'L____L_a.____L__ ° 6.s E; 7.0

E

Fig. 8. The depinning threshold K T as function of E. The threshold K T is calculated as the average value of 2 cos x for a stable solution x to eq. (5) satisfying the two point boundary condition x(t = 0) = x ° and x(t = 2 w / t o ) = x ° + 2-x. At the two edges EJo and E~ of the first harmonic RSJ step cos x is zero. Phase locking is impossible for K > K c. For K < K c the phase-locked region is [E~ , E~]. The ampli tude of the alternating field is A = 10 and the frequency is w = 2-rr.

served. Starting with random initial conditions the system will quickly develop to a metastable state resembling the incoherent state given by eq. (4) and eq. (5). This state exists for a long time before the system inevitably decays to the stable coherent state. This behavior is due to the fact that only a small fraction 2 / N of the principal directions are unstable in this part of parameter space.

is a first order solution to eq. (5). The external field E only enters this solution through the integration constant c. But in order to determine c, second order effects have to be considered. Substituting x I on the right hand side of eq. (5) gives the solution to the second order:

x 2 = E t - Acos(tot)to + c - fotSin(xt)dt . (16)

From the initial condition X2(2"ff/to)=x2(O)+ 2wn we have (T = 2rr/ to)

T( E -ton)

= foTSin[ntot- ( A / t o ) c o s ( w t ) + c] dt

= a] cos c + a 2 sin c, (17)

where

a 1 = ~J0Vsin [ ntot - (A / t o ) cos( tot)] dt

= ( ~,,TJ.( A/~o), n o d d ,

n even, (18)

and

= --joTCOS[ntot -- (A / t o ) cos(tot)] dt a2

= TJ. (A/ to) , n even, (19)

which is proved using the integral representation for Bessel functions

4. The high frequency limit 1

] c o s ( n t - x sin t) dt. (20) L ( x) = -4 Jo

We now consider the limit to >> 1, where an approximation can be found for the solution x of the RSJ equation (eq. (5)), and hence for cos x. The succeeding calculations correspond to the voltage biased model studied in ref. [13]. On the nth harmonic step,

A x I = ntot - - - cos ( to t ) + c (15)

to

We now have

forCOS(xl)dt=a2cosc a l s i n c , (21) COS X =

and

( ~ 1 2 + T 2 ( E _ o n ) 2 = a 2 + a22 = T2j2( A / o ) .

(22t

B. Christiansen et aL / Coupled oscillators with random pinning 31

7"00.L ......... '::::::::: .' .......... ' ..... I ' I ' I ,

6.75/-" ................... i ................................ 6 .50[ ......... Ep ....... E T

6.25

6.00

5.75

I i

i I i I t [ t' I t I I I 0.25 0.50 0.75 K1 1.00 Kc 1.25 30

K

Fig. 9. The effect on the depinning threshold E T and the switching threshold E r, of the first harmonic step when K is increased. The hysteretic part grows until K = K t where all of the step is hysteretic. For K = K c the step disappears. The amplitude of the alternating field is A = 11.5 and the frequency is to = 2av.

The phase-locked regions are de te rmined by

eq. (22) and eq. (14). We notice that these regions are symmetric in E a round E = ton while being asymmetric in A. The step size A E is given by

2 ¢ J , , 2 ( A / t o ) - ( K / T ) 2 / 4 = ~]K~ - K 2 ,

A E = K < K¢

O, K > Kc ,

(23)

where K¢ = 2 T J , ( A / t o ) is the critical coupling

strength. For small K the step size decreases slowly as K 2 f rom its value 2 J , ( A / t o ) at K = 0, while it as ment ioned above disappears square-

root-like when K is close to K c. As in the RSJ,

the maxima of the steps coincide with the maxima of J ~ ( A / t o ) . Hence the m t h island on the n th harmonic step disappears for K = 2 T M m, where M," is the m t h maximum of J , (x) . The longest surviving island is the par t of the zeroth step closest to the origin (E, A ) = (0,0), which disap-

pears exactly for K = 2. F rom eq. (14) it follows that no phase locking is possible for K > 2.

As seen f rom fig. 5 and fig. 6 both a coherent and a incoherent state exist in regions near the

edges of a step. This hysteresis is approximately

of the same size at the bot tom and the top of the steps. In fig. 3 and fig. 4 numerical calculations o f the threshold Ep, inside which no coherent state

exists, are shown. The thresholds E T and Ep seem to be nearly concentr ic and of the same

shape. In fig. 9, E T and Ep on the first harmonic

step are shown as funct ion of the coupling strength K for fixed A = 11.5. For small K the

size of the hysteretic region grows slowly from

zero, as there is no hysteresis in the RSJ. For increasing K the size grows faster until it covers all of the phase-locked step at K = K v The hys-

teresis disappears with the step at K = K c. We have not been able to f i n d a n analytical expres- sion for Ep.

5 . T h e c o h e r e n t s t a t e

The s t ruc tu re of the phase-locked state is sur- prisingly simple, as we have seen. The individual


oscillators all rotate according to eq. (5) and only differ by a constant phase. The subject of this section is the much more complicated structure of the coherent state encountered outside the phase-locked islands.

The time development of the coherent state close to the threshold Ep is shown in fig. 10 for the parameters E = 5.95, A = 5.0, and K = 0.2 by drawing xp as function of ap for several times t. We observe that xp is approximately a linear function of at, interrupted by a steep part. While both the position of the maximum of Xp moves along the a-axis, the average value ( x p ) = E~=lx~/N increases, and the general form of xp changes in time, we observe that after a full period T the original shape of xp is recovered translated along the a-axis leaving the relative position of xp and the line xp = ap unchanged. This observation suggests that xp be written as

101% ~ ~

8]% 3T

6 ~ ~ 2 T

X p

T /.11;-

0 2]%

2]% /.]% 6]% 8]% 101%

Fig. 10. The phase Xp as function of ap of a coherent state for a succession of times t. The point in parameter space is chosen close to Ep: E = 5.95, A = 5.0, K = 0.2 and to = 2"~.

xp = (Xp) + F ( t , ap - ~b), (24)

with the collective velocity

(Ycp) = (.~p) + f ( t ) = 21rR + f ( t ) , (25)

where both f and F are periodic: f ( t ) = f ( t + T),

F(t , a) --- F(t + T, a), and F(t , a) = F(t , a + 2rr). Furthermore, to assure that the position of xp is unchanged relative to x j, = ap after a period T we must have

2 ~ R - ~ = 2 a r m / T , (26)

1.00 I I I i I ; I '

0.75 r

0.50

e

0.25

0 . 0 0 -

= I i I a I i I I 2.0 I..0 6.0 8.0 10.0

t

Fig. 11. The order parameter r, the collective frequency 0 and the frequency ~ as functionof time for the same parameters as in fig. 10. Both 0 and ~ has for convenience been normalized with 2~rA. They are related by 0 - 4~ = 21r.


where m is an integer. Eqs. (24) and (26) are a generalization of the ansatz made in ref. [14] in case of a constant external field (A = 0). Fig. 11 shows the order parameter r, the collective frequency 0, and q~ as function of time for the same parameters as fig. 10. The frequency ~ has been determined by traci_ng the maximum of x, . The average frequency qb is negative and eq. (26) is

fulfilled with m = 0 (as 0 = (~ )=2" r rR) . The fluctuation of r is seen to be relatively large. Figs. 12 and 13 are the analogues to figs. 10 and 11 for parameters in a strongly coherent region between the steps. Now Xp seems t o have the form of a sine wave. The frequency q~ is positive and eq. (26) is fulfilled with m = 1. Note that in this situation the fluctuations of r are small. As re- gard to the value of the integer m in eq. (26) no at tempt has been made to find its variation with the parameters. We only note that in the case of strong coherence it takes the value zero (see the preceding section).

We emphasize that the coherent state only exists in the quasiperiodic regions. Although the symmetry is broken we still require the statistical properties of a single oscillator relative to its pinning phase to be equal for all oscillators. This

3 1 ' r , ' - - I ' I '

~ I i I i

2rt 3Tt 4Tt 0 [

Fig. 12. The phase Xp as function of ap of a coherent state for a succession of times t. The point in parameter space is chosen between the steps: E = 2.0, A =5.0, K = 0.2 and ~ = 2 1 " r .

means that the time average

h ( x p - a , ) = h ( ( x u) - a n + F( t , a p - d p ( t ) ) )

should be independent of p for every function h of period 2rr. The validity of this in the quasiperi-

1.00

0.75

0.50

0.25

i I i I ' I i I i

r

0.00

2.0 4.0 6.0 8.0 10.0 t

Fig. 13. The order parameter r, the collective velocity ~ and the frequency 4; as function of time for the same parameters as in fig. 12. Both ~ and 4~ has again been normalized with 2arA. They are related by ~ - 4; = 0.


odic regions where the rotation number R is not an integer can be seen by a translation of time as the periods are incommensurable. When R is an integer the periods are commensurable and this procedure is not feasible, ruling out the possibility for the coherent state on the phase-locked steps.

(28) with these simplifications we get

Xp = a ( t ) sin[ ap - th ( / ) ] ,

where

a( t ) exp[i~b(/)]

(29)

6. The limit of strong coherence = e x p ( - K t ) f o t e X p [ i x ( z ) +Kz] dr . (30)

For strong coupling, K >> 1, the oscillators are strongly coherent, and the order parameter r is close to unity. As previously noted, the oscillators are strongly coherent even for weak coupling in regions in (E, A) space between the steps. We now treat the limit of strong coherence by perturbation methods. We write x o = x + Xp where ~ = E + A sin(tot) and

g N

£p = - s i n ( a z - x - X p ) + ~ ~ s i n ( x j - X p ) . j=l

(27)

The right hand side of eq. (29) has the form of the general function F(t, ap - ¢) and describes a sine wave in oL space with time dependent amplitude a(t) and velocity ¢( t ) . By the definition of the order parameter r and the average phase 0 we find the relations O=x and r = 1 - ¼ a ( t ) z valid to first order. For strong coupling K >> E, to, the integrand in eq. (30) can be approximated by evaluating x at the time t. We then have a( t )= 1 / K and ~b(t)=x(t) , i.e. the amplitude is independent of time and decreases inversely propor- tional to K while the phase ~b synchronizes to the collective phase 0 .

Here x is independent of ap and represents a collective motion of the system. Expanding eq. (27) to first order in Xp we have

2p= - s i n ( a o - X ) - [ coS(ap-X) + K]Xp

K u Ext.

j=l (28)

Since the time average of c o s ( a p - x ) is zero, it only affects the transient behavior, hence this term can be ignored. Summing eq. (28) over p we find that EpN= ~ Xp is independent of time, and it is therefore chosen to zero without loss of generality. Accordingly, in the first order approximation, x is the total collective motion and Xp is pertur- bations to this motion. In higher order approxi- mations this result is not valid and Xp gives a contribution to the overall motion. Solving eq.

7. Summary and discussion

In summary, we have studied a large pool of coupled oscillators with random pinning in an alternating external field. We have found that the induced phase-locked steps form islands in the (E, A) space. On the phase-locked steps the individual oscillators rotates with a constant phase difference equal to the difference in the pinning phases, thereby preserving the randomness in the system and making the phase-locked states totally incoherent r = 0. Outside the phase-locked steps the states are coherent r > 0 and quasiperiodic. Around the edges of the steps a hysteretic region is found, where both kind of states coexist. The coherent state is born with a finite value of the order parameter r, which makes the behavior resemble that of a second order phase transition.


The phase-locked islands diminish as the coupling strength K is increased, until they vanish completely at a critical value of K. The complicated structure of the quasiperiodic, coherent state was studied numerically and an ansatz was proposed, including a relation between the rotation number and the velocity of a wave propagat- ing in a-space. In the limit of strong coherence we showed that this wave is sinusoidal with a velocity that for strong coupling equals that of the collective phase (R /2 r r ) .

The phenomenon of completely incoherent, phase-locked states is in contrast to what is found if the randomness is introduced to the system through a distribution of intrinsic frequencies in- stead of pinning phases. In this case oscillators with nearby intrinsic frequencies may lock both mutual and to the external frequency if the coupling is sufficiently strong [5]. Moreover, the phases of the entrained oscillators are strongly correlated giving a positive value of the order parameter. For weak coupling and no external field no entrainment takes place and all frequencies differ giving a homogeneous distribution of momentary phases and hence a vanishing order parameter. In the model studied here the randomness may be maintained although the oscillators are locked to the drive frequency. This is due to the fact that a random distribution of phases effectively decouples. While in the absence of an external ac field this state would be unstable for every K the phase-locking makes it stable for weak coupling.

If an inertial term is included in the equation describing the individual oscillators chaos is encountered in a supercritical region in parameter space. One could imagine, as the chaos is the result of overlapping steps, that the onset of chaos is delayed by introducing the coupling term.

This is left as an open question, as is the influence of chaotic motion on the coherence.

An important assumption for the analysis in the present paper is the infinite range couplings. Another subject for future research is the effect of short range couplings, which in most cases is the more realistic model.

Acknowledgement

This work was supported by the Danish Natu- ral Science Research Council.

References

[1] G.B. Ermentrout, J. Math. Biol. 22 (1985) 1; Physica D 41 (1990) 219.

[2] H. Daido, Prog. Theor. Phys. 75 (1986) 1460; J. Phys. A 20 (1987) L629; Phys. Rev. Lett. 61 (1988) 231.

[3] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79 (1984) 223. [4] Y. Kuramoto and I. Nishikawa, J. Stat. Phys. 49 (1987)

569. [5] H. Sakaguchi, Prog. Theor. Phys. 79 (1988) 39. [6] D.S. Fisher, Phys. Rev. B 31 (1985) 1396. [7] A.T. Winfree, The Geometry of Biological Time

(Springer, Berlin, 1990). [8] J. Buck and E. Buck, Sci. Am. 234 Nr. 5 (1976) 74. [9] G.B. Ermentrout and J.M. Rinzel, Am. J. Physiol. 246

(1983) R602. [10] R.E. Kronauer, Lectures in Mathematics in the Life

Sciences, Vol. 19 (Am. Math. Soc., Providence, RI, 1987) p. 63.

[11] K.Y. Tsang, R.E. Mirollo, S.H. Strogatz and K. Wiesenfeld, preprint.

[12] K.Y. Tsang and K. Weisenfeld, Appl. Phys. Lett. 56, 495 (1990).

[13] A. Barone and G. Paterno. Physics and Applications of the Josephson Effect (Wiley, New York, 1982), and references therein.

[14] S.H. Strogatz, C.H. Marcus, R.M. Westervelt and R.E. Mirollo, Physica D 36 (1989) 23.

collective dynamics of coupled modulated oscillators with random pinning

Documents