Phase Locking - Period Doubling

8/8/2019 Phase Locking - Period Doubling

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J . Ma th . B iolog y (1 9 8 2 ) 1 4 :1 - 2 3J o u r n a l o f

Ma t h ema t i c a l

B i o l o g y9 Springer-Verlag 1982

Ph ase L ock ing, Period D oubl ing Bifurcat ions and Chaos in a

M athem at ica l M odel o f a Per iodica lly Driven Osci lla tor: A

Theo ry for the Entrainmen t of Biolog ical O sci l lators and the

Generat ion of Cardiac Dysrhythmias

M i c h a e l R . G u e v a r a a n d L e o n G l a s s

Dep ar tmen t o f Phy siolog y , 3 6 5 5 D r u m m o n d S tr ee t, McG i l l Un iv e r s i ty , Mo n t r ea l, Qu eb ec H3 G 1 Y6 ,

C a n a d a

A b s t r a c t . A m a t h e m a t i c a l m o d e l f o r t h e p e r t u r b a t i o n o f a b i o lo g i c al o s c il la t o r

b y s i n g le a n d p e r i o d i c i m p u l s e s i s a n a l y z e d . I n r e s p o n s e t o a s i n gl e s t i m u l u s t h e

p h a s e o f t h e o s c i l l a t o r is c h a n g e d . I f th e n e w p h a s e f o l l o w i n g a s t i m u l u s i s

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

c u r v e o r P T C ( P a v l id i s , 1 9 73 ). T h e r e a r e t w o q u a l i t a t i v e l y d i f f e r e n t ty p e s o f

p h a s e r e s e t ti n g . U s i n g t h e t e r m i n o l o g y o f W i n f r e e ( 19 77 , 1 98 0) , la r g e p e r -

t u r b a t i o n s g i ve a t y p e 0 P T C ( a v e ra g e s l o p e o f t h e P T C e q u a l s ze r o ), w h e r e a s

s m a l l p e r t u r b a t i o n s g i v e a t y p e 1 P T C . T h e e f f ec t s o f p e r io d i c i n p u t s c a n b e

a n a l y z e d b y u s i n g t h e P T C t o c o n s t r u c t t h e P o i n c a r 6 o r p h a s e a d v a n c e m a p .

O v e r a l i m i t e d r a n g e o f s t im u l a t i o n f r e q u e n c y a n d a m p l i t u d e , t h e P o i n c a r 6 m a p

c a n b e r e d u c e d t o a n i n t e r v a l m a p p o s s e ss i n g a s in g le m a x i m u m . O v e r t h is r a n g e

t h e re a r e p e r i o d d o u b l i n g b i f u r c a t io n s a s w e ll as c h a o t i c d y n a m i c s . N u m e r i c a l

a n d a n a l y t i c a l s t u d ie s o f t h e P o i n c a r6 m a p s h o w t h a t b o t h p h a s e l o c k e d a n d

n o n - p h a s e l o c k e d d y n a m i c s o c cu r . W e p r o p o s e t h a t c a r d ia c d y s r h y t h m i a s m a y

a r is e f r o m d e s y n c h r o n i z a t i o n o f t w o o r m o r e s p o n t a n e o u s l y o s c i ll a ti n g r e g io n s

o f t h e h e a r t. T h i s h y p o t h e s i s s e r ve s t o a c c o u n t f o r t h e v a r i o u s f o r m s o f a tr i o -v e n t r i c u l a r ( A V ) b l o c k c l i n ic a l ly o b s e r v e d . I n p a r t i c u l a r 2 : 2 a n d 4 : 2 A V b l o c k

c a n a r i s e b y p e r i o d d o u b l i n g b i f u r c a t i o n s , a n d i n t e r m i t t e n t o r v a r i a b l e A V

b l o c k m a y b e d u e t o t h e c o m p l e x i r r e g u l a r b e h a v i o r a s s o c i a t e d w i t h c h a o t i c

d y n a m i c s .

K e y w o r d s : C a r d i a c d y s r h y t h m i a s - P h a s e l o c k i n g - C h a o s - P e r i o d

d o u b l i n g b i f u r c a t i o n s

I . I n t r o d u c t i o n

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

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

n e u r a l a n d c a r d i a c o s c i l la t o rs h a v e r e v e a l ed t h a t t h e i n tr i n si c r h y t h m m a y b e c o m e

e n t r a i n e d o r p h a s e l o c k e d t o t h e p e r i o d i c s t i m u l u s ( P e r k e l e t a l . , 1 9 6 4 ;

0303 - 68 12/82/0 014/00 01/$04 .60



Phase Locking, Period Dou bling Bifurcations and Chaos 3

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

e x p e r i m e n t a l o b s e r v a t i o n s , b o t h i n r e g a r d t o t h e r e s p o n s e t o a s in g l e s t i m u l u s a n d i n

r e g a r d t o t h e r e s p o n s e t o p e r i o d i c s t i m u l a t i o n . W h e n a p e r i o d i c s ti m u l u s i s a p p l i e d ,

t h e m o d e l d i s p l a y s p h a s e l o c k e d p a t t e r n s a s w e l l a s i r r e g u l a r , n o n - p h a s e l o c k e d

b e h a v i o u r t h a t a r i se s o u t o f c h a o t ic d y n a m i c s .

S i n ce o u r w o r k m a y b e o f i n t e r e s t t o e x p e r i m e n t a l i st s , w e h a v e w r i t t e n t h i s p a p e r

s o t h a t t h e r e s u lt s c a n b e a p p r e c i a t e d w i t h o u t u n d e r s t a n d i n g t h e t e c h n i c a l d e t a il s .

S e c t io n I I i s a g e n e ra l p r e s e n t a t io n o f t h e m a t h e m a t i c a l m o d e l a n d t h e p r i n c ip a l

r e s u lt s . S e c t i o n s I I I , I V a n d V a r e t e c h n i c a l a n d d i r e c t e d t o w a r d s t h e o r e t i c i a n s a n d

s h o w h o w t h e r e s u l t s w e r e o b t a i n e d , S e c t i o n V I is a n o n - t e c h n i c a l d i s c u s s i o n o f t h e

r e l e v a n c e o f t h is t h e o r e t i c a l w o r k t o e x p e r i m e n t a l a n d c l i n i ca l o b s e r v a t i o n s . T h e

A p p e n d ic e s c o n t a i n p r o o f s a n d c o m p u t a t i o n s .

I I . T h e M a t h e m a t i c a l M o d e l a n d a n O v e r v i e w o f t h e R e s u l t s

I n t h i s s e c t i o n w e d e s c r i b e t h e m a t h e m a t i c a l m o d e l a n d i t s l i m i t a t i o n s , a n d t h e

r e s p o n s e o f t h e m o d e l t o a s i n g le i s o l a t e d s t i m u l u s . N e x t w e il l u s t r a te t h e d i f f e r e n t

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

s t i m u l a t i o n .

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

prev ious ly been d i s cus sed by W inf re e (1975 , 1980) an d Sc o t t (1979). T he m od e l is

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

d ~- 2~z,

dt

dr (1 )- - = a r ( 1 - r ) ,

dt

w h e r e ~0 i s t h e a n g u l a r c o o r d i n a t e ( - ~ < # < o o) , r i s t h e r a d i a l c o o r d i n a t e a n d a

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

a t t r a c t i n g f o r a l l i n i t i a l c o n d i t i o n s e x c e p t f o r t h e e q u i l i b r i u m p o i n t a t t h e o r i g i n .

N o t e t h a t t h e u n p e r t u r b e d o s c i l l a t o r h a s u n i t p e r i o d a n d i t s s t a t e c a n b e

p a r a m e t r i z e d b y a n a n g u l a r c o o r d i n a t e ~ b :

4 = G ( m o d 1 ). ( 2)

Th e var iable ~b is ca l led th e p ha se o f the osc i l la t ion (0 ~< ~b < 1).

P e r t u r b a t i o n a w a y f r o m t h e l i m i t c y c l e r e s u l ts in r e l a x a t i o n b a c k t o t h e s t a b l e

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

o f a - * G o. I n t h i s ca s e, f o l l o w i n g p e r t u r b a t i o n a w a y f r o m t h e l i m i t c y cl e , t h e r e i s a n

i n s t a n t a n e o u s r e l a x a t i o n b a c k t o t h e l i m i t c yc le a lo n g a r a d i a l d i r e ct i o n . I n w h a t

f o l lo w s w e s h a l l c o n s i d e r p e r t u r b a t i o n s c o n s i st i n g o f i m p u ls e s o f m a g n i t u d e b w h i c h

a re d i rec ted pa ra l l e l to the x -ax i s (F ig . 1 ). Th us , the e f fec t o f a s ing le im pulse i s to

i n s t a n t a n e o u s l y r e s e t t h e p h a s e o f t h e o s c i ll a t o r . C a l l i n g q9 t h e o l d p h a s e o f t h e

o s c i ll a ti o n im m e d i a t e l y p r e c e e d in g t h e p e r t u r b a t i o n , a n d 0 th e n e w p h a s e o f t h e

o s c i ll a ti o n i m m e d i a t e l y f o l l o w i n g th e p e r t u r b a t i o n w e h a v e

0 = f ( ~ , b ) , ( 3)



4 M . R . G u e v a r a a n d L . G l a s s

t ~ - o

F i g . l . Sc h e m a t i c r e p r e s e n t a t i o n o f t h e e f f e c t o f

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

m o d e l o s c i l l a t o r . T h e u n i t c i r c l e fo r m s a l i m i t c y cl e

w h i c h i s g l o b a l l y a t t r a c t i n g f o r a l l p o i n t s e x c e p t t h e

o r ig i n . T h e p e r t u r b a t i o n i n s t a n t a n e o u s l y re s e ts t h e

p h a s e o f t h e o s c i l l a t o r f r o m p h a s e q5 p r i o r t o t h e

p e r t u r b a t i o n t o p h a s e 0 a f t e r th e p e r t u r b a t i o n . E v e r y

t i m e t h e o s c i l l a t o r p a s s e s t h r o u g h p h a s e q~ = 0 , w e

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

n e u r a l o r c a r d i a c a c t i o n p o t e n t i a l . I d e n t i f y i n g t h e x -

a x is v a r i a b l e w i t h m e m b r a n e p o t e n t ia l , p e r t u r b a t i o n s

d i r e c t e d a l o n g t h e x - a x i s ( b > 0 ) a r e a n a l o g o u s t o

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

o p p o s i t e d i r e c t i o n ( b < 0 ) a r e a n a l o g o u s t o h y p e r -

p o l a r i z a t i o n s

where the funct ionfi s called the phase transition curve (PTC). In Section III of this

paper we analytically compute the PTC for this model and discuss its properties.

Our choice of the model in Fig. 1 has been motivated by the fact that an analytic

expression can be found for the PTC. This is in sharp contrast to other simple

models of limit cycle oscillations such as the van der Pol oscillator for which

analytical expressions for the PTC are not available (Pavlidis, 1973; Winfree, 1980).

In addit ion to its analytic simplicity, the PTC of (1) displays certain qualitative

features which resemble some experimental observations. Consider a perturbation

directed along the positive x-axis (b > 0). Applying the per turbation during the firsthalf of the cycle (0 < q5 < 0.5) leads to a delay in the phase of the oscillation,

whereas application of the same perturbation in the second half of the cycle

(0.5 < q5 < 1.0) leads to an advance of phase. Also, for this model, the average slope

of the PTC is 1 (type 1 PTC) at low amplitudes of perturbation and 0 (type 0 PTC) at

higher amplitudes. Winfree (1977, 1980) reviewed data resulting from different

experimental preparations and showed that in many situations the PTC's are

biphasic and either type 1 or type 0.

Despite these parallels with experiment, the mathematical model is unrealistic

for many reasons. Since most realistic mathematical models for biological

oscillations are systems of differential equations of dimension greater than 2, thetopological dimension of the proposed system of equations is too low.

Furthermore, it is unrealistic to assume that d~b/d t is independent of r, and that the

relaxation back to the limit cycle is instantaneous. Finally, due to the rotational

symmetry inherent in the model, the PTC shows symmetries (see Section III) which

are no t observed experimentally (Pavlidis, 1973 ; Jalife and M oe, 1979; Scott, 1979;

Guevara et al., 1981). Possible changes in the behaviour of the system which arise

from relaxing one or more of our assumptions to make them more realistic have not

yet been studied.The PTC describes the response of the oscillator to an isolated impulse. The

PTC can be used to predict the response to periodic stimulation. In Section IV weshow how the PTC can be used to derive a mathematical function called the

Poincar6 map. Iteration of the Poincar6 map allows us to study the dynamics of the

model in response to periodic input. Although equivalent procedures have been

previously employed (Perkel et al., 1964; Moe et al., 1977; Pinsker, 1977; Glass



Phase Locking , Per iod Dou bl ing Bi furcat ions and Chaos 5

an d M ack ey , 1979 ; Sco t t , 1979; 1980 ; Lev i , 1981) , t he re ha s no t been a tho ro ug h

a n a l y s is o f t h e d i f f e r e n t p h a s e l o c k i n g b e h a v io r s d i s p l a y e d b y a m a t h e m a t i c a l

m o d e l d i s p l ay i n g b o t h t y p e 0 a n d t y p e 1 P T C ' s .

T h e m a j o r r e s u l t s o f t h i s p a p e r a r e g i v e n i n F i g . 2 w h i c h s h o w s t h e p r i n c i p a l

p h a s e l o c k i n g zo n e s a s a f u n c t i o n o f th e m a g n i t u d e b o f t h e p e r t u r b a t i o n a n d t h e

t i m e r b e t w e e n s u c c e s s iv e s t i m u l i . F i g u r e 2 a s h o w s t h e z o n e s o f N : M p h a s e l o c k i n g

fo r N ~> M , N ~ 3 . F ig u re 2b show s so m e o f the se zon es a s w e l l a s a l l t he 4 : M

l o c k i n g z o n e s o v e r a m o r e l i m i t e d r e g i o n o f ( r , b ) p a r a m e t e r s p a c e. I n t h e u n l a b e l l e d

r e g i o n s o f F i g . 2 , t h e r e a r e o t h e r p h a s e l o c k e d z o n e s , a s w e l l a s p o i n t s a t w h i c h p h a s e

l o c k e d d y n a m i c s d o e s n o t o c c u r ( se e S e c t i o n V ).

2 . 5

2 - 0

1 .5

I b l

I -0

0 - 5

( a )

I:0

i0 - 2 5"0 0 - 5 0

l"

I:1

-2:2

i

0 . 7 5 1 . 0

Fig . 2 . Phase locking zones resu l t ing f romperiodic pulsat i le inputs of magn i tude band f requency r - i . The areas not l abelledcon t a in bo t h phas e l ocked and non -phas elocked dynam ics (see Sect ion V) . (a) Phaselocking zones of the fo rm N ~> M , N ~< 3 .

(b) Phase locking zones of the formN i> M, N ~< 4 ove r a mo re l imi ted reg ionof (z , b) parameter space. Us ing the sym -met ry re la t ions in Sect ion IVB, phaselocking pat terns for o the r values of z canbe generated

1.4

1.3

2:1 ~ I:1

]bll'21.1 4:2 _ ~ ~ , ~ , , , ~ ~:2 43

1.00 - 5 5 0 -6 0 0 ' 6 5 0 ' 7 0 0 ' 7 5

1"



6 M . R . G u e v a r a a n d L . G l a ss

In o rde r to i l lus t r a te the d i f f e ren t phase lock ing pa t te rns , we a ssume tha t

crossing of the po si t ive x-axis (i .e . ~b passing t hro ug h 0) corre spo nd s to a n

obse rvab le even t ( fo r exam ple , an ac t ion po ten t ia l) . In F igs. 3 and 4 we show som e

o f t h e d i f fe r e n t c o u p l in g p a t t e rn s b e tw e e n th e p e r io d ic p e r tu r b a t i o n a n d t h e mo d e l

osc i l la to r a t seve ra l d i f f e ren t va lues o f f r equen cy and am pl i tude o f the pe r -

tu rba t ion . The pe r iod ic inpu t pu lses a re r epresen ted in F igs . 3 and 4 by heavy da rk

l ines and the t im es when the m ode l osc i lla to r passes th roug h the phase q5 = 0 ( ca lled

the f i r ing t imes) a re r epresen ted by l igh te r , shor te r l ines. A l l the pa t te rns show n a re

phase lock ed excep t fo r those in F ig . 3b and Fig . 4d . The r hy thm in Fig . 3b is an

exam ple o f quas ipe r iod ic dyn am ics (Glass e t a l. , 1980), and the rhy th m in Fig . 4d i s

a n e x a mp le o f a n i r re g u l a r p a t t e r n a r is in g o u t o f c h a o t i c d y n a m ic s (s e e Se c t io n V ) .

Th ere is a s t r ik ing qua l i ta t ive s imi la r i ty be twe en the pa t te rns obse rved in F igs . 3

a n d 4 a n d c o u p l in g p a t te r n s o b s e r v e d b e tw e e n p e rio d i c i n p u t a n d d r iv e n a c ti v it y i nexp er im enta l pre pa ra t io ns (P erkel e t a l. , 1964; Reid, 1969; Pinsk er , 1977; Ja l i fe an d

M oe , 1979; G ut tm an e t al ., 1980). M an y of the pa t te rn s in F igs. 3 and 4 r e semb le

c l in ica l ly obse rved c a rd iac a r rhy th m ias such a s AV b lock ( see Sec t ion VI) .

(a )

h h I , h L I ,0:0 3 :0

(r

t h I , I i i , l L I0 . 0 3: 0

( b )

I , i h i , I , I , I L i , h I , I I L I I I i , i6:0 0:0 3:0 6 : 0 9 : 0 12.0

(d)

I h I , I i o l o l , I I I , I I I , I I h i I I i6 : 0 9 : 0 - 3 : 0 6 : 0 9 : 0

T i m e

F i g . 3 . S c h e m a t i c r e p r e s e n t a t i o n o f t h e e f f e c ts o f p e r io d i c s t i m u l a t i o n o n t h e m o d e l o s c i l la t o r f o r

b = 0 .9 5 . T h e h e a v y d a r k b a r s h o w s th e p e r i o d i c p u ls a t il e i n p u t a n d t h e s h o r t e r l ig h t b a r s h o w s t h e f ir i n g

t i m e s ( t i m e w h e n o s c i l l a t o r p a s s e s t h r o u g h p h a s e ~b = 0 ) o f t h e m o d e l o s c i l l a t o r . ( a) ~ = 0 . 9 0, 1 : 1 p h a s e

l o c k i n g ; ( b ) T = 0 . 7 5, q u a s i p e r i o d i c d y n a m i c s ; (c ) z = 0 . 7 0, 4 : 3 p h a s e l o c k i n g ; ( d ) z = 0 . 6 5, 3 : 2 p h a s e

l o c k i ng . T h e p a t t e r n s o f ( c) a n d ( d ) d i sp l a y W e n c k e b a c h p e r i o d i c it y , s in c e t h e i n te r v a l b e t w e e n s t i m u l u s

a n d s u c c e e d i n g re s p o n s e g e t s p r o g re s s i v e ly l o n g e r u n t i l t h e d r i v e n o s c i l l a to r s k i p s o r m i s s e s a b e a t

(a) Ibl (e)

h I , h I , I , I , I , [ , I h I , h I i i , I , i~ I , I i h I I h i l , i , i , l h0:0 3 0 6:0 0:0 3:0 6:0 0-0 3:0 6:0

Id)

] ! l I I h l I , I , I h I i ~ L h l k I , i I , i I , I h I h I h I h l i ,0.0 3:0 6:0 9:0 IP.O 15.0 18.0

t e l ( f ) (g )

I , l I , I h L i , I I , L I , I I I I , I I I h l t I , l i , I t , l i , i L I I ,0:0 3:0 6:0 0.0 3:0 6:0 0 0 3:0 6.0

Time

Fi g . 4 . Sa m e as F i g . 3 bu t w i t h b = 1 .13 . ( a ) z = 0 .75 , 1 : 1 pha se l ock i ng ; (b ) ~ = 0 .69 , 2 : 2 ph as e l ock i n g :n o t e t h e a l t e r n a t i o n o f f i r i n g t i m e s ; ( c) 9 = 0 . 6 8, 4 : 4 p h a s e l o c k i n g : n o t e t h a t t h e r e a r e f o u r d i f f e r e n t

f i r in g t im e s ; ( d ) ~ = 0 .6 5 , i rr e g u l a r c o u p l i n g a r is ! n g f r o m c h a o t i c d y n a m i c s : n o t e t h e n a r r o w r a n g e o f

f i r in g t im e s a n d t h e i r r e g u l a r l y s k i p p e d b e a t s o f t h e d r i v e n o s c i l l a t o r ; ( e) ~ = 0 . 6 07 , 4 : 3 p h a s e l o c k i n g :

n o t e t h e a t y p i c a l W e n c k e b a c h p e r i o d i c it y ; (f ) z = 0 .6 0 , 4 : 2 p h a s e l o c k i n g : n o t e a g a i n t h e a l t e r n a t i o n o f

f i r in g t i m e s ; ( g ) ~ = 0 . 55 , 2 : 1 p h a s e l o c k i n g



P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u rc a t io n s a n d C h a o s 7

I l L T h e P h a s e T r a n s i t io n C u r v e ( P T C ) a n d I ts P r o p e r ti e s

T h e P T C g iv e s t h e n e w p h a s e 0 a s a fu n c t i o n o f t h e o l d p h a s e 4 ', f o l l o w i n g a

p e r t u r b a t i o n o f a m p l i t u d e b (F i g . 1 ). T h e c o m p u t a t i o n s w h i c h f o l l o w r e q u i r ea n a l y t ic a l e x p r e s s i o n s f o r t h e P T C . T h e P T C is w r i t te n u s i n g t h e p r in c i p a l v a l u e s o f

t h e i n v e rs e t a n g e n t a n d i n v e rs e c o s i n e f u n c t i o n s , d e n o t e d b y t a n - 1 x a n d c o s - 1 x

r e s p e c t i v e l y . F o r - ~ < x < ~ , - z ~ / 2 < t a n l x < ~ / 2 a n d f o r - 1 < x < 1 ,

0 < c o s - i x < z . F r o m t h e c o n s t r u c t i o n i n F ig . 1 t h e P T C is r e a d i ly c o m p u t e d .

D e f i n e

( s i nZ ~ z4 ' ) ( 4a )= z ~ z t a n - l \ c ~ o s 2 ~ - b '

]~ = l c o s - l ( - b ). ( 4 b )2 ~

W e f i n d

f( 4 , , b) = ~, 0 ~ 4, ~</?, ]b[ ~< 1, (5 a)

0 ~< 4, ~< 0.5, b > l ,

f( 4, , b) = 0.5 + c~, f l < 4 ' < 1 - / ~ , Ibl ~< 1, (5 b)

0 - . .< 4 ' < 1 , b < - 1 ,

f (4 ' , b) = 1.0 + c~, 1 - fl ~< 4' < 1, Ibl ~< 1, (5 c)

0 . 5 < 4 ' < 1 , b > l .

F i g u r e 5 s h o w s e x a m p l e s o f t y p e 1 P T C s ( w h i c h a r e f o u n d f o r Ib[ < 1 ) a n d t y p e 0

P T C s ( w h i c h a r e f o u n d f o r [bl > 1 ). T h e P T C s a r e c o n t i n u o u s o n t h e u n i t c ir c le a n d

h a v e c o n t i n u o u s d e r i v a ti v e s o f al l o r d e r s o n t h e u n i t c ir cle . M a t h e m a t i c a l m o d e l s

d i s p l a y i n g P T C s w h i c h a r e d i s c o n t i n u o u s o n t h e u n i t c i r c l e h a v e b e e n d e s c r i b e d

( G l a s s a n d M a c k e y , i 9 7 9 ; K e e n e r , 1 9 8 0 ; K e e n e r e t a l. , 1 9 8 1) .

T h e P T C s g i v en b y ( 5) a n d s h o w n i n F i g . 5, d i s p l a y th e f o l lo w i n g s y m m e t r ie s

f ( 1 - 4 ,, b ) = 1 - f ( 4 , , b ) , ( 6 a )f ( 4 , + 0 . 5 , - b ) = f ( 4 , , b ) + 0 . 5 , 4 , ~< 0 . 5 , ( 6 b )

f (4 , - 0 . 5 , - b ) = f ( 4 , , b ) - 0 . 5 , 4 ) ~> 0 . 5 . (6c )

F i g . 5. T h e p h a s e t r a n s i t i o n c u r v e ( P T C ) o f t h e m o d e l o s c i ll a t o r

f o r s e v e r a l v a l u e s o f b. F o r Ib < 1 t h e P T C h a s a v e r a g e s l o p e 1

a n d i s c a l l e d a t y p e 1 P T C , w h e r e a s f o r Ib[ > 1 t h e P T C h a s

a v e r a g e s l o p e 0 a n d i s c a l l e d a t y p e 0 P T C

I'O

0 0"5

0 . 0

b . I ' I~ '~

0 . 5 1 . 0



8 M . R . G u e v a r a a n d L . G l a s s

T h e s e s y m m e t r y r e l a t i o n s i n th e P T C w i ll b e u s e d l a te r t o d e r i v e s y m m e t r i e s i n

t h e p h a s e l o c k i n g z o n e s .

T h e s l o p e o f t h e P T C i s g i v e n b y

, ~3 f 1 + b co s 2 n ~b( 7 )

Oq5 1 + b 2 + 2b c o s 2~zq5

F o r Ib] < 1 , t h e P T C is a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o n t h e i n t e r v a l

( 0 , 1 ). F o r I b l > 1 t h e r e i s a u n i q u e l o c a l m i n i m u m a t ~bm in a n d a u n i q u e l o c a l

m a x i m u m a t q~ma~.

D e f i n e

1 1# - -- ~ - co s - 1 - -}bl' ( 8 a )

v = I t a n - i ( b Z - 1 ) - 1 /2 ( 8 b )21 r

F o r b > l , w e c o m p u t e

a n d f o r b < - I

~ m i n = 0 . 5 + ]A , f ( q ~ m i n , b ) = 1 - v , ( 9 a )

r x = 0 . 5 - - ] ./ , f ( q b . . . . b ) = v , ( 9 b )

~bmin:

]2 , f ( ~b min , b ) = 0 .5 - v ,

qSm,x = 1 - - # , f ( ~ b . . . b ) = 0 . 5 + v.

T h e s e f o r m u l a e w i ll b e u s e d i n t h e l a t e r a n a l y si s .

( 9 c )

( 9 d )

I V . T h e P o i n c a r 6 M a p a n d P h a s e L o c k i n g

I n th i s s e c ti o n w e u s e th e P T C t o c o n s t r u c t t h e P o i n c a r ~ m a p i n o r d e r t o i n v e s t i g a t e

t h e r e s p o n s e o f t h e m o d e l o s c i l l a t o r t o p e r i o d i c s t i m u l a t i o n . T h e l i m i t a - -, ~ i n ( 1)

w a s t a k e n i n o r d e r to r e d u c e th e P o i n c a r~ m a p t o a o n e d i m e n s i o n a l m a p .

I V A . F i xed P o in t s, P h a s e L o ck i n g , a n d th e R o t a ti o n N u m b e r

C a l l ~ bi- 1 t h e p h a s e i m m e d i a t e l y p r e c e e d i n g t h e ( i - 1 ) st s t i m u l u s , a n d ~ t h e t i m e

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

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

A ~ i , w h e r e

Aq ~i = f ( ~b i_ 1 ,b ) + z - ~b i - 1 ( 1 0 )

a n d f ( q S i _ t , b ) i s d e f i n e d b y ( 5) . C o n s e q u e n t l y , t h e p h a s e i m m e d i a t e l y p r e c e e d i n g

t h e i t h s t i m u l u s i s

q~, = f ( ~ b , - 1 , b ) + z ( r o o d 1 ). ( 1 1 )

T h e m a p

T : ~b i_ 1 ~ 4 h ( 1 2 )



Phase Lock ing, Period Dou bling Bifurcations and Chaos 9

Fig. 6. The Poincar6 map for b = - 1.30 and threevalues of

1.0

( • L + I0 ' 5

0 .0

c> /

0 . 5 1 .0

d e f i n e d b y ( 1 1 ) is c a l le d t h e P o i n c a r 6 o r p h a s e a d v a n c e m a p . F i g u r e 6 sh o w s t h e

P o i n c a r 6 m a p f o r b = - 1 .3 0 a n d f o r t h r e e v a l u e s o f r . T h i s f i g u r e i l lu s t r a t e s t h a t

v e r ti c a l t r a n s l a t i o n o f t h e P T C b y a n a m o u n t r g iv e s t h e P o i n c a r 6 m a p .

I t e r a ti o n o f th e P o i n c a r 6 m a p c a n b e u s e d t o c o m p u t e t h e e v o l u t io n o f t h e p h a s e

i n r e s p o n s e t o p e r i o d i c i n p u t ( A r n o l d , 1 9 73 ; G l a s s a n d M a c k e y , 1 9 79 ;

G u c k e n h e i m e r , 1 9 8 0 ; L e v i , 1 9 8 1 )

= T ( 4 ' 0 ,

~b,+m = T (q ~ , +z -1 )= T ' (~b , ) . ( 13 )

A p h a s e 0 * i s c a l le d a f ix e d p o i n t o f p e r i o d N i f

TN(dP*) = q~*' (14 )

T'(q~*) # q~*, 1 < i < N .

A f i x e d p o i n t 4 )* o f p e r i o d N is s t a b le i f

0TN(qS')~?~, *,=** < 1. (1 5 )

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

n u m b e r , O , d e f i n e d a s t h e a v e r a g e a d v a n c e i n p h a s e o f th e d r i v e n o s c i l l a to r f o r e a c hc y c le o f t h e p e r io d i c i n p u t . T h e r o t a t i o n n u m b e r is d e f i n e d

1 s

p = l im ~ aqS,, (16)J~azJ i=l

w he re A~b, i s g ive n in (10) .

S t a b l e f i x e d p o i n t s o n t h e P o i n c a r 6 m a p a r e a s s o c i a t e d w i t h p h a s e l o c k e d

d y n a m i c s . A s s u m e t h a t f o r s o m e v a l u e o f (7 , b ), t h e r e a r e N s t a b l e f ix e d p o i n t s o f

pe r i od N , q~o ,01 . . . . , q~N = q~o w he re ~bi+l = T (O i ) a s b e f o r e . D e f i n e

N

M = ~ A49,. (17), =1

Si nce ~bN = ~bo , M m us t be an i n t eg e r an d t he s equ enc e is a s soc i a t ed w i t h N : M

p h a s e l o c k i n g w i t h r o t a t i o n n u m b e r p = M / N . C o n s e q u e n t l y , f o r p h a s e l o c k e d



10 M .R . Gu ev a r a an d L . Gla ss

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

p h a s e l o c k i n g p a t t e r n . F o r e x a m p l e , b o t h 1 : 1 a n d 2 : 2 l o c k i n g h a v e p - - 1 a n d b o t h

2 : 1 a n d 4 : 2 p h a s e l o c k i n g h a v e p = 89 A l s o , f o r f i x e d v a l u e s o f b a n d z , p m a y n o t

b e u n i q u e a n d m a y d e p e n d o n t h e in i t i a l p h a s e .

I V B . S y m m e t r i e s o f t he P h a s e L o c k i n g Z o n e s

A s a c o n s e q u e n c e o f t h e s y m m e t r i e s o f t h e P T C d e t a i le d i n (6 ), t h e r e a r e s y m m e t r i e s

in th e p h a s e l o c k in g z o n e s . T h e s e s y m m e t r i e s a r e s u m m a r i z e d b y t h e f o l lo w i n g

p r o p o s i t i o n s , t h e f i r s t o f w h i c h is p r o v e n i n A p p e n d i x A .

P r o p o s i t i o n 1 . A s s u m e t h a t t h e re i s a s t a b le p e r i o d N c y c le w i th f i x e d p o i n t s

~bo , ~b1 . . . , ~ bN - i f o r b = Y , z = 0 .5 - 6 ( 0 < 6 < 0 .5 ) a s s o ci a te d w it h a n N : M p h a s el o c k in g p a t t e r n . T h e n , f o r z = 0 .5 + 3 , t here w i l l be a s t ab l e cyc l e o f per iod N

a s s o ci a te d w i th a n N : N - M p h a s e lo c k in g r a ti o. T h e N f i x e d p o i n t s

~o, ~ i , . . . , ~bN- i o f th i s cy c le are given by

qh = 1 - ~bi. (1 8)

Proposition 2 . S u p p o s e t h a t f o r b = 7 , z = 6 (0 < 6 < 1) there i s a s table cyc le o f

p e r i o d N w i t h f i x e d p o i n t s (a o, ~ i , . . . , ~ n - i a s s o c i a t e d w i t h a n N : M p h a s e l o c k in g

pa t t e rn . Th en , f or b = - 7 , z = 3 , t here w i l l be a st ab l e cyc l e o f per iod N as soc ia t ed

w i t h a n N : M p h a s e l o c k in g p a t t e r n w i t h N f i x e d p o i n t s ~ b o, ~ bi . . . . , ~ n - 1 w here

~ i = q~i + 0 .5 ,

~ i = q~i - 0 .5 ,

0 < qSi < 0. 5,

0. 5 ~< qSi < 1.0 .( 1 9 )

Proposition 3 . S u p p o s e t he r e i s a s t a b le c y c le o f p e r i o d N w i t h f i x e d p o i n t s

(ao, ~ 1 , . . . , ~ N - i f o r b = 7 , z = 6 (0 < 6 < 1) a s s o ci a te d w it h a n N : M p h a s e l o ck i ng

pa t t e rn . Then t here w i l l be a st ab l e cyc le o f per io d N w i th f i x ed p o in t s

q~ o, ~ b l , . . . , ~ N - i fo r b = 7 , z = 6 + K (where K i s any pos i t i ve in teger) ass ociate d

w i t h a n N : M + K N p h a s e l o c k in g r a t io .

I V C . T h e P o in c a rb M a p a s a n In t e rv a l M a p w i th a S in g le M a x i m u m

O v e r c e r t a in r e g i o n s o f t h e (z , b ) p a r a m e t e r s p a c e , th e P o i n c a r 6 m a p r e d u c e s t o a

m a p d e f in e d o n a n i n t e rv a l in w h i c h t h e r e is a s i ng le m a x i m u m o f t h e m a p . I n t e r v a l

m a p s w i t h a s in g le m a x i m u m h a v e b e e n t h e f o c u s o f c o n s i d e r a b le a n a l y s i s r e ce n t ly

( L i a n d Y o r k e , 1 9 7 5; M a y , 1 9 7 6 ; G u c k e n h e i m e r , 1 9 77 ; g t e f a n , 1 9 7 7) . T h e

b o u n d a r i e s o f t h e p h a s e l o c k i n g z o n e s s h o w n i n F i g . 2 c a n b e p a r t i a l l y u n d e r s t o o d

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

T h e r e d u c t i o n o f t h e P o i n c a r 6 m a p t o a n in t e r v a l m a p w i t h a s in g le m a x i m u m

c a n b e i l lu s t r a te d b y c o n s i d e r i n g a n e x a m p l e . F i g u r e 6 s h o w s t h e P o i n c a r 6 m a p f o r= 0 . 3 5 , b = - 1 .3 0 . F o r t h i s s i t u a t i o n , T : [ 0 , 1 ] ~ I -0 . 71 0 3 , 0 . 9 8 9 7 ] a n d i n t h i s

i n t e r v a l T h a s a s in g l e m a x i m u m ( F i g . 7 ). I n g e n e r a l , f o r b < - 1 a n d 0 ~< z ~< 0 .5

t h e f ir s t i t e ra t e o f t h e P o i n c a r 6 m a p w i ll b e a n i n v a r i a n t i n t e r v a l m a p w i t h a s in g l e

m a x i m u m p r o v i d e d t h e f o l lo w i n g i n eq u a l i t i e s h o l d :



P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t i o n s a n d C h a o s 11

I '0

0 " 9

C j~ L + I

0 - 8

0.7

0 . 7 0 ' 8 0 " 9 I -0

4 , ,

F i g . 7 . T h e P o i n c a r 6 m a p f o r ~ = 0 . 3 5 a n d

t h r e e v a l u e s o f b . F o r e a c h v a l u e o f b al li n i ti a l p h a s e s b e t w e e n 0 a n d 1 w i ll m a p

i n t o t h e i n v a r i a n t r e g i o n s h o w n f o l l o w i ng

o n e i te r a t io n o f t h e P o i n c a r 6 m a p . F o r

e a s e o f p r e s e n t a t i o n , q ~i+ l i s n o t g i v e n

m o d u l o 1

Z -5

2" 0

1.5

I b l

I- 0

0 . 5

i i

0 " 0 0 " 2 5 0 " 5 0 0 " 7 5 I ' 0

T

F i g . 8 . R e p r e s e n t a t i o n o f th e t h r e e r e g i o n s i n w h i c h q u a l i t a t iv e l y d i f f e r en t d y n a m i c s o c c u r . I n t h e

d i a g o n a l ly s t r i p ed a r e a t h e P o i n c a r 6 m a p r e d u c e s to a n i n t er v a l m a p w i t h a s i n gl e m a x i m u m ( S e c ti o n

I V C ) . F o r IbJ < 1 ( r e g i o n I) t h e P o i n c a r 6 m a p i s m o n o t o n i c , c o n t i n u o u s a n d d i f f e r e n t ia b l e o n t h e u n i t

c ir c le . I n t h i s re g i o n t h e r e w i l l e i t h e r b e q u a s i p e r i o d i c o r p h a s e l o c k e d d y n a m i c s . F o r [b] > 1 t h e r e a r e t w oq u a l i t a t i v e l y d i f f e r e n t r e g i o n s w i t h r e s p e c t t o t h e t y p e s o f t r a n s i t i o n s b e t w e e n n e i g h b o r i n g p h a s e l o c k i n g

z o n e s . R e g i o n I I I i s c o m p r i s e d o f t h e s m a l l s t ip p l e d a r e a s a s w e l l a s t h e 3 : 1 a n d 3 : 2 z o n e s , w h e r e a s

r e g i o n I I i s t h e r e m a i n d e r o f t h e a r e a f o r Ib] > 1. I n r e g i o n I I a r e f o u n d p e r i o d d o u b l i n g b i f u r c a t i o n s

l e a d i n g to p h a s e l o c k i n g r a t i o s o f t h e f o r m 2 i : M . T h e c o m p l e x d y n a m i c s o f r eg i o n I I I a r e d i s c u s s e d in t h e

t e x t ( S e c t i o n V C )



12 M .R. Guevara and L. Glass

z + f ( ~ b . . . b ) > q5 . . . (20a )

+ f(~b . . . b) < ~hmi,, (20 b)

z + f(q~m i., b) < ~bm,x. (2 0c )

A p p l y i n g s o m e s i m p l e t r ig o n o m e t r y , t h e i n e q u a li ti e s c a n b e r e w r i t t e n u s in g ( 8 ) a n d

( 9 )

z > 0 .25, (21a )

1 1 s i n - 1 b2 - 2< 2 - 2~ b2 , (21b)

w h e r e s i n - i x d e n o t e s t h e p r i n c i p a l v a lu e o f t h e in v e r s e s in e f u n c t i o n

( - n / 2 < s i n - 1 x < + r e /2 f o r - 1 < x < 1 ). F r o m t h e s y m m e t r i e s i n ( 6 ) , t h e

P o i n c a r 6 m a p i n o t h e r r e g i o n s o f (z , b ) p a r a m e t e r s p a c e c a n a l so b e r e d u c e d t o a n

i n t e rv a l m a p w i t h a s i n g le m a x i m u m . I n F i g . 8 t h e r e g i o n s o f (z , b ) p a r a m e t e r s p a c e

i n w h i c h t h e P o i n c a r 6 m a p r e d u c e s to a n i n t e r v a l m a p w i t h a si ng le m a x i m u m a r e

s h o w n b y d i a g o n a l s t r i p e s .

In F i g . 7 w e sho w t he Po i nc a r6 m ap fo r z = 0 . 35 , b = - 1 .1 , - 1 .3 , - 1 .5 ov e r

t h e re g i o n in w h i c h t h e m a p i s a n i n t e r v a l m a p w i t h a si ng le m a x i m u m . N o t e t h a t a s

Ibl a p p r o a c h e s 1 , t h e m a p b e c o m e s p r o g r e s s iv e l y st e e p e r i n t h e n e i g h b o r h o o d o f th e

f i x e d p o i n t ~bi+ 1 = ~bi. N o t e f r o m F i g . 6 t h a t c h a n g e s i n r w h i l e k e e p i n g b c o n s t a n t

a l s o c a n l e a d t o c h a n g e s i n t h e s l o p e o f t h e m a p a t t h e f i x e d p o i n t . A s w e w i ll s h o w

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

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

V . P h a s e L o c k i n g Z o n e s i n D i f fe r e n t R e g i o n s o f ( ~ , b ) P a r a m e t e r S p a c e

O u r m a j o r g o a l h a s b e e n t o p r o v i d e q u a l i t a t iv e a n d q u a n t i t a t i v e a n a l y s is o f t h e

d i f f e r e n t t y p e s o f d y n a m i c s t h a t a r is e f r o m p u l s at il e p e r i o d i c i n p u t s t o a s i m p l e t w o

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

s t u d y o f t h e e f f e c ts o f i te r a t i o n o f t h e P o i n c a r ~ m a p a t d i f f e re n t v a l u e s o f ( z, b ) a s

w e l l a s a n a l y t i c d e r i v a t i o n o f b i f u r c a t i o n b o u n d a r i e s w h e n t h is h a s b e e n f e a s ib l e.T h e a n a l y t ic c o m p u t a t i o n s a r e d i sc u s s ed in A p p e n d i x B . T h e s e c o m p u t a t i o n s w e r e

c r o s s - c h e c k e d w i t h n u m e r i c a l c o m p u t a t i o n . F o r a g i ve n v M u e o f b , t h e v a lu e s o f t a t

t h e b o u n d a r i e s o f t h e p h a s e l o c k i n g z o n e s a r e a c c u r a t e to + 0 .0 1 i n F ig . 2 a a n d t o

+ 0 . 003 i n F i g . 2b .

T h e a n a l y si s s h o w s t h a t t h e r e a r e t h r e e d i f fe r e n t re g i o n s o f (z , b ) p a r a m e t e r

s p a c e t h a t c a n b e i d e n t if i e d w i t h r e s p e c t t o t h e t y p e s o f t r a n s i ti o n s b e t w e e n

n e i g h b o r i n g p h a s e l o c k i n g z o n e s . T h e t h r e e r e g i o n s a r e s h o w n i n F ig . 8 . T h e t y p e s

o f b i f u r c a t i o n s t h a t o c c u r i n e a c h r e g i o n w i ll n o w b e b r i e f ly d is c u ss e d .

V A . R e g i o n I : ]bJ < 1

I n t h is r e g io n t h e P o i n c a r 6 m a p is a c o n t i n u o u s , m o n o t o n i c a n d d i f fe r e n t ia b l e m a p

o f t h e u n i t c i r c le o n t o i ts e lf . T h e q u a l i t a ti v e p r o p e r t i e s o f th e d y n a m i c s c a n b e

c o m p l e t e l y d e s c r i b e d b y t h e r o t a t i o n n u m b e r p . I f p i s r a t i o n a l t h e n t h e r e is



Phase Locking, Period Doubling Bifurcations and Chaos 13

s t a b l e p h a s e l o c k i n g ( F i g s . 3 a , 3 c , 3 d ) w h e r e a s i f p is i r r a t i o n a l , q u a s i p e r i o d i c

d y n a m i c s r e s u l t ( F ig . 3 b ) . I n q u a s i p e r i o d i c d y n a m i c s , t h e r e is a c o n t i n u a l s h i ft o f t h e

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

I n r e g i o n I , t h e r o t a t i o n n u m b e r i s a c o n t i n u o u s f u n c t i o n o f "c a n d b w h i c h is

p i ec e w i se c o n s t a n t o v e r t h e s e t o f r a t io n a l s . T h e m e a s u r e o f t h e s e t o n w h i c h t h e

r o t a t i o n n u m b e r is i r r a ti o n a l i s p o s i ti v e ( H e r m a n , 19 77 ). T h u s , t h e d y n a m i c s f o u n d

i n re g i o n I a r e to p o l o g i c a l l y e q u i v a le n t t o t h e d y n a m i c s f o u n d i n o t h e r m o d e l s o f

p h a s e l o c k i n g in t h e li m i t o f " s m a l l " p e r t u r b a t i o n s ( A r n o l d , 1 9 65 ; K n i g h t , 1 97 2 ;

G l a s s a n d M a c k e y , 1 9 7 9 ; K e e n e r e t a l. , 1 98 1). T h e s e r e f e re n c e s s h o u l d b e c o n s u l t e d

f o r a f u r t h e r d i s c u s s i o n o f th e d y n a m i c s i n r e g i o n I .

Fig. 9. The stable fixed points as a function of b for v = 0.35. Onlythe first two p eriod doubling bifurcations and the period 3 orbitare show n. Lying in the blank space between the p eriod 4 orbitand the period 3 orbit are orbits of all other periods (May , 1976;Guck enheim er, 1977). For ease of p resentation, q5 is not givenmodulo 1

V B . R e g i o n H : P e r i o d D o u b l i n 9 B i f u r c a t i o n s w i t h Jb[ > 1

I n r e g i o n I I , t h e r e a r e o r b i t s o f p e r i o d 2 i ( i a n o n - n e g a t i v e i n t e g e r) . T h e r e a r e t w o

d i f fe r e n t ty p e s o f b o u n d a r i e s b e t w e e n a d j a c e n t p h a s e l o c k i n g z o n e s : b o u n d a r i e s

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

n u m b e r . T h e a n a l y t ic c o m p u t a t i o n o f b o u n d a r i e s i n r e g i o n I I i s d i sc u s se d i n d e ta i l

i n A p p e n d i x B .

O n a p e r io d d o u b l i n g b o u n d a r y , t h e r e is a p e r i o d d o u b l i n g b i f u r c a t i o n o f t h e

a s s o c ia t e d P o i n c a r6 m a p w i t h o u t a n y c h a n g e in r o t a t i o n n u m b e r . F o r e x a m p l e ,

s u c h p e r i o d d o u b l i n g b i f u r c a t io n s a r e r e s p o n s ib l e f o r t h e b o r d e r s b e t w e e n t h e 1 : 0

a n d 2 : 0 z o n e s a n d t h e 1 : 1 a n d 2 : 2 z o n e s o f F i g . 2 a , a n d b e t w e e n t h e 2 : 1 a n d 4 : 2

z o n e s a n d t h e 2 : 2 a n d 4 : 4 z o n e s o f F i g . 2 b . I f q~* i s a f i x e d p o i n t o f t h e P o i n c a r 6m a p T N, t h e n a p e r i o d d o u b l i n g ( o r " p i t c h f o r k " ) b i f u r c a t i o n t o a n o r b i t o f p e r io d

2 N w i l l a r i s e w h e n

0TN = - 1 . (22)

I n F i g . 9 w e s h o w t h e s t a b l e f i x e d p o i n t s f o r ~ = 0 . 35 a s a f u n c t i o n o f b. F o r

- 1 . 8 < b < - 1 .0 t h e r e i s t h e f a m i l i a r c a s c a d i n g s e q u e n c e o f fi x e d p o i n t s

c u l m i n a t i n g i n th e e m e r g e n c e o f a n o r b i t o f p e r i o d 3 , a n a l o g o u s t o t h e s e q u e n c es

f o u n d i n s tu d i es o f th e i t e r a ti o n o f o n e p a r a m e t e r f a m i l ie s o f i n t e rv a l m a p s w i t h a

s in g le m a x i m u m ( M a y , 1 9 76 ; G u c k e n h e i m e r , 1 97 7).

1-05 / -

0 . 9 0

0 .7 6

\0 " 6 0

-1"8 -1"4 - I '0

b



14

I: 0 I: 1

M. R. G uevara and L . Glass

2 :0

4:0 4:1

2:1 2 : 2

4 : 2 ~ 4 : 2 4 :3 4 :4

8~1 8:3 8:5 8:7

8 ; 0 8 : 2 8 : 4 8 : 4 8 : 6 8 : 8

Fig . 10 . Hypothes ized ar rangement of

the phase locking zones in reg ion I I of

Fig. 8. Th e vertices repres ent the pha se

locking zones and the edges represent

t he boundar i e s be t ween ne i ghbo r i ng

zones . Alon g each hor izontal l ine the

ro ta t ion number increases . Ver t ical

edges represent phase locking boun da-

r ies corresponding to pe r iod doub l ing

bi furcat ions of the Poincar+ m ap

D e l i c a t e l y i n t e r l a c e d w i t h t h e s e p e r i o d d o u b l i n g b o u n d a r i e s a r e b o u n d a r i e s

a c r o s s w h i c h t h e r o t a t i o n n u m b e r c h a n g e s , b u t a c r o s s w h i c h t h e n u m b e r o f f ix e d

p o i n ts o n t h e P o i n c a r6 m a p r e m a i n s c o n s ta n t . F o r e x a m p l e , s uc h b o u n d a r i e s o c c u r

b e t w e e n t h e 1 : 0 a n d 1 9 1 zon es (F i g . 2a ) , t h e 2 : 1 an d 2" 2 zon es (F i g . 2a , 2b ) , t he

4" 2 and 4 : 3 zone s (F i g . 2b ) , a nd t he 4 : 3 and 4 : 4 zon es (F i g . 2b ) . I f ~b* i s a f i xed

p o i n t o f t h e m a p T N f o r b < - 1 , t h e n s u c h a b o u n d a r y o c c u r s w h e n

q~* = 0. (23 )

N ot e t ha t in F i g . 9, ~b* = 0 ( = 1 , m o d 1 ) f o r b -~ - 1 .220, ~ = 0 . 35 . Th i s po i n t li e s

o n t h e b o u n d a r y o f t h e 4 " 0 a n d 4 1 p h a s e l o c k i n g re g i o n s.

T h e t w o d i s t in c t t y p e s o f b o u n d a r i e s i n t e r s e c t a t s i n g u l a r p o i n t s a t w h i c h ( 22 )a n d ( 2 3 ) a r e s i m u l t a n e o u s l y s a t i s f i e d . T h e s i n g u l a r p o i n t s r e p r e s e n t p o i n t s o n t h e

c o m m o n b o u n d a r i e s o f f iv e d if f e r e n t p h a s e l o c k i n g z o n e s . T h e p o i n t [b[ = 2 ,

= 0 . 5 , w h i c h i s o n t h e b o u n d a r i e s o f th e 1 " 0 , 1 : 1 , 2 : 2 , 2 : 1 , a n d 2 " 0 p h a s e l o c k i n g

r e g i o n s ( F i g . 2 a ) r e p r e s e n t s s u c h a s i n g u l a r p o i n t . A p p l y i n g ( 2 2 ) a n d ( 2 3) e n a b l e s u s

t o co m pu t e t he s i ngu l a r p o i n t a t Ib[ = 2 , z = 0 . 5 a s w e l l a s a pa i r o f s i ngu l a r po i n t s

loc ate d a t Ibh = 51/2 - 1 , z -~ 0 .5 +__ 0 .14 (Fig . 2b , A pp en di x B ) . In a dd i t io n to the se

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

po i n t s a t w h i ch pe r i od 8 o rb i t s a r i s e l oc a t ed a t t he va l ues Ibh ~ 1 . 17 , r ~ - 0 . 5 _ 0 . 16

and [b[ -~ 1.06, ~ -~ 0.5 + 0.10.

O n t h e b a s is o f th e s e re s u lt s, w e p r o p o s e t h a t t h e t o p o l o g y o f t h e d i f fe r e n t p h a s el o c k i n g z o n e s a n d s i n g u la r p o i n t s i n r e g io n I I c a n b e d e p i c t e d u s i n g t h e g r a p h i c a l

r e p r e s e n t a t i o n s h o w n i n F ig . 1 0. T h i s g r a p h is t h e g e o m e t r ic d u a l o f t h e p h a s e

l o c k i n g z o n e s i n th e p e r i o d d o u b l i n g r e g i o n o f F ig . 2 . T h u s v e r ti c e s o f t h e g r a p h

r e p r e s e n t t h e p h a s e l o c k i n g z o n e s , w h i le t h e e d g e s o f t h e g r a p h r e p r e s e n t t h e

b o u n d a r i e s b e t w e e n p h a s e l o c k i n g z o n e s. N o t e t h a t e a c h r o w o f th e g r a p h i s r e l a t e d

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

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

b e e n p e r f o r m e d t o c h e c k t h e v a l i d i ty o f th e e x t e n s i o n o f t h e s c h e m e i n F i g . 1 0 t o

orb i t s o f p er i od 2% i >~ 4 .

V C . R e g i o n I I I : S t i p p l e d A r e a s P l u s 3 : 1 a n d 3 " 2 Z o n e s in F ig . 8

I n r e g i o n I I I , t h e r e i s a n o v e r l a p b e t w e e n t h e r e g i o n s w h e r e t h e P o i n c a r 6 m a p

r e d u c e s t o a n i n t e r v a l m a p w i t h a s i ng le m a x i m u m ( t h e d i a g o n a l l y s t r ip e d r e g i o n s o f



Phase Locking, Period Do ubling Bifurcations and Chaos 15

F i g . 8 ) a n d t h e r e g i o n s w h e r e s t a b l e p e r i o d 3 o r b i t s e x i st ( c o r r e s p o n d i n g t o 3 : 1 a n d

3 : 2 p h a s e l o c k i n g ) . A c o n s e q u e n c e o f t h i s f a c t is t h a t " c h a o s " e x i s t s a t v a l u e s o f

(z, b ) wh ere such a pe r iod 3 o rb i t ex i s ts , in tha t the re a re in i t i a l phase s s t a r t ing f ro m

w h i c h i t e r a t i o n o f t h e P o i n c a r 6 m a p w i ll p r o d u c e o r b i ts o f a r b i t r a ri l y l ar g e p e r i o d a s

we l l a s ape r iod ic o rb i t s (ga rko vsk i i , 1964 ; g te f an , 1975; L i an d Yo rke , 1975).

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

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

o f u n s t a b l e o r b i ts , b u t o n l y t h e a t t r a c t i n g o n e o f p e r io d 3 .

W e e x p e c t t h a t t h e a r e a o f (z , b ) s p ac e l y i n g b e t w e e n t h e p e r i o d d o u b l i n g z o n e o f

r e g i o n I I a n d t h e p e r i o d 3 o r b i ts ( i.e . t h e s t i p p l e d a r e a o f F i g . 8 ) w i l l c o n t a i n N : M

p h a s e l o c k i n g z o n e s w i t h a r b i t r a r y N , a s w e l l a s p o i n t s w h e r e a p e r i o d i c p a t t e r n s

e x i st ( M e t r o p o l i s e t a l ., 1 9 73 ; M a y , 1 9 76 ; G u c k e n h e i m e r , 1 9 77 ; H o p p e n s t e a d t ,

1 98 1). F o r m o s t v a l u e s o f (~ , b ) in t h i s r e g i o n , w e f i n d c o m p l i c a t e d d y n a m i c s w i t h a ni r r e g u l a r a p p e a r a n c e s u c h a s t h a t s h o w n i n F i g . 4 d ( b - - 1 .1 3, z = 0.6 5). T h i s

p a t t e r n s h o w s n o s i g n o f p e r i o d i c i t y a f t e r 1 0 00 i t e r a t io n s . N o t e t h a t t h e i r r e g u l a r i t y

is m a n i f e s t b y a s e e m i n g l y r a n d o m d r o p p i n g o r s k i p p i n g o f b e a ts o f t h e d r i v e n

o s c i l la t o r , e v e n t h o u g h t h e i n t e r v a l s b e t w e e n s t i m u l i a n d f i r in g t im e s a r e r e s t r i c t e d

t o a c o m p a r a t i v e l y n a r r o w r a n g e .

T h e t e r m " c h a o t i c d y n a m i c s " is a p p r o p r i a t e t o d e s cr ib e t h e e x tr e m e l y c o m p l e x

b e h a v i o u r d i s p l a y e d b y th e m o d e l i n t h e s t i p p le d a r e a o f F ig . 8 . A s d i s c u s s e d a b o v e ,

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

c o m p l i c a t e d a n d m a y e v e n b e a p e r i o d i c. I n a d d i t i o n , s i nc e q u i t e s m a l l c h a n g e s in

o r b a w a y f r o m t h a t p a r t i c u l a r v a l u e o f (z , b ) w il l c a u s e a c h a n g e i n t h e p a t t e r no b s e r v ed , t h e e f f e c t o f f l u c t u a t i o n s o r " n o i s e " i n h e r e n t t o a n y e x p e r i m e n t a l s y s t e m

w i l l b e t o d e s t r o y p a t t e r n s i n z o n e s w i t h h i g h o r d e r p h a s e l o c k i n g r a t i o s ( C r u t c h f i e l d

a n d H u b e r m a n , 1 98 0; G l a s s e t a l. , 1 98 0; G u t t m a n e t a l ., 1 9 80 ). T h u s , h i g h e r o r d e r

z o n e s t h a t c o v e r s m a l l e n o u g h a r e a s o f (z , b ) p a r a m e t e r s p a c e w i ll n o t b e o b s e r v e d i n

t h e l a b o r a t o r y . I n s t e a d , t h e y w i ll b e r e p la c e d b y z o n e s t h a t a r e n o t p h a s e l o c k e d a n d

w h i c h d i s p la y i r r e g u l ar d y n a m i c s .

O r b i t s i n r e g i o n I I I ( s u c h a s t h e p e r i o d 3 o r b i t ) c a n a r i s e f r o m t a n g e n t

b i f u r c a t i o n s ( se e A p p e n d i x B ). T h e d e t a i l e d t o p o l o g y o f t h e p h a s e l o c k i n g z o n e s in

t h i s r e g i o n i s n o t w e l l u n d e r s t o o d .

V I . D i s c u s s i o n

W e a r e i n t e r e st e d i n m e c h a n i s m s f o r t h e g e n e si s o f c a r d i a c a r r h y t h m i a s , a n d h a v e

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

c e ll s a r e p e r i o d i c a l l y s t i m u l a t e d w i t h p u ls e s o f c u r r e n t ( G u e v a r a e t al ., 1 98 1) . O u r

d e v e l o p m e n t o f t h e s im p l e m a t h e m a t i c a l m o d e l i n th i s p a p e r w a s m o t i v a t e d b y a

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

s t i m u l a t io n . W e h a v e f o u n d t h a t t h e e x p e r i m e n t a l p r e p a r a t i o n b e h a v e s i n a m a n n e r

t h a t is s i m i l a r t o t h e b e h a v i o r o f t h e m o d e l . I n p a r t i c u l a r , t h e e x p e r i m e n t a l l yd e t e r m i n e d P o i n c ar 6 m a p s c a n s o m e t i m e s b e r e d u c e d t o i n t e r v a l m a p s c o n t a i n i n g a

s in g le m a x i m u m . W h e n i t er a t e d , th e s e m a p s s h o w e v i d en c e o f p e r i o d d o u b l i n g

b i f u r c a t io n s a n d c h a o t i c d y n a m i c s . F u r t h e r m o r e , in r e s p o n s e t o p e r io d i c s t i m u -

l a t i o n , p a t t e r n s d u e t o p e r i o d d o u b l i n g b i f u r c a t i o n s ( s u c h a s 2 : 2 a n d 4 : 4 p h a s e



16 M.R. Guevara and L. Glass

locking) are experimentally found, as well as irregular patterns which may be due to

underlying "chaotic dynamics".

The experimental observation of these phenomena in cardiac tissue, coupled

with the predictions of a rather general mathematical model, leads us to believe that

the results of this study have direct implication for the analysis of both normal and

pathological cardiac rhythms. Under normal physiological conditions, the primary

pacemaking site in the heart is located in the sinoatrial (SA) node. The cardiac

impulse originates in the SA node, spreads to the atrial musculature, proceeds to the

atrioventricular (AV) node, and then passes through the bundle of His, the bundle

branches, and the Purkinje network to the ventricular muscle (Mandel, 1980).

Thus, firing of the SA node leads to an almost immediate contraction of the atria,

followed after a small delay by contraction of the ventricles.

There are at least two ways in which to interpret the delay between activation ofthe atria and the ventricles. The traditional and most widely held view is that the

delay is due to slow conduction of the cardiac impulse when passing through the AV

node (Tsien and Siegelbaum, 1978).

One alternate view is based on the hypothesis that the specialized electrical

conduction system of the mammalian heart contains two or more autonomous

oscillators (van der Pol and van der Mark, 1928). Normally, one oscillator (situated

in the SA node) is the driving oscillator that entrains another (situated in the region

of the AV node) in a 1:1 fashion. Recent work has provided simultaneous

anatomical and electrophysiological evidence that the site of the driven (subsidiary

or latent) oscillator is at the border of the AV node and the bundle of His (James etal., 1979). Since the driven oscillator has an intrinsic frequency that is lower than

that of the SA node (Nadeau and James, 1966; Urthaler et al., 1973 ; Urthaler et al.,

1974), there will be a phase shift between the two oscillators when synchronization

in a 1 : 1 pattern occurs (Fig. 3a, 4a). This phase shift would appear as a delay

between the contraction of the atria and the contract ion of the ventricles. Indeed, if

the intrinsic frequency of the SA node is decreased sufficiently by pharmacological

intervention, the phase shift reverses its sign, and contraction of the ventricles

precedes contraction of the atria (Nadeau and James, 1966; Roberge et al., 1968).

Modeling the heart as a system of two or more coupled nonlinear oscillators was

pioneered by van der Pol and van der Mark (1928) using an electrical circuit. In theintervening 50 years, several other investigators have extended their work using

electrical and electronic analogues and computer simulations (Grant, 1956;

Roberge et al., 1968; Sideris and Moulopoulos, 1977; Katholi et al., 1977) as well as

physiological experimentation (Nadeau and James, 1966; Roberge et al., 1968;

Roberge and Nadeau, 1969; Urthaler et al., 1973; Urthaler et al., 1974). The

modeling work shows that the disturbances of atrioventricular conduction (AV

blocks) seen in the electrocardiogram can be simulated by changing either of the

intrinsic frequencies of the two oscillators or by altering the degree of inter-

oscillator coupling. The physiological work also demonstrates that pharmacologi-

cal manipulat ion of the intrinsic frequencies of the oscillators or of the level of blockat the AV node can produce dysrhythmias when 1 : 1 synchronization is lost. Prior

modelling largely relied on electrical analogues. However, there has not been a

theoretical investigation and an analysis of the topology of the phase locking zones

such as that presented here.



Phase Locking, Period Doubling Bifurcations and Chaos 17

I n e l e c t r o c a r d i o g r a p h y , e l e c tr i c a l e v e n t s a s s o c ia t e d w i t h t h e c o n t r a c t i o n o f t h e

a t r i a ( P w a v e ) a n d v e n t r i c l e ( Q R S c o m p l e x ) c a n b e o b s e r v e d b y r e c o r d i n g e l e c tr i ca l

p o t e n t i a ls o n t h e s u r f a ce o f t h e b o d y . T h e c o u p l i n g p a t t e r n s b e t w e e n t h e s t im u l u s

a n d o s c i l l a to r o b s e r v e d i n F i g s . 3 a n d 4 a r e s i m i l ar t o c l in i c a l ly o b s e r v e d p a t t e r n s o f

A V b l o c k . I n p a r t i c u l a r , w e m a k e t h e f o l l o w i n g i d e n t i fi c a t io n s :

i) N o r m a l s i nu s r h y t h m o r fi rs t d e g re e A V b l o c k - F i g . 3 a, F ig . 4 a .

i i) 1 : 1 A V c o n d u c t i o n w i t h a l t e r n a t i n g P R i n t e r v a l s - F i g . 4 b ( 2 : 2 b l o ck ) .

i i i ) S e c o n d d e g r e e A V b l o c k w i t h W e n c k e b a c h p e r i o d i c i t y - F i g . 3 c ( 4 : 3

b l o c k ) , F i g . 3 d ( 3 : 2 b l o c k ) .

i v) S e c o n d d e g r ee A V b l o c k w i t h a t y p i c a l W e n c k e b a c h p e r i o d i c i t y - F i g . 4 e

(4 : 3 b loc k) .

v ) S e c o n d d e g r e e A V b l o c k w i t h v a r i a b l e c o n d u c t i o n o r t h i r d d e g r e e A V

b l o c k w i th a c c r o c h a g e - F ig . 4 d.v i) H i g h g r a d e se c o n d d e g re e A V b l o c k - F i g . 4 g ( 2 : 1 b lo c k) .

v i i) S e c o n d d e g r e e A V b l o c k (2 :1 b l o c k w i t h a l t e r n a t i n g P R i n t e r v a l s ) - F i g . 4 f

( 4 : 2 b l o c k ) .

v iii) T h i r d d e g r ee o r c o m p l e te A V b l o c k - F i g . 3 b.

T h e c u r r e n t a n a l y s i s p r e d i c t s t h e ex i s te n c e o n p u r e l y t h e o r e t i c a l g r o u n d s o f s u c h

p a t t e r n s a s 2 : 2 a n d 4 : 2 p h a s e l o ck i n g , w h e r e a s p r io r t h e o r e t ic a l w o r k d i d n o t

i d e n t i f y t h es e p a t t e r n s . A s t r i k i n g d e s c r i p t i o n o f a 2 : 2 p a t t e r n i s i n a n e a r l y c a n i n e

s t u d y b y L e w i s a n d M a t h i s o n ( 1 91 0) o f t h e r e s ul ts o f a s p h y x i a o n t h e h e a r t b e a t :

" A t o r a b o u t t h e t im e w h e n t h e h e a r t a p p e a r s t o w a v e r b e t w e e n a c o n d i t i o n o f 2 : 1h e a r t b l o c k a n d r e g u l a r s e q u e n t ia l c o n t r a c t i o n a c c o m p a n i e d b y p r o l o n g a t i o n o f t h e

P - R i n t e r v a l , i t n o t i n f r e q u e n t l y h a p p e n s t h a t p a s s i n g i n t o t h e l a t t e r s t a te i t e x h i b i ts

a r e g u l a r a l t e r n a t i o n o f t h e P - R i n t e r v a l s " ( se e F i g . 5 o f L e w i s a n d M a t h i s o n ( 1 91 0)

a n d c o m p a r e w i t h F i g. 4 b o f th i s p a p e r ). A r e c en t r ev i ew ( W a t a n a b e a n d D r e i f u s,

1 98 0) h a s e m p h a s i z e d t h a t t h e m e c h a n i s m s f o r g e n e r a t i n g p a t te r n s w i t h a l t e r n a t i n g

P R i n t er v a ls ( 2 : 2 r h y t h m s ) a r e n o t w e ll u n d e r s t o o d a n d a r e c o n t r o v e rs i a l. N o t e

t h a t f o r 51/2 - 1 < b < 2 , a s t h e f r e q u e n c y o f t h e s t i m u l a t i o n is i n c r e a s e d o n e p as s e s

i n t u r n f r o m 1 : 1 t o 2 : 2 t o 2 : 1 p h a s e l o c k i n g ( F i g . 2 ). T h u s , w e h y p o t h e s i z e t h a t t h e

p a t t e r n w i t h a l t e r n a t i n g P R i n te r v a ls e x p e r i m e n t a l l y o b s e r v e d c o r r e s p o n d s t o a 2 : 2

p h a s e l o c k i n g p a t t e r n a n d a r i se s a s a c o n s e q u e n c e o f a p e r i o d d o u b l i n g b i f u r c a t i o n .T h e 4 : 2 p a t t e r n s c l i n ic a l l y o b s e r v e d ( S e ge r s, 1 95 1) m a y b e d u e t o a s e c o n d p e r i o d

d o u b l i n g b i f u r c a t i o n ( F ig . 4 f ) .

A n o t h e r f e a t u r e o f o u r a n a l y s i s i s t h a t c h a o t i c d y n a m i c s c a n a r i se a s a

c o n s e q u e n c e o f p e r i o d i c i n p u t s t o a l i m i t cy c l e o s c i ll a t o r. A r e v ie w o f c li n i c a ll y

o b s e r v e d d y s r h y t h m i a s r e v ea ls t h a t t h e r e a r e m a n y d y s r h y t h m i a s w h i c h h a v e a n

e x t r e m e l y i r re g u l a r a p p e a r a n c e . T o c it e a f e w ex a m p l e s o f ir r e g u l ar A V b l o c k :

i) " A d v a n c e d s e c o n d d e g re e b l o ck w i t h f lu c t u a t i o n in th e d e p t h o f p e n e t r a t i o n

i n to t h e A V n o d e " - F ig . 1 6 - 1 7 i n W a t a n a b e a n d D r e if u s (1 98 0).

ii) " I n t e r m i t t e n t 2 : 1 A - V b l o c k a n d a n a r e a o f 3 : 2 M o b i t z t y p e I I A - V

blo ck " - F ig . 291 in Ch un g (1971) .i ii ) "W en ck eb ac h A-V b loc k o f va r y in g degree (6 : 5 , 5 : 4 , 4 : 3 , 3 : 2 )" - F ig . 295

in Chung (1971) .

iv ) " A t r i a l f lu t t e r w i t h v a r y i n g s e c o n d d e g re e A V b l o c k a n d p h a s i c a b e r r a n t

v e n t r i c u l a r c o n d u c t i o n " - C a s e 8 , p . 3 4 9 i n S c h a m r o t h ( 1 97 1) .



18 M .R . Guevara and L. Glass

v ) " Pa r t ia l A V h e a r t b l o c k " - F i g . 1 -1 6E in B el le t ( 19 71 ).

These examples serve to i l lust ra te the fac t tha t c l in ica l ly observed cardiac

dysrh y thm ias a re f requen t ly qu i t e i r regu la r. On the bas i s o f our ana lys is , we

prop ose tha t one source o f i r r egu la r i ty in e l ec t roca rd iograph ic pa t t e rns o f AV

block may we l l be chao t i c dynamics re su l t ing f rom in t e rac t ion be tween d i f fe ren t

au ton om ou s pac em aker s ite s in the hea r t (F ig . 4d). Tes t ing o f t h is hypo the s i s

requ i re s co l l ec t ion and ana lys i s o f l ong e l ec t roca rd iograph ic records , a s we l l a s a

b e t t e r u n d e r s ta n d i n g o f t h e d i ff e re n t wa y s in wh i c h c h a o t ic d y n a m i c s m a y b e c o m e

mani fe s t i n ma themat i ca l mode l s .

Ou r i d e n ti f ic a ti o n o f c ar d i a c d y s r h y t h m i a w i t h d e sy n c h r o n i z a t io n o f a u t o -

nomous osc i l l a to r s i s h igh ly specu la t ive . The reade r shou ld be aware tha t t hese

ideas run coun te r t o conven t iona l c l in i ca l t each ing , and a l so tha t a l t e rna t ivetheore t i ca l mode l s based on fa t igue and re l a t ive re f rac to r iness (Landah l and

Gri f fea th , 1971 ; Ke ener , 1981a, 1981b) display som e qual i ta t ive fea tures s im i lar to

the p rope r t i e s desc r ibed he re . H ow ever , t he de ta i l ed topo log ica l s t ruc tu re o f t he

phase lock ing p resen ted in these o the r mo de l s is com ple te ly d i f fe ren t f rom tha t o f

our m ode l (Keene r , 1980 , 1981 a , 1981 b ) . Fo r exam ple , o the r m ode l s do no t i nc lude

p h a se lo c k i n g z o n e s o f t h e f o r m 2 N: 2 M a n d t h e " c h a o s" o b se r v e d b y Ke e n e r d o e s

no t a r i se f rom a sequ ence o f pe r iod do ub l ing b i fu rca t ions . F ina l ly , a l though the

theory i s gene ra l, t he ab ov e d i scuss ion has cen te red o n dy srhy thm ias a r is ing in the

AV c o n d u c t i o n sy s te m . Fo r i n s ta n c e, i n t e r ac t io n s b e t we e n n o d a l p a c e m a k e r s a n d

ec top ic spon taneo us ly ac t ive foc i wo uld l ead to s imi l a r dynam ics and d ysrhy thmias(Moe e t a l . , 1977; Ja l i fe and Moe, 1979) .

In the ma the mat i ca l l it e ra tu re , a l thou gh i t has long been kno w n tha t a pe r iod ic

so lu t ions can be fou nd in re sponse to pe r iod ic inpu t s t o two d im ens iona l osc i l l a to r s

(Car twr igh t and L i t t l ewoo d , 1945 ; Lev inson , 1949) the topo lo gy o f t he d i f fe ren t

phase lock ing zones und e r changes in the s t imula t ion pa ram ete r s i s no t com ple te ly

un ders too d (Gu ckenhe im er , 1980 ; Lev i, 1981). The nove l con t r ibu t ion o f our w ork

is t o show tha t ov e r ce rt a in ranges o f pa ram ete r space , t he phase lock ing p rob lem

reduces to a p rob lem in the ana lys i s o f in t e rva l maps . W e show tha t t r ans i ti ons

be tween phase lock ing zones a re due to pe r iod doub l ing b i fu rca t ions , t angen t

b i f u r c a t i o n s , o r s i m p l y c h a n g e s i n r o t a t i o n n u m b e r . W e h a v e a l so p r o p o se d ascheme fo r t he topo log y o f the phas e lock ing zones (F ig . 10) in the pe r iod dou b l ing

r e gi o n. Pe r i o d d o u b l i n g b i f u r c a ti o n s a n d c h a o t i c d y n a m i c s h a v e b e e n o b se r v e d in

prev ious wo rk on s inuso ida l ly fo rced non l inea r osc i l l a to r s (To mi ta an d Ka i , 1978 ;

Huberman and Cru tchf i e ld , 1979) . In add i t ion , "chao t i c dynamics" have been

o b se r v e d f r o m t w o m u t u a l l y c o u p l e d t u n n e l d i o d e o sc i ll a to r s b u t n o e v id e n c e o f

p e r i o d d o u b l i n g b i f u r c a t io n s w a s f o u n d ( Go l l u b e t a l. , 1 9 8 0 ) . Ho we v e r , a s t u d y o f

m utua l en t ra inme nt o f two e l ec t ron ic mode l s o f ca rd iac pacem aker ce ll s appa re n t ly

d id no t show chao t i c dynamics (Ypey e t a l . , 1980) .

One o f t he ma in ways fo r s tudy ing b io log ica l osc i l l a to r s i s t o sub jec t t he

osc i l l a to r s t o pe r iod ic pu l sa t i l e i npu t s . Th i s pape r has desc r ibed the qua l i t a t iveprope r t i e s o f phase lock ing fo r a s imple ma thema t i ca l m ode l o f a two d im ens iona l

osc i ll a to r . On the bas i s o f t h is t heore t i ca l wor k w e p red ic t t ha t pe r iod d oub l ing

b i fu rca t ions l ead ing to chao t i c dynam ics a s a conseq uence o f pe r iod ic s t imula t ion

of spon taneo us ly osc i ll a ting b io log ica l sys t ems wi ll be widespread .



P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t io n s a n d C h a o s 1 9

A c k n o w l e d g m e n t s . T h i s r e s e a r c h h a s b e en s u p p o r t e d b y g r a n t s f r o m t h e N a t u r a l S c ie n c es a n d

E n g i n e er i n g R e s e a r c h C o u n ci l o f C a n a d a a n d t h e C a n a d i a n H e a r t F o u n d a t i o n . W e t h a n k M . M a c k e y

a n d C . G r a v e s f o r h e l p f u l d i s c u s s io n s , S . J a m e s f o r t y p i n g t h e m a n u s c r i p t , a n d B . G a v i n f o r d r a f t i n g t h e

f i g u r e s .

A p p e n d i x A

I n t h i s a p p e n d i x w e p r o v e P r o p o s i t i o n 1 o f S e c t io n I V B . T h e p r o o f s o f t h e o t h e r t w o p r o p o s i t i o n s f o l lo w

i n a s i m i l a r f a s h i o n .

P r o o f o f P r o p o si t io n 1 . F r o m t h e s t a t e m e n t o f th e p r o p o s i t i o n a n d t h e d e fi n it i on o f t h e P o i n c a re m a p w e

k n o w t h a t

4 , = f ( O , 1,7) + 0 .5 - 6 , ( A 1 )

A~b~ =f (~ b~ _l ,7 ) + 0 .5 - 6 - qSi 1 , (A 2)N

M = Z A~) ,. ( A 3 )

i - I

S t a r t i n g f r o m p h a s e ~b~ ~ = 1 - ~b~ ~ a n d a s s u m i n g z = 0 . 5 + ~ w e c o m p u t e t h e n e w p h a s e , y /

r = f ( 1 - q ~ - l , Y ) + 0 . 5 + 6 ( r o o d 1 ). ( A 4 )

A p p l y i n g t h e s y m m e t r y in E q . ( 6 a ) a n d s u b s t i t u t i n g f r o m E q . ( A 1 ) , w e f i n d

= 1 - q~i. (A S)

T h u s , t h e s e q u e n c e # o , ~ bl . . . . f l u - 1 w i l l f o r m a c y c l e f o r z = 0 . 5 + 6 w h e r e ~ = 1 - ~ bl. T h e p h a s e

l o c k i n g p a t t e r n i s c o m p u t e d b y s u m m i n g t h e A ~kl, w h e r e

A ~/~ = f O P ~ - l , 7 ) + 0 .5 + 6 - r ( A 6 )

O n c e a g a i n u s i n g t h e s y m m e t r y in ( 6 a ) a n d s u b s t i t u t i n g f r o m ( A 2 ), w e f i n d

9 A ~ = 1 - A~b~. (A 7)

S u m m i n g a n d a p p l yi n g ( A 3) , w e c o m p u t e

N

Z A ~ b l = X - M . ( A S )

T h e r e f o r e , t h e s e q u e n c e tP o, ~ P ~ , . . . , ~ N - 1 i s a s s o c i a t e d w i t h a n N : N - M p h a s e l o c k i n g p a t t e r n .

A p p e n d i x B

I n t h i s a p p e n d i x w e g i v e a b r i e f s u m m a r y o f t h e a n a ly t i c c o m p u t a t i o n o f t h e b o u n d a r i e s o f th e p h a s e

l o c k i n g z o n e s a n d s i n g u l a r p o i n t s i n ( ~ , b ) p a r a m e t e r s p a c e . I n t h i s a n a l y s i s w e u ti l iz e r e s u l t s f r o m l o c a l

b i f u r c a ti o n t h e o r y i n c o n j u n c t i o n w i t h a c o n s i d e r a t i o n o f t h e r o t a t i o n n u m b e r .

A s s u m e t h a t 0 * i s a p e r i o d i c p o i n t o f p e r io d n o f t h e P o i n c a r ~ m a p ( 1 3 ). T h e n , c h a n g e i n t h e s t a b i l it y

o f t h e p e r i o d i c p o i n t ( b i f u r c a t i o n ) w i ll o c c u r i f

= _ 1 . (B1 )

I f t h e d e r i v a t i v e i n ( B 1 ) i s + 1 t h e b i f u r c a t i o n i s c a l le d a s a d d l e - n o d e o r t a n g e n t b i f u r c a t i o n a n d i f t h ed e r i v a t i v e i s - 1 t h e n t h e b i f u r c a t i o n i s c a l le d a p i t c h f o r k o f f l ip b i f u r c a t i o n ( M a y , 1 9 7 6; G u c k e n h e i m e r ,

1 97 7) . I n a d d i t i o n t o s a t i s f y i n g ( B 1 ) t h e r e a r e a d d i t i o n a l n o n - d e g e n e r a c y c o n d i t i o n s ( G u c k e n h e i m e r ,

1 97 7) w h i c h w e d o n o t c o n s i d e r in t h e c o m p u t a t i o n s w h i c h f o l lo w . A t a p i t c h f o r k b i f u r c a ti o n t h e n u m b e r

o f s t a bl e f i x ed p o i n t s i s d o u b l e d b u t t h e r o t a t i o n n u m b e r i s i n g e n e r a l th e s a m e . A t a t a n g e n t b i f u r c a t i o n

t h e n u m b e r o f s t ab l e fi x ed p o i n ts c h a n g e s a n d t h e r o t a ti o n n u m b e r m a y o r m a y n o t c h a n g e .



2 0 M . R . G u e v a r a a n d L . G l a s s

I n a d d i t i o n t o c h a n g e s i n r o t a t i o n n u m b e r a s s o c i a t e d w i t h t a n g e n t b i f u r c a t io n s , t h e r o t a t i o n n u m b e r

o f t h e s ta b l e p h a s e l oc k e d o r b i t m a y c h a n g e w i t h p a r a m e t r i c c h a n g e s in b a n d z , ev e n t h o u g h t h e n u m b e r

a n d s t a b il it y o f t h e f i x e d p o i n t s r e m a i n u n c h a n g e d . S u c h c h a n g e s i n r o t a t i o n n u m b e r o n l y o c c u r f o r

Ibb > 1 . T o s h o w h o w t h e s e c h a n g e s a r i s e c o n s i d e r t h e c a s e i n w h i c h b > 1 , a n d a s s u m e t h a t a l o n g t h el o c u s o f p o i n t s

= o ( b ) (B2)

~b* = 0 .5 i s a s t a b le f i x e d p o i n t o f p e r i o d N . A s s u m e t h a t i n t h e n e i g h b o r h o o d o f t h e c u r v e i n ( B 2 ) th e r e

e x i s t s a s t a b l e f i x e d p o i n t qS* o f t h e m a p TN

4 " = 0 . 5 - ~ ( b , T ) , ( a 3 )

w h e r e ~ is a c o n t i n u o u s d i f f e r e n t i a b l e f u n c t i o n

e(b, g(b)) = 0 (B4)

a n d e is p o s it i v e o n o n e s i d e o f t h e l in e i n ( B 2) a n d n e g a t i v e o n t h e o t h e r s id e . A s s u m e t h a t i n t h e r e g i o n i n

w h i c h e i s n e g a t i v e t h e r e i s N : M p h a s e l o c k in g . T h e n f r o m a d i r e c t a p p l i c a t i o n o f ( 1 7 ) th e r e w i ll b e

N :M + 1 p h a s e l o c k i n g o n t h e o t h e r s id e . C o n s e q u e n t l y t h e l o c u s o f p o i n t s in ( B 2 ) w il l c o n s t i t u t e t h e

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

S e c t i o n I V , a l o c u s o f p o i n t s a l o n g w h i c h ~b* = 0 c o n s t i tu t e s a b o u n d a r y o f a p h a s e l o c k i n g r e g i o n f o r

b < - 1 .

I n t h e r e m a i n d e r o f t h is A p p e n d i x w e g i v e a b r i e f s u m m a r y o f t h e c o m p u t a t i o n o f th e b i f u r c a t io n

b o u n d a r i e s a n d s i n g u l a r p o i n t s i n (~ , b ) p a r a m e t e r s p a c e . T h e r e s u l ts a r e a p p r o p r i a t e f o r b < 0 a n d

0 <~ z ~< 0 . 5. U s i n g t h e s y m m e t r i e s i n S e c t i o n I V a p p r o p r i a t e f o r m u l a e f o r s y m m e t r i c a l l y p l a c e d

b o u n d a r i e s c a n b e c o m p u t e d . S i n c e t h e c o m p u t a t i o n s e n t a i l s im p l e a l g e b r a a n d t r i g o n o m e t r y , o n l y t h e

m a i n r e s u l t s o f th e c o m p u t a t i o n s a r e g iv e n . T h e f o r m u l a e p r e s e n t e d h a v e b e e n c h e c k e d a g a i n s t r e s u l ts

f r o m n u m e r i c a l i t e r a t i o n o f t h e P o i n c a r 6 m a p .

T h e 1 : 0 p h a s e l o c k i n g b o u n d a r y c o i n c i d e s w i t h a t a n g e n t b i f u r c a t i o n f o r ] bl < 1 , a p i t c h f o r k

b i f u r c a t i o n f o r 1 < I bl < 2 a n d a c h a n g e i n r o t a t i o n n u m b e r f o r [ b[ > 2 .

C o n s i d e r f i r s t t h e l o c u s o f t h e t a n g e n t b i f u r c a t i o n . S e t t i n g 3T /O~ = + 1 a n d u s i n g (7 ) w e f i n d

1qb = 1 - - - c o s - l ( - b ). ( B 5)

2~

N o t e t h a t t h e o n l y r e a l s o l u t i o n o f ( B 5 ) o c c u r s f o r Ibl < 1 , a n d t h u s t h e b o u n d a r y o f t h e 1 : 0 p h a s e

l o c k i n g c o i n c i d e s w i t h a t a n g e n t b i f u r c a t i o n o n l y f o r Ib l < 1 . A p p l i c a t i o n o f ( 5 ) a n d ( 1 1 ) g i v e s t h e

b o u n d a r y o f t h e 1 : 0 p h a s e l o c k i n g f o r - 1 < b < 0

z = l ~ c o s - l b - 0 .2 5 . ( B 6)27r

A p i t c h f o r k b i f u r c a t i o n i s c o m p u t e d b y s e t t i n g ~T/OO = - 1 . U s i n g ( 7 ), a p i t c h f o r k b i f u r c a t i o n o c c u r s

fo r

~b = 1 - l c o s - l ( - b 2 + 2 ~ ( B 7)

2~z 2 3b / "

N o t e t h a t t h e o n l y r e a l s o l u t i o n s o f ( B 7 ) o c c u r f o r 2 / > Ib ~> 1 . S u b s t i t u t i n g (B 7 ) i n t h e P o i n c a r 6 m a p

g i ves

1 1 [ ( b Z + 2 ~ + ( 1 - b2 / 4~1 / 2~z . . . . c o s - 1 - t a n 1 (B 8 )

2 27r 3 b J \ ~ / J "

T h i s i s t h e b o u n d a r y b e t w e e n 1 : 0 a n d 2 : 0 p h a s e l o c k i n g r e g i o n s fo r 0 < z < 89 nd - 2 < b < - 1 .

T h e b o u n d a r y b e t w e e n t h e 1 : 0 a n d 1 : 1 p h a s e l o c k i n g z o n e s f o r b < - 2 i s a s s o c i a t e d w i t h a c h a n g ei n r o t a t i o n n u m b e r . A s s u m i n g t h a t t h e r e i s a f i x e d p o i n t qS* = 0 o n t h e P o i n c a r 6 m a p ( 11 ) w e f i n d t h a t

z = 89 (B9)

T h e i n t e r s e c ti o n o f ( B 8 ) a n d ( B 9) g iv e s t h e s i n g u la r p o i n t o n t h e c o m m o n b o u n d a r i e s o f t h e I : 0 , 1 : 1 ,

_ 12 : 2 , 2 : 1 an d 2 : 0 ph ase l ock i ng r e g i ons . Th e i n t e r se c t i on o ccu r s fo r z - ~ , Ib l = 2 .



P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t i o n s a n d C h a o s 2 1

T h e b o u n d a r y b e t w e e n 2 : 0 a n d 2 : 1 p h a s e l o c k i n g o c c u r s w h e n

T2(0) = 0 . (B10)

S u b s t i t u t i n g in t h e P o i n c a r 6 m a p (1 1) w e f i n d

1 1 ( b ) 1 c o s _ l _ b ' ( B l l )z = - - - - - C O S - 1 - - =2 2~ 27r 2

N o t e t h a t t h e f i x e d p o i n t s a l o n g t h e b o u n d a r y i n ( B l l ) o c c u r f o r ~b = 0 , ~b = 0 . 5 + z .

T h e s i n g u la r p o i n t o n t h e c o m m o n b o u n d a r i e s o f t h e 2 : 0 , 2 : 1, 4 : 2 , 4 : 1 a n d 4 : 0 p h a s e l o ck i ng

r e g i o n s o c c u r s w h e n ( B 1 0 ) i s s a t i s f i e d a l o n g w i t h t h e c r i t e r i o n

63T2 _ OT c~T 4,=o.5+~

c~b c~b ,=0 c~ b = - 1, (B1 2)

w h e r e w e h a v e u s e d t h e c h a i n r u l e. U s i n g ( 7 ) w e f i n d t h e s i n g u l a r p o i n t o c c u r s f o r

b = 1 - 5 1 / 2 , r = - - c o s - 1 - ~0 .3 5 6.2~

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Phase Locking - Period Doubling

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