s Trogat z 1991 Stability

7/25/2019 s Trogat z 1991 Stability

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Journal o f Statistical Physics Vo l. 63 Nos. 3/4 1991

S t a b i l i t y o f I n c o h e r e n c e i n a P o p u l a t i o n o f

o u p l e d O s c i l l a t o r s

S t e v e n H S t r o g a t z 1 a n d R e n a t o E M i r o l l o 2

Received June 26 1990; fin al January 8 1991

We analyze a mean-field model of coupled oscillators with randomly distributed

frequencies. This system is known to exhibit a transition to collective oscilla-

tions: for small coupling, the system is incoherent, with all the oscillators

running at their natural frequencies, but when the coupling exceeds a certain

threshold, the system spontaneously synchronizes. We obtain the first rigorous

stability results for this model by linearizing the Fokker-Planck equation about

the incoherent state. An unexpected result is that the system has pathological

stability properties: the incoherent state is unstable above threshold, but

neutrally

stable below threshold. We also show that the system is singular in the

sense that its stability properties are radically altered by infinitesimal noise.

K E Y W O R D S : Nonlinear oscillator; synchronization; phase transition;

mean-field model; bifurcation; collective phenomena; phase locking.

1 I N T R O D U C T I O N

Collective sync hroni zat ion is a remar kable phe nom en on which occurs at

practi cally every level of biological organ izat ion. (1 2) Example s range from

epileptic seizures in the bra in (g) and electrical sync hro ny a mon g cardiac

pa ce ma ke r cells (4) to sy nc hr on ou s flashing in swarm s of fireflies, (5~ the

chirpi ng of crickets in unis on, (6) and the mut ua l syn chr oni za tio n of

men st rua l cycles in groups of women. (7)

Winfr ee (1) was the first to empha siz e the ubi qui ty of collective syn-

chronization, and also to reduce the problem to its mathematical essence.

He model ed each me mber of the popul ati on as a nonline ar oscillator with

1Department of Mathematics, Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139.

2 Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02167.

613

0022-4715/91/0500-0613506.50/0 9

9 9 P l e n u m P u b l i s h in g C o r p o r a t i o n


2/23

6 4 S t r o g a t z a n d M i r o l l o

a g l o b a l l y a t t r a c t i n g l i m i t cy c le . T h e o s c i ll a to r s w e r e a s s u m e d t o b e w e a k l y

c o u p l e d a n d t h e ir n a t u r a l f r eq u e n c ie s w e re a s s u m e d t o b e r a n d o m l y d is -

t r i b u t e d a c r o s s t h e p o p u l a t i o n . W i n f r e e d i s c o v e r e d t h a t s y n c h r o n i z a t i o n

o c c u r s

c o o p e r a t i v e l y ,

i n a m a n n e r s t r i k i n g l y r e m i n i s c e n t o f a t h e r m o -

d y n a m i c p h a s e t r a n s i ti o n . I n t h e a b s e n c e o f c o u p li n g , th e p o p u l a t i o n

b e h a v e s i n c o h e r e n t l y , w i t h e a c h o s c i l l a t o r r u n n i n g a t i t s n a t u r a l f r e q u e n c y .

A s t h e c o u p l i n g i s i n c r e a s e d , t h e p o p u l a t i o n c o n t i n u e s t o b e i n c o h e r e n t

u n t i l a c r i ti c a l c o u p l i n g i s e x c e e d e d - - t h e n t h e sy s t e m s p o n t a n e o u s l y c o n -

d e n s e s i n t o a p a r t i a l l y s y n c h r o n i z e d st a te .

K u r a m o t o (8'9) p r o v i d e d t h e n e x t m a j o r a d v a n c e i n t h e s t u d y o f

p o p u l a t i o n s o f c o u p l e d o s c i ll a to r s , H e p r o p o s e d a n a n a l y t i c a l l y t r a c t a b l e

m o d e l w h i c h e l u c i d a t e d t h e c o n n e c t i o n b e t w e e n c o l l e c t i v e s y n c h r o n i z a t i o n

a n d p h a s e t r an s i ti o n s . T h e g o v e r n i n g e q u a t i o n o f t h e m o d e l is

N

0 i

= D i

~

~ j 2 1 = s in O j 0 ~ ) ,

i = l , . . . , N 1 . 1 )

H e r e Oi is the ph ase o f t he i t h o sc i l l a t o r , co i s i ts na tu ra l f r e quen cy , and

K ~ > 0 is th e c o u p l i n g s t r e n g th . T h e f r e q u e n c i e s a r e c h o s e n a t r a n d o m f r o m

a s y m m e t r i c , o n e - h u m p e d d i s t r i b u t i o n w i t h d e n s i t y g (c o). B y g o i n g i n t o a

r o t a t i n g f r a m e i f n e c e s s a r y , w e m a y a s s u m e t h a t g(c o) h a s m e a n z e r o .

In t he mode l (1 . 1 ) , each osc i l l a t o r i s coup l ed equa l l y t o a l l t he o the r s .

T h i s is a c r u c ia l s i m p l i fy i n g a s s u m p t i o n , c o r r e s p o n d i n g t o t h e m e a n - f i e l d

a p p r o x i m a t i o n i n s t a ti s ti c a l m e c h a n i c s . K u r a m o t o (8'9) s h o w e d t h a t (1 .1 )

c o u l d b e s o l v e d f o r m a l l y in t h e i n f i n i t e - N l im i t , a s f o ll o w s . C o n s i d e r a

c o m p l e x o r d e r p a r a m e t e r d e f i n e d b y

1 N

re iO = ~ ~ e i~ (1.2)

j = t

H e r e

r t ) > ~ O

m e a s u r e s t h e p h a s e c o h e r e n c e o f t h e o s c i l la t o r s, a n d O ( t )

m e a s u r e s t h e a v e r a g e p h a s e . T h e n , b e c a u s e o f a t r i g o n o m e t r i c i d e n t i t y , t h e

g o v e r n i n g e q u a t i o n s m a y b e r e w r i t t e n a s

0 i = co s + K r

sin(~b - 0,)

1 . 3 )

fo r i = 1 .... N . E qu a t i o n (1.3 ) show s t ha t t he o sc i l l a t o r s a re cou p l ed on ly

t h r o u g h t h e m e a n - f i e l d q u a n t i t i e s r a n d 0 . T h e c o u p l i n g t e n d s t o s y n -

c h r o n i z e th e o s c i l l a t o r s ~ a c h p h a s e 0 i is p u l le d to w a r d t h e av e r a g e p h as e

b y a r e s t o r i n g f o r c e o f s t r e n g t h

K r .

T h e k e y i n s i g h t i s t h a t i n t h e i n f i n i t e -N li m i t, t h e r e a r e s t e a d y s o l u t i o n s

in wh ich a l l f l uc tua t i ons van i sh . In pa r t i cu l a r , t he re a re s e l f - cons i s t en t

so lu t i ons o f (1 . 3 ) i n wh ich

r t ) a n d O t ) a r e c o n s t a n t .

F o r e x a m p l e , o n e


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C o u p l e d O s c i l la t o r s 6 5

s u c h s o l u t i o n i s t h e c o m p l e t e l y incoherent so lu t i on wi th a l l t he o sc i l l a t o r s

d i s t r ib u t e d u n i f o r m l y a r o u n d t h e c ir cle a n d r o t a t i n g a t t h e i r n a t u r a l f re -

q u e n c i e s . T h e i n c o h e r e n t s o l u t i o n h a s r ( t ) - 0 and ex i s t s fo r a l l va lues o f

t h e c o u p l i n g s t r e n g t h K . K u r a m o t o (8'9~ s h o w e d t h a t a s e c o n d f a m i l y o f

s o l u t i o n s b r a n c h e s o f f t h e i n c o h e r e n t s o l u t i o n a t a c r i ti c a l c o u p l i n g s t r e n g t h

2

K c = 1 . 4 )

=g O)

T h e s e n e w s o l u t i o n s h a v e c o h e r e n c e r > 0 a n d a r e p a r t i a ll y s y n c h r o n i z e d ,

i n t h e s e n s e t h a t t h e p o p u l a t i o n o f o s c i ll a t o rs s p l it s i n t o a m u t u a l l y s y n -

ch ro niz ed gr ou p w i th to01 ~ Kr. I n o t h e r

w o r d s , t h e o s c i l la t o r s l y i n g n e a r t h e c e n t e r o f t h e f r e q u e n c y d i s t r i b u t i o n

b e c o m e l o c k e d , w h e r e a s t h e o u t l y i n g o s c il la t o rs r e m a i n d e s y n c h r o n i z e d .

J u s t a b o v e t h e t r a n s i ti o n , t h e c o h e r e n c e r g r o w s c o n t i n u o u s l y a s

( K - K c ) 1/2. T h i s r e s u l t su g g e s t s th a t t h e o n s e t o f s y n c h r o n i z a t i o n is s im i l a r

t o a s e c o n d - o r d e r p h a s e t r a n s i t i o n .

A l t h o u g h K u r a m o t o ' s a n a l y s i s (8'9) o f t h e i n f i n i t e - N li m i t is e l e g a n t a n d

s u c c e s s f u l i n m a n y r e s p e c t s , i t f a i l s t o a d d r e s s t w o v e r y i m p o r t a n t s e t s o f

q u e s t i o n s . T h e f ir s t s e t o f q u e s t i o n s c o n c e r n s

the differences between infinite

and large bu t f i n i t e N . F o r a n y f in i te n u m b e r o f o s c i ll a to r s , th e r e a r e

i n e v i t a b l e f l u c t u a t i o n s i n

r(t) .

D a i d o (1~ h a s s h o w n t h a t t h e s e f l u c t u a t i o n s

a re t yp i ca l l y o f s i ze O ( N 1 /2 ), bu t t h ey ca n b e am pl i f i ed c lo se t o t he on se t

o f syn ch ron i za t i on , i.e ., fo r K c lose t o Kc . Ther e h ave bee n s evera l r ecen t

a t t e m p t s (~~ t o d e v e l o p a t h e o r y o f t h e f l u c t u a t i o n s n e a r t h e o n s e t o f

s y n c h r o n i z a t i o n , b u t t h e m a t t e r r e m a i n s c o n t r o v e r s i a l .

T h e r e i s a l s o t h e p o s s i b i l i ty o f m u c h l a r g e r f l u c t u a t i o n s , a s p o i n t e d o u t

b y N a n c y K o p e l l ( p e r s o n a l c o m m u n i c a t i o n ) . F o r e x a m p l e , i n t h e a b s e n c e

o f c o u p l i n g ( K - - 0 ) , t h e s y s t e m (1 .1 ) is k n o w n t o e x h i b it P o i n c a r 6

recu r ren ce , (15) l ead ing t o f l uc tua t i o ns o f size O (1 ) fo r a l l N . A l tho ug h t hese

l a r g e f l u c t u a t i o n s a r e e x t r e m e l y r a r e , t h e y a r e g u a r a n t e e d t o o c c u r e v e n -

t u a l l y . I t i s u n k n o w n w h e t h e r t h e c o u p l e d s y s t e m a l s o e x h i b i t s P o i n c a r 6

r e c u rr e n c e. A r i g o r o u s t r e a t m e n t o f t h e se f l u c tu a t i o n s w o u l d b e a m a j o r

t h e o r e t i c a l c o n t r i b u t i o n .

T h e s e c o n d c l as s o f o p e n q u e s t i o n s c o n c e r n s t h e stability o f t h e f o r m a l

s o l u t io n s o b t a i n e d b y K u r a m o t o i n t h e i n f i n i te - N l im i t. K u r a m o t o (9) c o n -

j e c t u r e d t h a t t h e in c o h e r e n t s o l u t i o n is s ta b l e fo r K < K,, a n d u n s t a b l e f o r

K > K c , b u t m e n t i o n e d t h a t , s u r p r is i n g l y e n o u g h , t h is s ee m i n g l y o b v i o u s

f a c t s e e m s d i f fi c u lt t o p r o v e . T h e f i rs t s te p s h a v e b e e n t a k e n b y K u r a m o t o

an d N i sh ik aw a , (~3'~4) w ho recen t l y p rese n t ed two d i f f e ren t ana lyses o f t he

s t a b i li t y o f t h e i n c o h e r e n t s o l u t i o n . B o t h o f t h e s e a n a l y s e s i n v o l v e s o m e

ap pr ox im at io ns w hos e va l i d i t y is u nce r t a in . Th e f i rs t ana ly s i s (13~ is bas ed


4/23


o n a n a p p r o x i m a t e e v o l u t i o n e q u a t i o n f o r t h e c o h e r e n c e

r t ) ,

whi l e t he

r e v is e d a n a l y s i s 14) i s b a s e d o n a n a p p r o x i m a t e l i n e a r i n te g r a l e q u a t i o n f o r

t h e g r o w t h o f f l u c t u a t i o n s i n

r t ) .

A c c o r d i n g t o K u r a m o t o a n d

N i s h i k a w a , 141 t h e i n c o h e r e n t s o l u t i o n g o e s f r o m l i n e a r ly s t a b le t o l i n e a r ly

u n s t a b l e a s K i n c r e a s e s t h r o u g h K c.

I n t h i s p a p e r w e p r e s e n t t h e f ir s t r i g o r o u s s t a b i li t y a n a l y s is o f t h e

i n c o h e r e n t s o l u t i o n f o r th e i n f i n i t e - N s y s te m . O n e o f o u r m a i n r e s u lt s is a n

e x a c t f o r m u l a f o r t h e e i g e n v a l u e c h a r a c t e r i z i n g t h e g r o w t h r a t e o f

c o h e r e n c e f o r K > K c . B y f i n d in g w h e r e t h is e i g e n v a l u e v a n is h e s , w e a ls o

o b t a i n a n e w d e r i v a t i o n o f th e c r i ti c a l c o u p l i n g K c g i v e n b y 1 .4 ).

T h e a n a l y s i s re v e a ls t h a t t h e i n f i n i te - N s y s t e m h a s p a t h o l o g i c a l

s t a b i l i t y p r o p e r t i e s : t h e i n c o h e r e n t s o l u t i o n i s u n s t a b l e f o r

K > K c ,

b u t

n eu t r a l l y

s t ab l e fo r a l l

K < K c .

T h i s r e s u l t i s u n u s u a l ; i n a n o r d i n a r y

s e c o n d - o r d e r p h a s e t r a n s i t i o n , t h e d i s o r d e r e d s t a t e i s s t a b l e o n o n e s i d e o f

t h e tr a n s i t i o n a n d u n s t a b le o n t h e o th e r . F u r t h e r m o r e , f r o m t h e p o i n t o f

v i e w o f d y n a m i c a l s y s t e m s t h e o r y , o n e u s u a l l y e x p e c t s n e u t r a l s t a b i l i t y t o

h o l d o n l y a t a s p e c i a l p a r a m e t e r v a l u e , r a t h e r t h a n f o r a w h o l e i n t e r v a l o f

p a r a m e t e r s .

W e a l s o f in d t h a t t h e i n f i n i t e - N s y s t e m is s i n g u l a r w i t h r e s p e c t t o t h e

a d d i t i o n o f i n f i n i t e s i m a l a m o u n t s o f n o i s e : f o r

K < K ~ ,

a n d a n y n o i s e

s t r e n g t h g r e a t e r t h a n z e r o , t h e i n c o h e r e n t s o l u t i o n c h a n g e s f r o m n e u t r a l l y

s t ab l e t o l i ne a r ly s tab l e . The s ingu l a r l im i t o f ze ro n o i se is a lso as so c i a t ed

w i t h a b r e a k d o w n o f u n i q u e n e s s : w h e r e a s i n t h e n o i s y c a s e t h e r e i s a

u n i q u e s o l u t i o n w i t h r t )= - O for a l l t , in the noise-f ree case there i s an

i n f in i te n u m b e r o f s u c h s o l u t i o n s . T h u s i f w e s t u d y t h e p r o b l e m f o r s m a l l

n o i s e , a n d t h e n l o o k a t t h e l i m i t i n g b e h a v i o r a s t h e n o i s e t e n d s t o z e r o , t h e

r e s u lt s a r e c o m p l e t e l y d if f e r e n t f r o m t h o s e o b t a i n e d f o r z e ro n o i s e . T h i s

s i n g u l a r b e h a v i o r i s s i m i l a r t o t h a t s e e n i n f l u i d m e c h a n i c s i n t h e l i m i t o f

z e r o v i s c o s i t y - - i n b o t h c a s e s t h e h i g h e s t o r d e r d e r i v a t i v e i s l o s t f r o m t h e

g o v e r n in g N a v i e r - S t o k e s o r F o k k e r - P l a n c k ) e q ua t io n .

T h e p a p e r i s o r g a n i z e d a s f o l lo w s . I n S e c t i o n 2 w e p r e s e n t t h e

g o v e r n i n g e q u a t i o n s f o r t h e s y s t e m , u s i n g t h e F o k k e r - P l a n c k f o r m a l i s m .

A f t e r d e f i n i n g t h e i n c o h e r e n t s o l u t i o n , w e o b t a i n t h e l i n e a r e q u a t i o n s

g o v e r n i n g t h e g r o w t h o f s m a l l p e r t u r b a t i o n s a b o u t t h e i n c o h e r e n t s ta te .

F o u r i e r m e t h o d s a r e t h e n u s e d t o s h o w t h a t t h e fir st h a r m o n i c o f t h e p er -

t u r b a t i o n p l a y s a s p e c ia l r o le ; i n S e c t i o n 3 w e s t u d y t h e e v o l u t i o n o f t h e

f u n d a m e n t a l m o d e , a n d t h e r e b y o b t a i n a f o r m u l a f o r t h e g r o w t h r a t e o f t h e

c o h e r e n c e

r t ) .

T h e r e s u l t s a r e i l l u s t r a t e d w i t h a n a l y t i c a l , g r a p h i c a l , a n d

n u m e r i c a l e x a m p l e s . W e a l s o s tr e ss t h e p a t h o l o g i c a l s t a b i l it y p r o p e r t i e s o f

t h e n o i s e- f re e s y s t e m . S e c t i o n 4 d e a l s w i t h t h e e v o l u t i o n o f th e h i g h e r h a r -

m o n i c s o f th e p e r t u r b a t io n . T h e s e h i g h e r h a r m o n i c s d e c a y e x p o n e n t i a ll y

fas t i n t he p resence o f no i se , bu t ca n p e r s i s t fo rever i n t he a bsence o f no i se .


5/23

Cou pled Osci l lators 617

F i n a ll y , i n S e c t i o n 5 w e d i sc u ss s o m e o p e n p r o b l e m s a n d c o m p a r e o u r

a p p r o a c h t o t h e s e l f -c o n s is t en t m e t h o d o f K u r a m o t o . (8'9) W e p o i n t o u t

t h a t o u r t e c h n i q u e s c a n b e u s e d t o a n a l y z e p r o b l e m s w i t h n o n s i n u s o i d a l

c o u p l i n g , f o r w h i c h t h e s e l f- c o n si s te n t m e t h o d b r e a k s d o w n .

2 . G O V E R N I N G E Q U A T I O N S

T h e f ir s t p r o b l e m is t o d e c i d e w h a t w e m e a n b y t h e i n f i n i t e -N l im i t o f

( 1 . 1 ) . O n e a p p r o a c h i s t o r e p l a c e O i t ) , i = 1 . . . . N , w i t h a f u n c t i o n O t , c o ) ,

w h e r e co r a n g e s o v e r t h e s u p p o r t o f g (c o ). W e w i l l n o t f o l l o w t h is s e e m i n g l y

r e a s o n a b l e a p p r o a c h , f o r t w o r e a s o n s . F i r s t , w e a r e i n t e r e s t e d i n a d d i n g

n o i s e t o ( 1 . 1 ) ; i n t h i s c a s e i t i s m o r e n a t u r a l t o d e s c r i b e t h e s y s t e m i n t e r m s

o f a d e n s i t y

p O , t , c o ) ,

a s d i s c u s s e d b e l o w . S e c o n d , w e w a n t t o a l l o w f o r t h e

p o s s i b i l i t y t h a t t w o o r m o r e o s c i l l a t o r s h a v e t h e s a m e f r e q u e n c y , a s i n t h e

c a s e w h e n t h e r e a r e d e l t a f u n c t i o n s i n g ( c o ) . I n t h i s c a s e 0 c o u l d n o t

po ss ib ly be a s in g le - va lue d f u nc t ion o f co.

E v e n i f g ( co ) d o e s n o t c o n t a i n d e l t a f u n c t i o n s , i t is s ti ll m o r e n a t u r a l

t o d e s c r i b e t h e i n f i n it e s y s t e m i n t e r m s o f a d e n s i t y p O , t , c o ) r a t h e r t h a n

a f u n c t i o n

O t , c o ) .

F o r e x a m p l e , c o n s i d e r w h a t o n e m e a n s b y a r a n d o m

i n i ti a l c o n d i t i o n f o r t h e l a r g e - N s y s t e m . F o r i = 1 ,..., N , w e p i c k a f r e q u e n c y

c o g

a t r a n d o m f r o m g (c o) , a n d t h e n w e p i c k a r a n d o m p h a s e 0 i f r o m a

u n i f o r m d i s t r i b u t i o n o n t h e c i rc le . F o r s i m p l i c it y s u p p o s e t h a t g ( c o) i s t h e

u n i f o r m d e n s i t y o n t h e i n t e r v a l I = [ - 7 , 7 ] - T h e n t h e p a i r (c o~ , 0 i) i s a

p o i n t o n t h e c y l i n d e r I x S 1. F o r l a r g e N , t h e p o i n t s a r e e s s e n t ia l l y

u n i f o r m l y d i s t r i b u t e d o n t h e c y l in d e r . T h u s , a s N ~ 0 t h e d e n s i t y b e c o m e s

i n c r e a s i n g l y w e l l b e h a v e d , w h e r e a s t h e f u n c t i o n

O t , c o )

b e c o m e s

i n c r e a s i n g l y i r r e g u l a r.

W e n o w d i s c u s s t h e d e n s i t y a p p r o a c h i n m o r e d e t a i l .

2 .1 . D e n s i t ie s a n d t h e F o k k e r P l a n c k E q u a t io n

T h e i n f i n i t e s y s t e m s h o u l d b e v i s u a l i z e d a s f o l l o w s : f o r e a c h f r e q u e n c y

co, t h e r e i s a c o n t i n u u m o f o s c i l l a t o r s d i s t r i b u t e d a l o n g t h e c i rc le . S u p p o s e

t h a t t h i s d i s t r i b u t i o n i s c h a r a c t e r i z e d b y a d e n s i t y p O , t , c o ) . H e r e

p O , t , c o ) d O

g i ve s t h e f r a c t i o n o f o s c i ll a t o r s o f n a t u r a l f r e q u e n c y co w h i c h

l ie b e t w e e n 0 a n d 0 + d O a t t i m e r T h e n t h e a p p r o p r i a t e n o r m a l i z a t io n

c o n d i t i o n i s

re

f o p O , t , c o ) d O = l

(2 .1)

f o r al l co a n d a ll r F u r t h e r m o r e , t h e d e n s i t y p is r e q u i r e d t o b e 2 ~ - p e r i o d i c

in 0.


6/23

618 Strog atz and Mi rot lo

As men t ioned above , we a re in te res ted in add ing no i se to (1 .1 ) .

Cons ide r the fo l lowing genera l i za t ion o f (1 .1 ) , cons ide red p rev ious ly by

Sakaguchi(16):

K N

O i ~ - C O l ~ - ~ i ~ - ~ jE I =

i n 0 j - 0 , ,

i = 1,.. ., N (2.2)

where the va r iab les r a r e indepe nden t wh i te no i se p roces ses tha t s a ti s fy

( r = 0 ( 2.3 a)

( ~ , ( s ) ~ j ( t ) ) = 2 D 6 ~ 6 ( s - t )

2.3b)

In (2 .3 ) , the no i se s t r eng th D i s nonnega t ive and the angu la r b racke t s

den o te an average over r ea l i za tions o f the no ise. In the con tex t o f b io log i -

ca l o sc i l l a to r s , the no i se t e rms can be in te rp re ted as r ap id f luc tua t ions in

the in t rin s ic f r eque ncy o f the osc i l la to rs . In the c on tex t o f the rm ody nam ic

sys tems , the no i se t e rms r ep resen t the rmal f luc tua t ions a t a t empera tu re

p r o p o r t i o n a l t o t h e p a r a m e t e r D .

S ince (2 .2 ) is a sys tem o f coup led L angev in equa t ions , the evo lu t ion o f

p ( O , t , c o )

is gov erned by the fo l lowing F ok ke r -P lan ck equation (16 17):

8p 82p 8

8 t - P 8 0 2 8 0 ( p v )

(2.4)

where the ve loc i ty v ( O , t , c o ) i s g iven by

v ( O , t , c o ) = c o + K r

sin(t~ - 0)

(2.5)

a n d t h e o r d e r p a r a m e t e r a m p l i t u d e

r ( t )

and phase q / ( t ) a r e now def ined by

r e g ~ = d ~ t , c o ) g ( c o ) d c o d O

(2.6)

- - o 3

N ote tha t the in te rac t ion be tw een osc i l la to r s o f d i ff e ren t f r equenc ies occur s

so le ly th rou gh the o rde r pa ram ete r (2.6 ).

2 2 I n c o h e r e n t S o l u t i o n

Eq ua t io ns (2 .4 ) - (2 .6 ) govern the evo lu t ion o f the dens i ty

p ( O , t , c o ) .

Our goa l i s to ana lyze the evo lu t ion o f

p ( O , t , c o )

i n t h e n e i g h b o r h o o d o f

the incoherent so lu t ion , def ined as fo l lows .


7/23

Cou pled Osc i llators 619

D e f i n i t i o n T h e i n c o h e r e n t s o l u t i o n o f 2 .4 ) i s g i v e n b y

1

p o ( O ,

t , c o )= 2--~ 2.7 )

for a l l 0 , t , and co .

T h e i n c o h e r e n t s o l u t i o n P 0 c o r r e s p o n d s t o a s t a t e i n w h i c h , fo r e a c h

co , a l l t h e o s c i l l a t o r s a r e u n i f o r m l y d i s t r i b u t e d a r o u n d t h e c ir c le . I t i s e a s y

t o v e r i fy t h a t P o is a s t a t ic s o l u t i o n o f t h e F o k k e r - P l a n c k e q u a t i o n 2 . 4) :

s u b s t i t u t i o n o f P o i n t o 2 . 6 ) s h o w s t h a t t h e c o r r e s p o n d i n g r ( t ) v a n i s h e s

i d e n t i c a l ly a n d h e n c e t h e v e l o c i ty 2 .5 ) r e d u c e s t o v ( O , t , c o ) = c o . I n

p a r t i c u l a r , v i s i n d e p e n d e n t o f 0 a n d t h e r e f o r e

o 0 2 p 0 ~

D - - - p e r ) = 0

~ t c~02 ~30

T hu s 2 . 4 ) i s s a t i sf i e d .

2 3 L i n e a r iz e d S y s t e m

N o w c o n s i d e r t h e e v o l u t i o n o f a s m a l l p e r t u r b a t i o n a w a y fr o m t h e

i n c o h e r e n t s t a t e : l e t

1

p ( O , t , c o ) = ~ + e t/ 0 , t, c o) 2 . 8 )

w h e r e 8 4 1 . T h e n o r m a l i z a t i o n c o n d i t i o n 2 .1 ) i m p l ie s t h a t t l ( O , t , c o )

sa t i s f ie s

2zr

f o t l( O t , c o ) d O = 0 2 .9)

f o r a ll co a n d t, a n d t h e F o k k e r - P l a n c k e q u a t i o n 2 .4 ) i m p l i es

0 1

8 ~ t = 8 D 302 32q ~0 I 2 -~ + 8 r / ) v ] 2.1 0)

C o n s i d e r 2 . 1 0 ) a t l o w e s t o r d e r i n e. T o f i n d t h e 0 8 ) c o n t r i b u t i o n

f r o m t h e b r a c k e t e d t e r m i n 2 .1 0 ), w e o b s e r v e th a t r ( t ) i s O e ) , a n d h e n c e

v = c o + 0 8 ) . M o r e s p e c i fi c a ll y , w e f i n d

r ( t ) = 8 r l / ) + 0 8 2 )

w h e r e

r ~ e iO e i ~ t , c o ) g c o ) d c o d O 2 . 11 )

822/63/3 4-13


8/23

6 2 0 S t r o g a t z a n d M i r o l io

Thus , (2 . 5 ) imp l i es t ha t

d v / 8 0 = - e K G

c o s ( 0 - 0 ) , a n d so a t O (e ) th e

e v o l u t i o n e q u a t i o n ( 2 . 1 0 ) b e c o m e s

/ 8 2 1

8 r

K

? 7 = D g g - co 5 0 + r , c os , - 0 )

(2.12)

T o a n a l y z e ( 2 . 1 2 ) i t i s c o n v e n i e n t t o u s e F o u r i e r m e t h o d s . S i n c e t h e

fu nc t ion 7(0 , t, co) is rea l an d 2re-periodic in 0 , we see k so lu t io ns of (2 .12)

o f th e f o r m

t l O , t , c o ) = c t , co ) e i ~ c o ) e - : ~ 1 7 7 co )

(2.13)

w h e r e t h e s t a r d e n o t e s c o m p l e x c o n j u g a t i o n a n d r /a ( 0 , t , co) c o n t a i n s t h e

s e c o n d a n d h i g h e r h a r m o n i c s o f ~/. [ N o t e t h a t r/ a u t o m a t i c a l l y h a s z e r o

m e a n , b y ( 2 . 9 ) . ]

T h e p a r t i c u l a r f o r m o f ( 2 .1 3 ) is m o t i v a t e d b y t h e o b s e r v a t i o n t h a t t h e

f i r s t

h a r m o n i c o f t / i s d i s t in g u i s h e d . F o r e x a m p l e , t h e f ir s t h a r m o n i c is th e

o n l y t e rm t h a t c o n t r ib u t e s t o t h e c o h e r e n c e

r t ) .

C o n s e q u e n tl y , r / m a k e s n o

c o n t r i b u t i o n t o t h e f in a l t e r m i n ( 2.1 2) e x c e p t t h r o u g h

c t , c o )

a n d i t s

c o m p l e x c o n j u g a t e . T o s e e t h i s , n o t e t h a t

r~ c o s g , - O ) = R e [ q e i O e ~ o ] .

Now subs t i t u t i ng (2 . 13 ) i n to (2 . 11 ) y i e ld s

r 1e i o = 2 r t c * t , c o ) g c o ) d o )

o

a n d s o

(2.14)

w h e r e c .c . d e n o t e s t h e c o m p l e x c o n j u g a t e o f t h e p r e c e d i n g t e r m . T h u s t h e

f i n a l t e r m i n ( 2 . 1 2 ) d e p e n d s o n l y o n

c t , o ) )

a n d i t s c o n j u g a t e , a s c l a i m e d .

W hen (2 . 13 ) and (2 . 14 ) a re subs t i t u t ed i n to (2 . 12 ) , we ob t a in two

q u a l i t a t i v e l y d i f fe r e n t e v o l u t i o n e q u a t i o n s , o n e f o r t h e f u n d a m e n t a l

a m p l i t u d e

c t , c o )

an d ano the r fo r r / t, co) . The nex t s ec t i on is con cern ed

w i t h t h e a m p l i t u d e e q u a t i o n f o r t h e f u n d a m e n t a l m o d e

c t , o ) ) ,

a n d

S ec t i on 4 dea l s w i th t he ev o lu t i on o f r / t , co).


9/23

Co upled Os cil lators 621

3 E V O L U T I O N O F T H E F U N D A M E N T A L M O D E

T h e a m p l i t u d e e q u a t i o n f o r

c t , c o )

is o b t a i n e d b y i n s e rt i n g ( 2 .1 3 ) a n d

( 2 . 1 4 ) i n t o ( 2 . 1 2 ) a n d t h e n e q u a t i n g t h e c o e f f i c i e n ts o f e i~ o n b o t h s i d e s o f

t h e re s u l t in g e q u a t i o n . W e f in d

O c=ot - D + i ~ o ) c + K I L c t , v ) g v ) d v (3 .1)

T h e l in e a r e q u a t i o n ( 3 .1 ) h a s a n i n te r e s t in g s t r u c tu r e . F o r a n y g iv e n

f r e q u e n c y c o , t h e e v o l u t i o n o f c t , c o ) d e p e n d s o n a l l t h e o t h e r f r e q u e n c i e s

t h r o u g h t h e t e r m s c t , v ) i n t h e in t e g ra l . H o w e v e r , th e d e p e n d e n c e is t h e

s a m e f o r a ll f r e q u e n c i e s, b e c a u s e t h e i n t e g r a l i s i n d e p e n d e n t o f co T h i s

c o n v e n i e n t p r o p e r t y s te m s f r o m t h e m e a n - fi e ld c h a r a c t e r o f t h e o r ig i n a l

m o d e l ( 2 . 2 ) .

B e f o r e a n a l y z i n g t h e s p e c t r u m o f ( 3.1 ) , w e m a k e a f e w r e m a r k s . T h e r e

i s n o n e e d t o w r i te t h e e v o l u t i o n e q u a t i o n f o r c* , s in c e it is j u s t t h e

c o n j u g a t e o f (3 .1 ). N o t e a l s o t h a t t h e c o h e r e n c e r t ) is d e t e r m i n e d a t t h is

o r d e r b y c t , c o ) , v i a ( 2 . 1 4 ) . I n p a r t i c u l a r , i f c t , c o ) g r o w s e x p o n e n t i a l l y , s o

d o e s r t ) .

3 1 D i s c r e t e S p e c t r u m

E q u a t i o n ( 3.1 ) h a s b o t h a d i s cr e te a n d a c o n t i n u o u s s p e c tr u m . T o f in d

t h e d i s c r e t e s p e c t r u m , w e s e e k s o l u t i o n s o f (3 . 1) o f t h e f o r m

c t , co) = b(co) e ;" (3.2)

w h e r e t h e e i g e n v a l u e 2 is i n d e p e n d e n t o f co. S u b s t i t u t i n g ( 3 .2 ) in t o ( 3 .1 )

y i e l d s

2 b = - D + i c o ) b + ~ b v ) g v ) d v (3 .3)

T h i s e q u a t i o n is e a s y t o s o lv e , b e c a u s e t h e i n t e g r a l i n ( 3 . 3 ) is j u s t s o m e

u n k n o w n c o n s t a n t t o b e d e t e r m i n e d s e lf -c o n s i st e n tl y . T h u s , l e t

K f ~

= - ~ - ~ b v ) g v ) d v (3 .4)

S o l v i n g ( 3 . 3 ) f o r b ( c o ) , w e f i n d

A

b c o ) - 3 . 5 )

2 + D + i c o


10/23

6 S t r o g a t z a n d M i r o l l o

Now we invoke se l f - cons i s t ency : (3 . 5 ) mus t be cons i s t en t wi th (3 . 4 ) .

S u bs t i t u t i on o f (3.5 ) i n to (3.4 ) y i e lds e i t he r A = 0 o r

= - - f ~ 1 7 6 g (v ) d v

3 . 6 )

2 o~ 2 + D + i v

T h e s o l u t i o n A = 0 is n o t a ll o w e d , b e c a u s e ( 3.2 ) a n d (3 .5 ) w o u l d t h e n

i m p l y t h a t

c(t, co) ==_0

fo r a l l co , and t h i s i s no t cons ide red an e igen func t i on .

T h u s , (3 .6 ) i s t h e e q u a t i o n f o r t h e d i s c r et e s p e c t r u m o f t h e l i n e a r s y s t e m

3 . 1 ) .

F r o m n o w o n , w e s u p p o s e t h a t g ( c o ) i s a n e v e n f u n c t i o n , i . e . ,

g (co) = g ( - c o ) fo r a l l co. W e a l so as su m e tha t g (co) is " no n in c re as in g" on

[0 , oe ) , i n t he s ense t ha t

g ( ~ o ) < < , g ( v )

fo r a l l co~>v. These two p rop er t i e s

h o l d f o r t h e G a u s s i a n , L o r e n t z i a n , a n d u n i f o r m d i s t r i b u t i o n s , a s w e l l a s

m a n y o t h e rs o f p r a c t ic a l i n te r es t. U n d e r t h e s e tw o a s s u m p t i o n s

o n e c a n

p r o v e t h a t ( 3 . 6 ) h a s a t m o s t o n e s o l u t i o n f o r 2 , a n d i f s u c h a so l u t i o n e x i s t s,

i t i s n e c e s s a r i l y r e a l

( see t he p r oo f o f Th eo rem 2 i n r ef . 18 ) . Th en (3.6 )

b e c o m e s

K oo 2 + D g ( v ) d v (3.7)

l = ~ f ~ ( Z + D ) 2 + v 2

E q u a t i o n (3 .7 ) i s t h e o n e o f t h e m a i n r e s u l t s o f t h is p a p e r . I t s h o w s

h o w t h e e i g e n v a l u e 2 d e p e n d s o n t h e n o i s e s t r e n g t h D , t h e c o u p l i n g

s t r e n g t h K , a n d t h e f r e q u e n c y d e n s i t y g ( c o ) . T h i s e i g e n v a l u e g o v e r n s t h e

l in e a r s t ab i li ty o f t h e f u n d a m e n t a l m o d e : w h e n 2 > 0 , t h e f u n d a m e n t a l

m o d e i s u n s t a b l e , a n d t h e c o h e r e n c e g r o w s l i k e r ( t ) ~ ro e~ .

I t i s v e r y i m p o r t a n t t o r e c o g n i z e t h a t a n y s o l u t i o n 2 o f ( 3 . 7 ) m u s t

s a ti s fy t h e i n e q u a l i t y

2 > - D (3.8 )

s ince o the rw i se t he r i gh t -ha nd s ide o f (3 .7 ) i s ~ 0 .

E qu a t i on (3.7 ) a l l ows us t o f i nd t he c r i t i ca l coup l ing Kc. a t w h ich s t ab i l i t y

i s l o s t . The c r i t ica l c on d i t i o n i s 2 = 0, w h ich imp l i es

K ~. = 2 D 2 + v 2 g ( v ) d v

(3.9)

T h e c r i t i c a l c o u p l i n g g i v e n b y ( 3 . 9 ) w a s o b t a i n e d p r e v i o u s l y b y

S a kag uc h i (16) v i a an ex t en s ion o f K u ra m o to ' s (8'9) s e l f - cons is t ency a rgu m en t .


11/23


H e f i r s t f o u n d s e l f - c o n s i s t e n t s t a t i c s o l u ti o n s o f t h e F o k k e r - P l a n c k e q u a -

t i o n 2 .4 ), a n d t h e n s h o w e d t h a t s o l u t i o n s w i t h c o h e r e n c e r > 0 b i fu r c a t e

f r o m t h e i n c o h e r e n t s o l u t i o n a l o n g t h e l o c u s d e f i n ed b y 3 .9 ). W e s t re s s

t h a t , i n c o n t r a s t t o t h e a n a l y s i s g i v e n h e r e , t h e a n a l y s i s o f S a k a g u c h i 16)

d o e s n o t p r o v i d e a n y i n f o r m a t i o n a b o u t s t a b i l i t y .

O u r a n a l y s i s p r o v i d e s t h e f i rs t p r o o f t h a t f o r t h e n o i s e - f re e c a s e, t h e

i n c o h e r e n t s o l u t i o n g o e s u n s ta b l e f o r K > K c = 2 / [ ~ g 0 ) ] , a s c o n je c t u r ed

b y K u r a m o t o . 9~ T o s e e t h is , l e t D = 0 i n 3 . 7) a n d l e t 2 -- * 0 + . T h e k e r n e l

f u n c t io n 2 / 2 2 + v 2) b e c o m e s m o r e a n d m o r e s h a r p ly p e a k e d a b o u t v = 0 ,

y e t i ts i n t e g r a l o v e r t h e r e a l l i n e e q u a l s ~ f o r al l p o s i t i v e 2 . T h u s t h e k e r n e l

f u n c t i o n a p p r o a c h e s ~ 6 v ) a s 2 --* 0 + , a n d s o t h e r i g h t - h a n d s i d e o f 3 . 7 )

t e n d s t o K / 2 )7 c g 0 ) . H e n c e 2 > 0 f o r K > 2 / [ ~ g 0 ) ] , a s r e q u i r e d .

I n S e c t i o n 3 . 4 t h e s e r e s u l t s w i l l b e i l l u s t r a t e d f o r p a r t i c u l a r d e n s i t i e s

g c o ) f o r w h i c h t h e e i g e n v a l u e 2 c a n b e f o u n d e x p l ic i tl y . B u t f i r st w e

c o m p u t e t h e r e m a i n i n g p a r t o f t h e s p e c t r u m o f 3 .1 ).

3 2 C o n t i n u o u s S p e c t r u m

T h e l in e a r o p e r a t o r L a s s o c i a t e d w i t h th e r i g h t - h a n d s i d e o f 3 . 1 ) is

g i v e n b y

L b = - D + ic o)b + ~ _ ~ b v ) g v ) dv

3.10)

R e c a ll t h a t t h e c o n t i n u o u s s p e c t r u m o f L is d e fi n e d a s t h e s et o f c o m p l e x

n u m b e r s 2 s u c h t h a t t h e o p e r a t o r L - 2 I is n o t s u rj ec ti v e. T h u s w e a re le d

t o c o n s i d e r t h e e q u a t i o n

K ~

- 2 + D + ic o)b + - ~ f _ ~ b v ) g v ) d v = f 3.11)

f o r f i x ed 2 a n d f o r a n a r b i t r a r y f u n c t i o n f c o ) . I f 3 . 1 1 ) c a n a l w a y s b e

so lv ed fo r b co), th en 2 is n o t i n t h e c o n t i n u o u s s p e c t r u m .

N o t i c e t h a t t h e i n t e g r a l in 3 . 11 ) i s i n d e p e n d e n t o f co. A s i n 3 . 4) , w e

de no te t h i s i n t e g r a l b y A . I f 2 + D + ico = 0 f o r so m e c o in t h e su pp or t o f g ,

t h e n 3 . 1 1) is n o t s o l v a b l e in g e n e ra l , l i t is s o l v a b l e o n l y f o r c o n s t a n t

f u n c ti o n s f c o ) - A . ] H e n c e t h e c o n t i n u o u s s p e c t ru m c o n t a in s t h e s et

{ - - D - i co : co ~ S u pp or t g ) } 3 .12 )

In fac t , 3 .12) i s al l o f t h e c o n t i n u o u s s p e c t r u m , f o r s u p p o s e t h a t 2 is

no t i n 3 .12 ). T he n 3 .11 ) i s so lva b le : f r om 3 .11 ),

A - f c o )

b co) 2 + O + ice 3.1 3)


12/23

6 4

S t r o g a t z an d M i r o l l o

A l l w e h a v e t o d o n o w i s s h o w t h a t A c a n b e d e t e r m i n e d s e l f - c o n s i s t e n t l y .

S u bs t i t u t ing 3 .13 ) i n to 3 .4 ) y i e ld s

g v) - 2 -o~ 2 D i v

1 - _ 2 D i v d V = 3 .14)

B y a s s u m p t i o n , 2 is n o t i n t h e d i s c r e t e s p e c t r u m , a n d s o th e c o e f f ic i e n t o f

A is n o n z e r o , b y 3 .6 ). T h u s 3 . 1 4 ) c a n b e s o l v e d f o r A .

H e n c e t h e s e t 3 . 1 2 ) i s t h e c o n t i n u o u s s p e c t r u m . N o t e t h a t i t l ie s o n

the im a g ina r y a x i s i f D = 0 , bu t i n t he l e f t ha l f - p l a n e i f D > 0 .

3 3 G r a p h s o f t h e S p e c t r u m

N o w w e s k e t c h t h e d i s cr e te a n d c o n t i n u o u s s p e c t r a f o r t h e c as e o f a n

e v e n , n o n i n c r e a s i n g g c o ) w i t h s u p p o r t [ - - 7 , 7 ] , w h e r e 7 > 0. T h e a i m i s t o

g i v e a m o r e c o n c r e t e p i c t u r e o f t h e s p e c t r u m , a n d a l s o t o h i g h l ig h t t h e

p a t h o l o g i c a l f e a t u r e s o f t h e n o i s e - f r e e c a s e D = 0 .

continuous

spectrum

Re ,. = - D

m X a)

discrete

]t / spectrum

e X

- y

b)

c )

d )

F i g . 1. T h e c o n t i n u o u s a n d d i s c r e t e s p e c t r a fo r th e li n e a r o p e r a t o r 3 .1 0) , f o r t h e n o i s y c a s e

D > 0 . a ) K > K c . T h e f u n d a m e n t a l m o d e i s u n s t a b l e s i n ce th e d i s c re t e s p e c t r u m 2 > 0 . b )

K = K c. T h e f u n d a m e n t a l m o d e i s n e u t r a l ly s ta b l e. c ) K * < K < K , .. T h e f u n d a m e n t a l m o d e

i s s t a b l e . d ) K


13/23


F i r s t c o n s i d e r t h e n o i s y c a s e D > 0 , a n d s u p p o s e t h a t D i s f i x e d . T h e n

t h e l o c a t i o n o f t h e s p e c t r u m d e p e n d s o n l y o n t h e c o u p l i n g s t r e n g t h K .

F i g u r e 1 s h o w s t h a t t h e d i s c r e t e s p e c t r u m i s e i t h e r a s in g l e p o i n t ( f o r

K>K ,

i n F ig s. l a - l c ) o r e m p t y ( fo r

KKc

( F ig . l a ) t h e f u n d a m e n t a l m o d e is u n s t a b l e s in c e 2 > 0 . A s K

d e c r e a s e s , t h e e i g e n v a l u e m o v e s t o t h e l e f t - - t h e f u n d a m e n t a l m o d e

b e c o m e s n e u t r a l l y s t a b l e ( F i g . l b ) a n d t h e n l i n e a r l y s t a b le ( F ig . l c ) .

F i n a l ly , i n F i g . l d , t h e d i s c re t e s p e c t r u m is a b s o r b e d b y t h e c o n t i n u o u s

s p e c t r u m a n d d i s a p p e a r s .

F i g u r e 2 s h o w s t h a t t h e p i c t u r e s a r e d r a m a t i c a l l y d i f f e r e n t w h e n D = 0 .

N o w t h e c o n t i n u o u s s p e c t r u m l i e s e x a c t l y o n t h e i m a g i n a r y a x i s . F u r t h e r -

m o r e , t h e r e a r e o n l y t w o p i c t u r e s i n s t e a d o f f o u r a s in F i g . 1, b e c a u s e K *

a n d K c c o i n c i d e w h e n D = 0 N o t e t h a t t h e f u n d a m e n t a l m o d e is u n s t a b l e

w h e n

K>K,.

( F ig . 2a ) , bu t

neutrally

s t a b l e w h e n

K


14/23


a) Identical Oscillators. Sup pose tha t g ( (o )= 6 (co ) , so tha t a ll the

osc i l l a to r s have the s ame na tu ra l f r equency . Then the mode l (2 .2 ) i s

equ iva len t to the mean- f i e ld theo ry fo r a magne t i c sp in sys tem known

as the c lass ica l

X Y

model . The no i s e s t r eng th D i s in te rp re ted as the

tempera tu re and K i s the f e r romagne t i c coup l ing s t r eng th . Equa t ion (3 .7 )

y ie lds

K

2 = ~ D

(3.15a)

which r ecover s the we l l -known resu l t tha t the incoheren t o r

pa ram agne t i c s t a t e lo ses s t ab i l ity a t a c r it ica l t em pera tu re D~=K/2 .

B e l o w th i s te m p e r a t u r e , t h e s y s te m u n d e r g o e s s p o n t a n e o u s m a g n e t i z a ti o n

(o r synchron iza t ion , in ou r con tex t ) .

b ) Un i f o rm D i s t r i bu t i on , Sup pose tha t g (o 9)= (27 ) -1 fo r Icol - D , a s exp la ined abo ve in (3 .8 ). The s am e

is t rue for the next two resul ts .

F igu re 3 shows a g raph o f Eq ua t io n (3.15b) . T he e igenva lue 2 is

p lo t t ed vs. K , wh i le D and 7 a re he ld f ixed . To show the r e la tion be tw een

2 a n d t h e c o n t i n u o u s s p e c tr u m , w e a ls o p l o t t h e re a l p a r t o f t h e c o n t i n u o u s

spec t rum. In F ig . 3a , the d i s c re te spec t rum 2 emerges f rom the con t inuous

s p e c tr u m a t K = K * = 4 7 / r c . T h e o n s e t o f i n st a bi li ty o c c u rs l a te r a t K =

Kc=27/ [ tan -~ 7 /D)] . F i g u r e 3 b sh o w s th a t o n c e a g a in t h e c a se D = 0 is

pecu l i ar ; the p o in t s K * and Kc co incide . Thus , the on se t o f in s tab il i ty

occur s exac t ly when the d i s c re te spec t rum i s bo rn .

c ) Lo ren t z i an o r Cauc hy D i s t r i bu t i on .

I f

g o ) =

( ~ / / T c ) ( y 2 - - F c o 2 ) - i ,

then (3 .7 ) becomes

K

, ~ = ~ - D - 7 ( 3 . 1 5 c )

d) Gaussian Distribution.

dev ia t ion a , we find that 2 sa tisf ies the im pl ic it equ at io n

1 = ( 8 ) ~ / 2 K e x p ((2+D )2~-a~- e r/ c ) - / 2 + D \

where e r f c deno tes the complemen ta ry e r ro r func t ion .

F o r a n o r m a l d i s t r i b u t i o n w i t h s t a n d a r d

(3.15d)


15/23

Cou pled Osc illators 627

e X

0

-D

a )

discrete

t s p e c t r u . j ~

/

continuous

spectrum

K

I 0

K

Fig. 3. G rap h of 3.15b). The eigenvalue 2 is plotted vs. K, for fixed D and 7. The re al part

of the continuous spectrum is also show n. a) The n oisy case D > 0. The d iscrete spectrum )~

emerges from the continuous spectrum at K = K * . The onset o f instabili ty occurs later at

K = K c. b) For the noise-free case D = 0, the points K * and KCcoincide. Hence the onset of

instability occurs just as the discrete spectrum is born.

3 5 N u m e r i c a l E x p e r i m e n t s

A s a b r i e f c h e c k o f t h e a n a l y t i c a l r e s u lt s a b o v e , w e p e r f o r m e d t h e

f o l l o w i n g n u m e r i c a l e x p e r i m e n t s . T h e s e e x p e r i m e n t s a r e f o r t h e s a k e o f

i l l u s t r a t i o n a n d a r e c e r t a i n l y n o t c o m p r e h e n s i v e .

T h e s y s t e m 1 .1 ) w a s s tu d i e d f o r t h e c a s e o f N = 4 8 0 o s c i ll a to r s , w i t h

c o u p l i n g s t r e n g t h K = 1 . T h e f re q u e n c ie s w e r e u n i f o rm l y d i s t r ib u t e d o n

[ - 7 , 7 ] , w h e r e 7 = 0 . 2 . T h e r e w a s n o n o is e , i.e., D = 0 . T h e g o a l w a s t o

s i m u l a t e t h e e v o l u t i o n o f t h e s y s t e m s t a r t in g n e a r t h e i n c o h e r e n t s o l u t io n .

F o r t h e se p a r a m e t e r s , e q u a t i o n 3 . 1 5 b ) p r e d i c t s t h a t t h e c o h e r e n c e r t )

s h o u l d i n i ti a l ly g r o w e x p o n e n t i a l l y a t a r a t e 2 ~ 0 .4 7 3 0 4 .

S t r i c tl y s p e a k i n g , t h e i n c o h e r e n t s o l u t i o n 2 . 7) e x is ts o n l y f o r i n fi n it e

N . T o a p p r o x i m a t e t h e in c o h e r e n t s t a te f o r f in i t e N , w e c h o s e M e v e n l y

s p a c e d f re q u e n c i e s o n t h e i n t e rv a l [ - 7 , 7 ] , a n d a s s ig n e d N / M o s c i l l a t o r s

t o e a c h f r e q u e n c y . F o r i n s t a n c e , i n th e c a s e s h o w n i n F ig . 4 , t h e p o p u l a t i o n

w a s p a r t i t i o n e d i n t o 4 0 f re q u e n c i e s , w i t h 1 2 o s c i ll a to r s a t e a c h f r e q u e n c y .

T h e n , f o r e a c h f r e q u e n c y , t h e o s c i l l a t o rs w e r e e v e n l y s p a c e d a r o u n d t h e


16/23


c i r c l e . F o r a c o m p u t e r w i t h i n f i n i t e - p r e c i s i o n a r i t h m e t i c , t h i s i n i t i a l

c o n d i t io n w o u l d h a v e c o h e r en c e r - 0 . F u r t h e r m o r e , r t ) w o u l d r e m a i n

z e r o fo r a ll t im e . T o b r e a k t he s y m m e t r y , w e a d d e d a r a n d o m n u m b e r o f

s iz e O ( 1 0 - l ~ t o e a c h o f t h e i n it ia l p h a s e s 0 i, r e s u l ti n g i n a n o n z e r o i n it ia l

c o h e r e n c e .

F i g u r e 4 s h o w s t h a t t h e c o h e r e n c e i n i t i a l l y g r o w s e x p o n e n t i a l l y f a s t :

r t ) ~ r o e x p ( 2 M , N t)

w h e r e ) o M ,N ~ 0 . 4 7 1 6 4 . T h e e x p o n e n t i a l g r o w t h o f r t ) b r e ak s d o w n w h e n

r ~ 0 .1 b e c a u s e t h e s y s t e m i s n o l o n g e r c l o s e t o t h e i n c o h e r e n t s o l u t i o n . T h e

m e a s u r e d v a l u e o f 2M , N is w i t h i n 1 o f t h e 2 p r e d i c t e d b y th e t h e o r y f o r

i n f i n i t e N .

F i g u r e 5 i n d i c a te s t h a t t h e d e v i a t i o n o f M N f r o m )~ i s a f i n i t e - s i ze

e ff ec t. T o d e m o n s t r a t e t h is , w e r e p e a t e d t h e s i m u l a t i o n s f o r d if f e re n t

n u m b e r s o f f r e q u e n c ie s M w h i le h o l d i n g t h e t o t a l p o p u l a t i o n f ix e d a t

N - 4 8 0 o s c i l la t o rs . F i g u r e 5 s h o w s t h a t

2M N 2 ~ O 1 / M )

a s M i n c r e a s e s .

T h i s is p r e c is e ly t h e d e p e n d e n c e o n M t h a t o n e w o u l d h a v e e x p e c t e d .

B y c h o o s i n g t h e f r e q u e n c i e s t o b e e v e n l y s p a c e d , w e h a v e e s s e n t i a l l y

.47164

0

5

10

15

20

2 5

3 0

0

In r

20 40 60 80

t i m e

Fig. 4. Linear grow th o f In r, corresponding to expon ential initial growth o f the coherence

r , for K > Kc. The measured exponential growth rate is

~m,u~'0.47164.

Equation (1.1) was

integrated num erical ly us ing the fourth-order Runge-Kutta m ethod. Parameters: K = I ;

N = 480; uniform g ~ o) with 7 = 0.2; M = 40 d istinct frequencies; D = 0. See text for description

of simulation and choice of nearly incoherent initial condition.


17/23

oupled Oscil lators 6 2 9

. 0 1

;L X

M N

. 0 0 1

s l o p e = -1

. 0 0 0 1

10 1O0 1000

F i g . 5 . L o g - l o g p l o t o f 2 - - ~ M N s a f u n c t i o n o f n u m b e r o f d i st i n ct f r e q u e n ci e s M .

P a r a m e t e r s a n d s i m u l a t i o n m e t h o d a s i n F i g . 4 , e x c e p t th a t M v a r i e s M v a l u e s : 1 0, 2 0, 30 ,

4 0, 6 0 , 80 , 1 2 0 , s h o w n a s s o l i d d o t s ) . A d a s h e d l i n e o f s l o p e - 1 i s s h o w n f o r c o m p a r i s o n .

r e p l a c e d g ~ o) w i t h a c o m b o f M d e l t a f u n c t i o n s . T h i s m e a n s t h a t t h e

i n t e g ra l i n 3 .7 ) is r e p l a c e d b y a R i e m a n n s u m - - b u t t h e d e v i a t i o n b e t w e e n

a R i e m a n n s u m a n d i t s l i m i t i s

O 1 / M ) .

T h i s d e v i a t i o n i n t h e i n t e g r a l i n

3 .7 ) l e a d s t o a d e v i a t i o n o f t h e s a m e o r d e r i n 2 .

I n c o n t r a s t , t h e d e p e n d e n c e o f 2M , N o n N w i t h M h e l d f ix e d ) w a s

f o u n d t o b e m u c h w e a k e r . I n o t h e r w o r d s , f o r t h e s i m u l a t i o n p r o t o c o l u s e d

h e r e , i t i s m o r e i m p o r t a n t t o s a m p l e t h e f r e q u e n c i e s d e n s e l y t h a n i t i s t o

h a v e m a n y o s c i l l a t o r s p e r f r e q u e n c y .

4 E V O L U T I O N O F H I G H E R H A R M O N I C S

N o w w e c o m p l e t e t h e l i n e a r s t a b i l i ty a n a l y s i s o f t h e i n c o h e r e n t s o l u -

t io n . N e a r t h e e n d o f S e c t i o n 2 w e d e r i v e d th e e v o l u t i o n e q u a t i o n 2 .1 2 )

g o v e r n i n g t h e g r o w t h o f a s m a l l p e r t u r b a t i o n r/ 0 , t, co) a b o u t t h e

i n c o h e r e n t s o l u t i o n . I t w a s c o n v e n i e n t t o e x p r e s s t/ i n t e r m s o f t w o f u n c -

t i ons ,

c t , co )

an d r / t , co), as in 2 .13). Th e ev olu t io n of

c t , co )

was d is -

c u s s e d i n t h e l a s t s e c t io n , a n d is c lo s e l y r e l a t e d t o c h a n g e s i n t h e c o h e r e n c e

r t ); n o w w e s t u d y t h e e v o l u t i o n o f t h e h i g h e r h a r m o n i c s c o n t a i n e d i n

~ / t , c o). P e r t u r b a t i o n s c o r r e s p o n d i n g to t h e s e h i g h e r h a r m o n i c s d o

n o t

l e a d t o c h a n g e s i n

r t ) .

T h e m a i n r e s u l t i s t h a t t h e n o i s e - f r e e c a s e a g a i n h a s p a t h o l o g i c a l

s t a b i li t y p r o p e r t i e s : f o r D > 0 t h e h i g h e r h a r m o n i c s d e c a y e x p o n e n t i a l l y


18/23

630 Strogatz and M irol lo

f a s t, w h e r e a s f o r D = 0 t h e y p e r si s t f o r e v e r a s n e u t r a l ly - s t a b l e r o t a t i n g

w a v e s . F u r t h e r m o r e , t h e r e i s a b r e a k d o w n o f u n i q u e n e s s w h e n D = 0 : i t

t u r n s o u t t h a t

a n y

f u n c t i o n o f t h e f o r m

~ i t , c o ) = r i O - c o t , co )

i s a solu-

t i o n o f t h e r e l e v a n t e v o l u t i o n e q u a t i o n . T h e s e r o t a t i n g w a v e s o l u t io n s a l s o

o c c u r i n t h e f u l l s y s t e m , a n d a r e n o t j u s t a n a r t i fa c t o f th e l i n e a r i z a t io n .

4 1 L i n e a r S t a b i l i t y A n a l y s i s

To der ive t he ev o lu t i on eq ua t i on fo r r /~ 0 , t , co ) , we sub s t i t u t e 2 .13 )

an d 2 . 14 ) i n to 2 .12 ) an d co l lec t t e rm s i nvo lv in g ~ / Th e r esu l t i s

G ~

0 2 ~ + ~ / ~ 4.1)

0t =D co

E q u a t i o n 4 .1 ) is m u c h s im p l e r t h a n th e e v o l u t i o n e q u a t i o n 3 .1 ) st u d i e d i n

the l a s t s ec t i on , because t he re i s

n o c o u p l i n g

be tween osc i l l a t o r s o f d i f f e ren t

f requ enc ies in 4 .1).

W e s o lv e 4 .1 ) u s i n g F o u r i e r m e t h o d s . R e c a l l t h a t t/ h a s z e ro m e a n ,

by 2 .9 ), a nd ze ro f i rs t ha rm on ic , by 2 .13 ). H enc e t he F ou r i e r se r ie s fo r ~/

s t a r t s w i t h t h e s e c o n d h a r m o n i c :

t l ( O , t , co ) = L ak ( t , o9 ) e i k~

4.2)

Ikl >/2

w her e ak = a_ k)* , s ince ~/ i s rea l . Su bs t i tu t ion of 4 .2) in to 4 .1) y ie lds

t h e a m p l i t u d e e q u a t i o n

~a---3~= ( - k2 D - i kco ) a~

O t

w h i c h h a s t h e g e n e r a l s o l u t i o n

ak ( t , o9 ) = ak ( O , co ) e ( - k 2 D - ik~

H e n c e

~I t , co ) = ~ ak ( O , co ) e k2~176 4.3)

I M > ~

2

Th e so lu t i on 4 .3 ) show s t ha t r / t , co) deca ys t o ze ro exp on en t i a l l y

f a s t f o r a n y D > 0 . B e c a u s e t h e h i g h e r h a r m o n i c s d i e o u t s o r a p i d l y , t h e

e v o l u t i o n o f th e t o t a l p e r t u r b a t i o n t/ is e s s e n t ia l ly c o n t r o l l e d b y i t s

f u n d a m e n t a l m o d e .

I n c o n t r a s t , w h e n D = 0 , E q . 4.3 ) s h o w s t h a t t/ is a n u n d a m p e d

ro t a t i ng wav e , i.e ., a func t i on o f 0 - cot a l one . O f cou rse , t h i s r e su l t fo l l ows

di re ct ly f ro m 4 .1) , s ince

a n y

f u n c t i o n o f th e f o r m

q t , c o ) = f ( O - c o t, c o)

s o lv e s 4 .1 ) w h e n D = 0 . T h e f a c t t h a t a n y f u n c t i o n o f 0 - co t is a c c e p t a b l e

m e a n s t h a t t h e s e r o t a t i n g w a v e s a r e

n e u t r a l l y s t a b l e

t o p e r t u r b a t i o n s

i n v o lv i n g o n l y h i g h e r h a r m o n i c s .


19/23

Co upled Os cillators 631

4 2 N e u t r a l S t a b i l i ty f o r t h e F u ll S y s t e m

I f D = 0 , n e u t r a l l y - s t a b l e r o t a t i n g w a v e s o c c u r e v e n in t h e o r i g in a l

n o n l i n e a r s y s t e m ( 2 . 4 ) - (2 . 6 ) . M o r e o v e r , t h is f o r m o f n e u t r a l s t a b il i ty h o l d s

f o r a ll v a l u e s o f K . T h i s n e u t r a l s t a b i l it y i s t h e r e f o r e c o m p l e t e l y d i ff e r e n t

a n d m u c h s t r o n g e r t h a n t h a t a s s o ci a te d w i t h th e f u n d a m e n t a l m o d e , w h i c h

is b a s e d o n a l i n e a r a n a l y s is , a n d w h i c h o c c u r s o n l y f o r K < K c .

W e f i r s t e x p l a i n t h e n e u t r a l s t a b i l i t y p h y s i c a l l y , a n d t h e n m a t h e m a t i -

c a ll y . S u p p o s e t h a t D = 0 a n d c o n s i d e r a f a m i l y o f d e n s i t ie s p O , t , r t h a t

sa t isf ies

~

0 = f o e i ~ t , ~ o ) d O (4 .4)

f o r a l l 09, a t som e f ixe d t im e t. P hys i c a l ly , t h i s m e a n s tha t f o r e a c h

f r e q u e n c y c,, th e o s c i l l a to r s o f t h a t f r e q u e n c y m a k e z e r o c o n t r i b u t i o n t o t h e

o r d e r p a r a m e t e r ( 2. 6) . T h e n t h e c o h e r e n c e r = 0 , a n d s o v O , t, co) = co. In

o t h e r w o r d s , e a c h o s c i l la t o r m o v e s w i th i n s t a n t a n e o u s v e l o c i ty ~ , i n d e -

p e n d e n t o f i ts p o s i t i o n o n t h e c i rc le . H e n c e f o r e a c h 09, t h e a s s o c i a t e d

p O , t , ~ o )

w i ll s i m p l y r o t a t e r i g i d l y a r o u n d t h e c i r cl e a t f r e q u e n c y ~o. B u t

th i s me a ns tha t ( 4 .4 ) w i l l c on t inue to be sa t i s f i e d a t a l l l a t e r t ime s

[ A r o t a t e d v e r s i o n o f p s t il l s a ti sf ie s ( 4 . 4 ) .] N o t e t h a t t h e d e n s i t y f o r e a c h

~o is o b li v i o u s to a l l t h e o t h e r s - - t h e y r o t a t e a r o u n d a n d a r o u n d w i t h n o

i n t e r a c t i o n b e t w e e n t h e m . S i n c e t h e d e n s i t ie s a r e e f fe c t iv e l y u n c o u p l e d , t h e

sys t e m i s n e u t r a l ly s t a b l e t o p e r t u r b a t i o n s i n v o l v in g o n l y h i g h e r h a r m o n i c s ;

c h a n g i n g t h e s h a p e o f o n e o r a l l o f t h e d e n s i t ie s w i ll l e a d t o n o r e s t o r i n g

f o r c e s - - a s l o n g a s n o f i r s t h a r m o n i c s a r e i n t r o d u c e d .

T o p u t i t m o r e m a t h e m a t i c a l l y , f ix o3 a n d t a n d l et

S = { p O , t , co): (4.4) is sa t is f ied} (4.5)

T h e n S i s i n v a r i a n t u n d e r t h e f l o w ( 2 . 4 ) , a s i s e a s i l y c h e c k e d :

O - ~ f 2 ~ e i ~e i ~ ~ 7 d O

= - f ~ e i ~

= I ~ ~ i e ~ ~ d O

7z

= ia ~ ~ 0 e i ~ d O

= 0


20/23

632 Strogatz and M irol lo

I n t h is c h a i n o f e q u a t i o n s w e h a v e u s e d ( 2 .4 ) ; i n t e g r a t i o n b y p a r t s ; v = co

s ince r = 0 a t t im e t ; an d f ina l ly (4 .4) .

T h e i n v a r i a n c e o f S is i m p o r t a n t b e c a u s e i t i m p l ie s t h e t y p e o f n e u t r a l

s t a b i li t y c l a i m e d a b o v e . T h e f a c t t h a t S i s i n v a r i a n t m e a n s t h a t v = co f o r

al l t . T h e n t h e s o l u t i o n s o f ( 2. 4) a r e r o t a t i n g w a v e s o f t h e f o r m

p O - c o t , c o ) ,

whe r e p i s

a r b i t r a r y .

T h i s i s t h e s t r o n g e s t p o s s i b l e f o r m o f

n e u t r a l s t a b i li t y : t h e d y n a m i c s r e s t r i c t e d t o S a r e e f fe c t iv e l y u n c o u p l e d .

A n o t e w o r t h y c o n s e q u e n c e o f t h es e re s u lt s is t h a t w h e n D - - 0 , t h e r e

a r e i n f i n i t e ly m a n y d i f f e re n t s o l u t i o n s w i t h r = 0 f o r a l l t im e . I n c o n t r a s t ,

w h e n D > 0 , a l l t h e h i g h e r h a r m o n i c s d e c a y a s t ~ o o, a n d s o t h e o n l y

s o l u t i o n w i t h r = 0 f o r al l t i m e is t h e i n c o h e r e n t s o l u t i o n ( 2 .7 ) .

5 D I S C U S S I O N

5 1 O p e n P r o b l e m s

I n t h i s p a p e r w e h a v e a n a l y z e d t h e l i n e a r s ta b i l it y o f t h e i n c o h e r e n t

s o l u t i o n ( 2 .7 ) . O n e i m p o r t a n t c o n c l u s i o n is t h a t , i n t h e a b s e n c e o f n o i s e,

t h e i n c o h e r e n t s o l u t i o n i s u n s t a b l e f o r K > K c , b u t n e u t r a l l y s t a b l e f o r a l l

K < K c . I t w o u l d b e d e s i r a b l e t o e x t e n d t h i s l i n e a r s t a b i li t y a n a l y s is t o

i n c l u d e h i g h e r - o r d e r t e r m s . F o r K < K c , d o t h e n o n l i n e a r t e r m s s t a b i l i z e o r

d e s t a b i l i z e t h e i n c o h e r e n t s o l u t i o n , o r l e a v e i t n e u t r a l l y s t a b l e ?

A r e l a t e d q u e s t i o n c o n c e r n s t h e l o n g - t e r m b e h a v i o r o f

r t )

f o r K < K c .

A l t h o u g h t h e i n c o h e r e n t s o l u t i o n i s l i n e a r l y n e u t r a l l y s t a b l e , i t i s s t i l l

p o s s i b l e t h a t r t ) - - * 0 a s t ~ o a s p r e d i c t e d b y K u r a m o t o a n d

N i s h i k a w a . (t4 1 T o g i v e a s i m p l e e x a m p l e ( k i n d l y p o i n t e d o u t b y

Y . K u r a m o t o ) , s u p p o s e K = 0 , s o th a t t h e o s c i ll a to r s a r e u n c o u p l e d . I n th i s

c a se ou r ne u t r a l s t a b i l i t y r e su l t s ho ld t r i v i a l l y . A t t he s a me t ime , i t i s t r ue

t h a t r ( t ) --* 0 a s t ~ o o b e c a u s e o f t h e R i e m a n n - L e b e s g u e L e m m a . U n f o r -

t u n a t e l y , th e e x t e n s i o n o f t h is s i m p l e i d e a t o t he c a se K > 0 r e m a i n s

p r o b l e m a t i c a l .

A n o t h e r q u e s t i o n c o n c e r n s t h e s t a b i li ty o f t h e p a r ti a ll y s y n c h r o n i z e d

s t a t e (9'13'14) w h i c h b r a n c h e s o f f t h e i n c o h e r e n t s o l u t i o n a t K = K c . C a n o n e

s h o w t h a t t h is s o l u t i o n i s l o c a l l y s t a b l e f o r K > K c ? [ - The pa r t i a l l y syn -

c h r o n i z e d s t a t e c a n n o t b e g l o b a l l y s t a b l e , b e c a u s e t h e s u b s p a c e S d e f i n e d

b y ( 4 .5 ) is i n v a r i a n t f o r a ll K . ]

A s m e n t i o n e d i n th e I n t r o d u c t i o n , t h e r e is a w h o l e c la ss o f u n s o l v e d

p r o b l e m s c o n c e r n i n g f l u c t u a ti o n s i n t h e f i n it e - N s y s te m (1 .1 ). P r o m i s i n g

s t a r ts h a v e b e e n m a d e b y D aid o(~ O ~2) a n d b y K u r a m o t o a n d

N i s h i k a w a , (~3'j4~ b u t m a n y d i ff ic u lt q u e s t i o n s r e m a i n . F o r e x a m p l e ,

D a i d o (~ l'x 2) h a s s h o w n t h a t t h e f l u c t u a t i o n s a r e c h a r a c t e r i z e d b y t w o d i f-

f e r e n t c r i ti c a l e x p o n e n t s o n e i t h e r s id e o f t h e p h a s e t r a n s i t i o n a t K = K c . I s


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Coupled Os ci l lators 633

this peculiar behavior related to the unusual properties of the bifurcation

at Kc, in which the incoherent state changes from unstable to neutrally

stable?

5 .2 . S e l f C o n s i s t e n t M e t h o d v s . F o k k e r P l a n c k M e t h o d

Finally, we would like to comment on some of the novel features of

the analysis used in this paper. We have introduced a method for studying

the onset of synchronization in mean-field models of coupled oscillators.

The strategy is to linearize the relevant Fokker-Planck equation about the

incoherent state, and then analyze the resulting linear stability problem.

Until now, the only tool available for analyzing mean-field systems of

coupled oscillators has been the self-consistency approach pioneered by

Kuramoto. (8 9) That approach has the advantage that it allows one to

study both incoherent and partially synchronized solutions in one

stroke-- in a sense, the self-consistent method is global, whereas our

method is local.

On the other hand, the self-consistent method has some important

limitations. First, it is concerned only with steady-state behavior and it

therefore provides no information about stability. Second, it depends cru-

cially on the sinusoidal form of the coupling in the model (1.1). Because of

a convenient trigonometric identity, the order parameter (1.2) appears in

the governing equation (1.3)--it is this coincidence which allows the order

parameter to be determined self-consistently.

In contrast, our method can be applied to systems with more general

coupling than (1.1) or (2.2). For example, consider the system

K N

O i = c o i + ~ j ~ i = l , . .. ,S (5.1)

where f(q~) is an arbitrary 27r-periodic function. Equation (5.1) shares an

important qualitative feature with the sinusoidal model (1.1): both systems

are coupled only through

phase dif ferences.

This sort of coupling arises

naturally from averaging theory applied to a system of weakly coupled

limit cycle oscillators. (9)

Using the notation of Section 2, we see that the appropriate infinite-N

limit of (5.1) is

Op 8

pv) (5.2)

Ct 30

where the velocity is given by

J

O , t , o )) = ~ o + X f q S - O ) p ~ , t ,v ) d O g v ) d v (5.3)

0:3


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Coupled Os ci l lators 6 5

R E F E R E N C E S

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Cooperative Dynamics in Complex Physical Systems

H. Ta kay am a , ed . Spr i nge r , Be r l i n , 1989) .

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